Elementary Lessons Sound - Forgotten Books

ELEMENTARY LESSONS

S O U N D .

DR . W. H . STONE,

N

LECTUR ER ON PHYSICS AT S'I

‘

. THOMAS 'S HOSPITAL.

WITH ILLUSTRATIONS .

Bionbun

M A C M I L L A N A N D C O.

The Right of Tran slation is Reserved .

R . CLAY ,SONS , AND TAY LOR

B READ STREETHILL .

Q 2 2 3

PREFACE.

THE ob je ct o f the presen t work , b e side s giving a

con cise ou tlin e o f su b je cts requ ired for examin ation ,is

to fu rnish in formation in termediate b etwe en Acou st i cs

an d Mu sic proper, su pplemen tary to b o th . Mu ch of

this material is on ly to b e fou n d in b u lky, e xpen s ive ,

an d fore ign treatises ; the rest is pract i cal l y con ceale d

in Memoirs an d Tran saction s o f Scien tific So cie tie s .

W. H . STONE.

CONTENTS .

INTRODUCTORY

CHAPTER I .

MODES or PRODUCTION or SOUND— V I B RATION or SONOROUSB OD IES

CHAPTER I I .

MODES OF PROPAGATION OF SOUND V ELOCITY IVAVE~

MOTION— REFLECTION— REFRACTION

CHAPTER III .

INTEN SITY , CONSONANCE, INTERFERENCE

CHAPTER IV.

rrrcn ~ rrs MEASUREMENT , LIMITS, VARIATION , STANDARDS,AND TONOMETRY

CONTENTS .

CHAPTER V .

PAGE

NATURE OF MUSICAL TONE QUALITY HARMON ICSSULTANT TONES

CHAPTER VI .

EFFECTS OF HEAT, ATMOSPHERIC PRESSURE,MOISTURE

,

DENS ITY

CHAPTER VI I .

SCALES,CHORDS, TEMPERAMENT, AND TUN ING

CHAPTER VI I I .

SPECIAL APPLICATIONS TO MUSIC— THE EAR AND VOICE

LIST OF ILLUSTRATIONS.

1 .

-V ib ration s of Stretched Strin g‘

2.

— Harmon ic Sou n ds . Nodes an d Ven tral Segm en ts o f a

V ib rating StringHarmon ics. Nodes an d V en tral Segmen ts o f a Vi

b ratin g Strin g4 .

— Son om eter5 .

— Longitu dinal V ib ration s o f Rods6 .

—Marloye’

s Harp7 .

— V ib ration s of a Metal Rod8 .

-Nail Fiddle9 .

—Zan ze

1 0 .— Marimb a

l l .

—ATu n ing-fork moun ted on a Sou n ding-b ox.

12 .

— V ib ration s of 3. Plate .

13 .

—Nodal Lin es o f V ib ratin g Circul ar, 0 1 Polygon al Plates ,according to Chladn i an d Savart .

1 4.-Nodes and Segmen ts of a V ib ratin g B el l

15 .-Section of a B ell

16 .

—Proof o f the V ib ration o f a G lass B e l l1 7 .

— Strikin g Reed18 .

—Free Reed19 .

— Egyptian Flu te20 .

— P1isn 1atic Son orou s Pipes

x LIST OF ILLUSTRAT IONS .

21 .-Cylindrical Son orou s Pipes

22.-Trevelyan

’

s In strumen t23. Cau se o f V ib ratory Movemen ts .

24 —Ph ilosophical Lamp or Chem ical Harmon icon25.

— Ku n dt’

s Tu b e

26.— Experimen tal Stu dy of the Laws o f Reflection of Sou n d—Son orou s Re fraction . M. Son dhau ss

’

s In strumen t28.

—Propagation of a Son orou sWave through an Un l im itedMedium

29.—M. He lmholtz’

s Reson an ce Glob e30 .

- B eats of Imperfect Un ison . Ordin ates of Ten Wave s

3 1 .

— Ordin ates o f Eleven Waves tran smitted in the sam e

time

32.

— Sums of the Correspon ding Ordin ates33 .

-Savart’

s Toothed Whee l34 .

—Se eb eck ’

s Siren35.

— Cagn iard de Latou r’

s Siren36 .

— In terior V iew of the Siren37 .

— He lmholtz’

s Dou b le Siren38.

—V ib roscope39.

— Comb in ation of two Paralle l V ib ratory Movem en ts

40.-V ib ration s of Compou nd Sou n ds

41.—Optical Stu dy o f V ib ratory Movemen ts

42.—Optical Cu rv es represen ting the Rectan g ular V ib ration s

o f Two Tu n ing -forks in Un ison43 .

— Comb in ation o f Two Rectan gu lar V ib ratory Movemen ts

44.— Optical Cu rves. The Octave , Fou rth , an d Fifth

45 .— Open Tu b e , with Man ometric Flame

46 .

—Apparatu s for th e Comparison of the V ib ratory Movemen ts of Two Sono rou s Tu b es

47 .

—Man ometric Flam es . Fu n dam en tal Note , an d th e Octaveab ove the Fu n damen tal Note

48.-Man ometric Flames simu ltan eou sly given b yTwo Tu b es

at the Octave

P AG E

LIST OF ILLUSTRATIONS .

49.

-Man ometric Flames of Two Tu b es o f aThird50 .

— Professor B lake ’

s Method of Photographin g V ib ration s51 .

—Cu rve represen ting a Sou n d-wave52.

— Perron et Thompson ’

s Keyb oard53 .

— Poole ’

s Keyb oard54 .

-B osan qu et’s Gen eralised Keyb oard

55.

-P1an of Natu ral Fingerb oard—Organ Stops

57 .

— The Flu te . Lon gitu din al an d Tran sversal Section of

the Mou thpiece58 .

—Nay or Egyptian Flu te59.

—Hau tb ois. Fron t an d Side V iew of Re ed

60.— Clarin et

61 .

— Section of Mou thpiece62.

—B assoon63 .

— Fren ch Horn

64.

— Trumpet an d Clarion65.

— Tromb on e66 .

— The Human Ear

67 .-The Human Voice

xiPAG E

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2 ON SOUND.

The pre sen to text-book is con ce rn ed main ly with the in

strumen tal apphan ce s which m in i ste r to sou n d. The physicalaspect o f Acou stics has be en lu cidly mapped ou t by Cle rkMaxwel l in the followin g man n er

VIB RATIONS AND WAVES.

Physical Asp ect, of Acou stics.

1 . S o u r ce s — V ibration s of variou s bodie s

Air

ReedStrIn gs

Membran e sPlate sRods

2. D is tr ib u to r s

Speakin g -tribe s, Ste thoscopes.

Sou n ding Rods .Wire s .

Pugg ing of floors,Dampers of Pian oforte s .

Re servoirs, Re son ators , Organ Pipe s, Sou n ding -boards.

Reg u lators Organ Swe ll .

The Ear, Sen sitive Flames,6. De tectors

Membran es,Phon au tog raphs , &c .

7 . Tun in g -fOrks,Pitch-Pipe s

,Mu sical Scale s.

To bridg e ove r the gap between this scien ce an d the art

of mu sic,slightly more exten ded treatmen t is n eeded

1. MODES OF PRODUCTION. VI B RATION OF SONOROUS B ODIES.2. MODES OF PROPAGATION . VELOCITY. WAVE-MOTION .

REFLECTION. REFRACTION.

3. INTENSITY. CONSONANCE . INTERFERENCE.

4 . PITCH. MODES OF DETERMINATION AND MEASUREMENT .

STANDARDS OF PITCH.

5. NATURE OF MUS I CAL TONE . QUALITY. RESULTANTTONES

5Organ pipes,re son ators, and

other win d in strumen ts.The Siren .

Harp,

Drum,& c.

Gon g , &c . ,

Tu n ing -fork, &c.

INTRODUCTORY.S

6 . EFFECTS or HEAT. ATM OSPHER IC PRESSURE . MOISTURE.

DENSITY.

7 . SCALES . TEMPERAMENT . TUNING .

8. TRE ‘

EAR AND VOICE. SPECIAL APPLICATIONS To MUSIC.

The whole sub ject n atu rally an d m ore con cise ly divide sitse lf in to — 1 . Mechan ical ; 2. Theoretical ‘

c‘

On sideration s ;3 . P ractical application s. It is

,howeve r, to b e n oted that

the se differen t aspe cts o f the facts can n ot b e separated on e

f rom the othe r : the re spe ctiv e influ en ce s of the art '

o f mu sicand of scien tific re search hav ing be en re ciprocal , g radu al ,an d in timately combin ed . At the ve ry ou tset of history w emee t with t he Monochord , n amed afte r Pythagoras, a machin en ot in tended for artisti c pe rforman ce , b u tWich at on ceyie lded immen se practical re su lts to mu sic . e then en te rin to a lon g period du rin g which in strumen tal applian ce sg rew,

withou t de sign and withou t theory. The discove rie sof n ew or improved in strumen t s were pu re ly te chn ical , oftenf ortu itou s althou gh eve ry in strumen t added was a pie ce

'

of

me chan i sm open to scien tific analysis. Du rin g the pre sen tcen tu ry a retu rn has been made to . a paratu s e ssen tially.

scien tific , for the explan ation of what had been mu sicallyin ven ted ; we find soon the reciprocal in flu en ce of in stru

m en ts on apparatu s—n o be tte r in stan ce of which can b e

g iven than the discovery of Tartin i’s Terzo Su on o , or thirdsou n d ; origin ally tau ght by the great violin ist to his pupilsas a mean s of accu rate tu n ing , b ut n ow Shown by He lmholtzto involve a n ew an d importan t acou stic prin ciple .

In many case s in strumen ts of mu sic actually stan d in theplace of apparatu s . Strictly con side re d , a mu sical n ote i sof itse lf a mathematical fact

,qu ite in depen den t of its

powe r of exciting emotion and pleasu re by its artistic produ ction . On the other han d

,tu n ing an d in ton ation

,orig in ally

left en tire ly to the accu rate and cu ltivated ear of a skilledpe rform er, have be come a bran ch of scien ce

,with defin ite

laws and practical ru le s ; in somu ch that the u n con sciou sdepartu re s from a fixed tu n ing , which older mu sician s madeby a kin d of in stinct

,are n ow explain ed ; an d even the dis;

position of variou s in strume nts,with difierent qu alities of

ton e , in an orche stra i s shown to b e corre ct,or the con trary

,

according as the harmon ics of each pecu liar qu ality are

con son an t or d1sson an t .

4 ON SOUND. [CHAR

CHAPTER I.

MODES OF PRODUCTION OF SOUND. VI B RATION OF

SONOROUS B ODIES.

Soun d may b e g en erated in variou s ways ; some of the sehave be en u tilized for the produ ction of mu sical or regu larv ibration s

,othe rs remain in the categ ory of me re n oise .

The s h o ck of two bodie s again st on e an othe r is pe rhapsthe common e st of all sou rce s , varying , howeve r, mate riallyaccording to the n atu re o f the soundin g masse s

‘

and the

mode o f con veyan ce to the ear. The simple u nmu sicaltap of a drumstick on a mem b ran e is of mu sical valu ein the side-drum

,to pre serve time an d indicate rhythm

,

while in the Morse te leg raph the click cau sed by the collisiono f magn et an d keepe r is, by a su itable code , made to fu rn ishan in te llig ible alphabe t.

Irr e gu lar V ib ration .

Fr ict io n in many form s fu rn ishe s a sou rce o f sou n d ; theirregu lar vibration s. which in its rou ghe r kin ds it em its

,

passing impe rceptibly in to m ore regu lar and mu sical ton e s .This g radation may Often b e n otice d in the sharpen ing of a

saw , an d still more n otably in the action of railway brake su pon a train in motion . In its more refin ed adaptation toin strumen ts friction orig in ate s the ten de r ton e s of the violinand its cong en ers ; while in the mu sical g lasses, a wettedfing eI m oving with friction alon g the edge of a con son an tbe ll

,produ ce s a qu ality o f sou n d almost pain fu l an d cloying

from its exce ssive swe e tn e ss . The f riction of air again stsolid bodies orig inates the m e lan choly moan ing of the win d

,

the sharp hiss,of a can e

,an d the lou d g en ial crack of a

carte r’s whip.

Exp l o s io n as of g u n powde r and o f explosive gases , g ive srise to lou d n oises ; in the pyrophon e of Wheatston e and

othe rs it has been made to se rve,though as yet rathe r im

perfe ctly,the fu n ction of a mu sical in strumen t. Of a kin dred

n atu re are the variou s kinds of sing ing and sen sitive flame s.

MODES OF PRODUCTION OF SOUND . 5

El e c tricity , whethe r in its disru ptive discharg e at highten sion

,cau sing the rolling of thu n de r, or in the le ss awf u l

man ife station s o f f rictional and in du ctive machin e s , is a

sou rce of soun d. In the form of a dyn amic cu rren t,it has

also the powe r of produ cing su ch an alteration in the m olecu le s of soft iron as to b e accompan ied by au dible mu sicalv ibration . Latte rly this has b een adopted in to the serv ice o f

mu sic in the variou s forms of te lephon e .

R e gu l ar V ib rat io n

The se being some of the common e r and more physicalsou rce s of vibration appre ciable by the ear, t here remain sa large r an d more distin ct class

,which

,from its Spe cial

adaptation to the produ ction of reg u lar wave s,fu rn ishe s the

bu lk of the in strumen ts an d con trivan ce s u sed f or e licitin gpleasan t ton e s

,an d f or bu ildin g u p the aesthetical art o f

mu sic. They w ill hav e n ext to b e con sidered at som e leng th ,and may b e en ume rated as follow s

Summary of Vibration s .

Tran sverse1. Of Strings LongItsdinal

lexcéted

Tors 10 n al y

B oth en ds fixed (approach to tho se

of strin gs).( 1 . Tran sverse On e en d fixed (n ail fiddle , mu sical

II. Of Rods 2. Longitu dmal b ox).3 . Tors ion al Cen tre fixed (tu n ing-fork)

Both en ds fre e , n odal poin ts su p

p orted (harmon icon ) .

111. Of P lates }As in Chladn i’s exp eriments (gong , cym b al).

(A. Excited b y b lows (clock chimes).Spherical B . B y tangen tial or radial friction (mu s ical

l glasse s).Of complex figure l give compoun d n otes with irrele van t

harmon ics).

IV . or Bells

V . orMem ( 1 . In dep enden t (tamb ou rin es , zamb omb a).b ran es (2. With as sociated air-chamb er (kettle drum ,

reson ators)

1 . Free (as in harmon iums).IA. Single (clarin et organ reed).

VI. Of Re ed s2° B eating

in Dou b le ( b assoon, o b oe ).A . The lip s in b rass in strumen ts.

3 . Memb ran ou s is . The 1a

{6‘

1 ON SOUND . [GEAR

A. Open p ip es .

B . Stopped p ip es.

II. Organ p ipe s C . Hal f-stormed pipes .VH ’ orcolumn s

\D. Pipes with reeds .

Of air

I E Mixtu res an d mu tation stops .

2. Consonan ce b oxes and vessels.

1 . Trevelyan’

s rocker.

V I I I .V‘i

lb ration 2. Son dhaus

’

S exp erimen t.cau se y h eat 11 . Ch emical harmon icon

, pe pho n e .3 ° Of flames12. Sen sitive and smgmg flames .

IX. Cau sed b y ( 1 . Cu rren t“

In iron b ar,Reiss

’

s Te lephon ic rece iver.

electricity 12. Te lephon e , M1crophon e .

S trin g s .—Amon g the common e st and earliest modes of

e l icitin g mu sical sou nds may b e n amed s trings. They hav econ tribu ted the larg e st share to in strumen ts of all tim e s andall cou n trie s

,an d we re early employe d for more accu rate

determ in ation s. The ir theory 1s simple comparative ly to thatof othe r sou n d-produ ce rs .

The string itse lf is su pposed to b e a pe rfectly u n iform an d

flexible thread of solid matte r,stretched be twe en two fixed

poin ts . Although this ideal i s n o t actu ally attain ed in

practice,the dev iation s f rom it are n ot so g reat as to preven t

n ece ssary corre ction s be ing made . Its vibration s may b e

divided in to transverse,longitudinal, an d torsional, the f orme r

be ing the form more u su ally stu died,in which

,if the

stretchin g du e to late ral displacemen t b e small in comparisonwith that to which the strin g is already su bje cted

,it may b e

n eg le cted .

Tran s v e rs e V ib ratio n s o f S trin g s .

The laws are as follows

1 . For a g iven string and a g iv en ten sion, the time of a

vibration varie s dire ctly,the vib ration “

n umbe r inve rse ly , asthe length .

2. When the leng th of the strin g is g iven , the vibrationn umbe r varies dire ctly, the time of vibration inve rse ly, as thesqu are root of the ten sion .

3 . Strin g s of the same len g th an d ten sIOn vibrate in time swhich are proportional to the squ are root of the lin earden sity

,the vibration n umbe r be in g In inverse ratio to this

The motion s of a string thu s fixe d at its en d and excited at

s ome in termediate poin t, radiate from that poin t to the fixed

L ] MODES OF PRODUCTION OF SOUND.

i7

extremities, whence they are reflected in the opposite direction ,

travelling ove r twice leng th o f the string .

1 The simple st

The velocity with which transverse vib ration s ru n alon g a flexib le string 13

gi1 en b y the formu la where t denotes ten sion, m the mass o f

u n it length . The p eriod o f a complete vib iation is the i efore the time requ iredfor a p u lsation to travel over twice its length i.e . n

_2

1° l

the length of the string , and n the numb er o f vib ration s in a secon d orthe frequ ency. Fe 1led and frequency are there fore reciprocals

8 ON SOUND.

f orm of vibration is that in which the strin g v ibrate s as a

whole , an d produ ce s its lowe st or fu ndame n tal n ote ; b u t itmay also b e broken u p in to two or more ven tra l segmen ts ,

separated by n odes or poin ts of rest, the rapidity o f v ib ration

b eing proportion al to the n umbe r of the se segmen ts,and

produ cing partial-ton e s , which will b e described furthe r o n .

10 [a'

m

s trin g will sou n d the u ppe r partial ton e be lon g ing to the

div ision . Thu s if the damper b e applied in the middle , a

sing le n odal poin t , will b e formed,the f

string will vibrate . in

two halves , and g ive the octave o f its fu ndame n tal n ote . If

it b e cl

amped at a poin t on e -third from the en d , and excitedm idway between the en d and the damping , a secon d n odew il l b e e stablished in the free part the string will vibrate inthre e seg men ts , g iving the twe lfth of its fundamen tal n ote .

The highe r n ote s thu s obtain ed are te rmed harmon ie s,and

w il l b e con side red late r o n . They can e asily b e shown bythrow ing a strong l ight on the string , or by mean s o f smallpape r ride rs set astride on it

,which are immediately thrown

o if at the h em,b u t n ot at the n ode s .

String s may b e excited in variou s ways,an d with corre

spon din g variety in the sou n d produ ced . The oldest and

s imple st mode i s obviou sly that of plu cking them,draw

in g the ten se cord ou t o f its position of e qu ilibrium and

su dden ly letting i t go . This is . the plan adopted in manyan cien t and m odern in strumen ts , su ch as the harp and

gu itar. Or the finge r may b e armed with a qu il l o r ple ctrum ,

as in the'

case of the zithe r and in some orien tal in strumen ts .

In the harpsichord this qu ill was in se rted in to a smallm ov eable piece of me chan ism term ed the jack , which was

itse lf actu ated from the keyboard of the in strumen t.'

A third and most importan t improvemen t is e ffe cte d in

the pian oforte , where a hamme r of comparative ly soft materialstrike s a blow on the string in stead of twanging it like theharpsichord jack. B u t an en tire ly diffe ren t cou rse , earlyadopted , con sisted in brin g ing a bow in to f rictional con tactwith the string , and by a succe ssion of impu lse s conveyed to it,produ cing a con tin u ou s instead of an evan esce nt ton e . In all

the in n ume rable an d an cien t varietie s of the viol an d violinfam ily, the b ow i s made of horsehair rubbed w ith re sin

, keptat a moderate ten sion by the stick to which it is attached .

Many attempts , on ly partially su cce ssfu l f or the most part , havebeen made to adapt this method of excitation to in strumen tsof the pian o spe cie s , amon g which

,e lastic rolle rs , and

rotating bows of parchmen t or horsehair passing over rolle rs ,may b e n amed .

1 A cu rren t of air dire cted again st a stu nghas long been kn own to b e competen t to excite it, an d the

o lian harp has been con stru cted on this prin ciple . Latte rlyattempts have be en made to ren de r this comb in ation less

l The an cien t vielle ,n ow degraded in to the b u rdy

-

‘gur

‘

dt the oldest,‘

and perhaps the b es t repre sentative of this class.

MODES OF PRODUCTION OF SOUND. 11

vagu e an d fortu itou s than it is in that primitive an d in tractablein strumen t, b u t hitherto withou t produ cing any ve ry practicalre su lt;It was shown by Delezenn e in 1842 that it is impossible to

make a strin g sou n d if it b e excited in the cen tre by a bow .

Du hame l was of opin ion that in a string which is g iving itsfoun dation ton e the first partial is vibrating also

,an d that

sin ce the b ow preven ts this f orm Of motion,sou nd can n ot b e

produ ced . To v e rify this hypothesis, he en deavou red tosou n d a strin g by mean s of two bow s m ovin g in the same

dire ction , to the right an d lef t respective ly of the m iddlepoin t of the strin g . Still n o sou nd was produ ced . B u t on

the othe r han d , if the position of the bows We re retain edu n chan g ed , an d an Opposite dire ction of motion with equ alve locity were g iven them

,the foun dation ton e came o u t

in stan tly,accompan ied by the first u ppe r partial . If the

strin g b e attacked su cce ssive ly close to each of the con se cu tiv eharmon ic poin ts , so as to produ ce the fu ndamen tal ton e

,the

corre spon ding u pper partial i s re in forced . At on e -third of

the leng th the tw elfth has abou t e qual in ten sity w ith the

fun damen tal , at on e -fou rth the dou ble octave , at on e—fif th themaj or seven te en th . The harmon ic always slightly pre cede sthe fun damen tal ton e . Speakin g g en eral ly , a strin g vibratingtran sversally can on ly sou n d on the condition that it"g iv e stwo tran sve rsal ton e s , the sharpe r of which depends on the

poin t of attack,or the mode of excitemen t.

So n om e t e r .

The most conven ien t apparatu s for the performan ce o f

expe rimen ts on string s is the mon ochord or son ometer. Thisi s a device of g reat an tiqu ity . It con sists

,in its m ost mode rn

form,of a long re sonan t box bearing on its u ppe r su rface

wrest-pin s , an d two bridg e s set at a fixed distan ce,u su ally

on e or two metre s apart. The space between the se is occu piedby a g raduated scale , an d a trave lling bridg e slide s alon g thewhole distan ce . By mean s of the wre st pin s

,on e or m ore

w ire s are strain ed ove r the fixed b ridge s to the requ iredten sion ,

an d any g iven length of string can b e cu t off by thesl iding bridge . Be side s the w re st-pin s

,the re i s u su ally at on e

en d a pu lley grooved . to rece ive the wire,to which we ights

can b e attached so as to ve rify the secon d law g iven above .

A modification of this latte r con trivan ce,su ited to bear a ve ry

con siderable load , ov er. two hu n dredweight‘in some in stan ces ,

12 ON SOUND. [en s P.

yie lde d practical re su lts in the han ds of Colon e l Perron e tThompson

,an d will b e de scribed in a su bsequ en t chapte r.

Mon s . Me ld e has in trodu ced a striking me thod of illu stratingthe vibration s o f a string , by fixing on e of its extremitie s to

MODES OF PRODUCT ION OF SOUND . 13

the -pron g of a tu n ing -fork, and the othe r to a w re st pin , by

mean s of which its te n sion can b e varied . The f ork be ingmade to vibrate ,and the strin g prope rly stretched , the latte roscillate s in u n ison to the forme r in a man n er easy to see froma distan ce . When the ten sion i s g radu ally le ssen ed

,the

sing le segmen t at first formed divide s in to two,then in to

three portion s,each separated by motion le ss n ode s .

1

Lo n g itu d in al an d To rs io n al V ib ration s .

Lo n g itu d in al V ib rat io n s .— Eve ry strin g which vibrate s

tran sv erse ly betwe en two poin ts mu st also'

vibrate longitudinally. This is eviden t from the fact that it can n ot deviatefrom a straight lin e withou t leng then ing , n or re tu rn to itw ithou t shorten ing . The re su ltan t sou nd is u su al ly difficu ltto hear in ordin ary son ometers

,be cau se the two bridg e s at

the extrem ities of the string be in g u n able to stop the lon gitu ~

d in al vibration s, they are tran sm itted to the lengths of wireb eyon d , and thus stifled by in te rfe ren ce . In the Violon ce llothe effect of long itu din al vibration s is sometime s very u n

pleasan tly sen sible , f orm ing on e o f the varietie s of false n oteor “w olf .

”

Longitudin al vibration s have been l ittle u til ized in the

produ ction o f mu sical sou n ds. The ir f re qu en cy of vibrationvarie s in v ersely as the leng th of string .

Ev en in the mon ochord a powe rfu l acu te sou n d can b eobtain ed by rubb ing the string lengthw ise with a piece o f

rosin ed leather. By damping the string in the cen tre , a n oteis heard an octave highe r, an d by ste ppin g it at on e third

,

on e a twelfth above the fu n damen tal. These lon g itudinalv ib ration s are n ot affected by ten sion , as are the tran sverse

,

b u t a re mate rially influ e n ced by the su bstan ce o f which thew ire ismade

,and the vel ocity with which it tran sm its sou n d ; 2

in deed , the w ire in this form approximate s in its behaviou r toa rod or b ar. In the case of the fu n damen tal n ote each ofthe tw o halve s of the string is altern ate ly extende d and

compre ssed . At the middle poin t the re i s n o compre ssionb u t great amplitu de of movemen t .

A b eau tifu l apparatu s o f this kind was exh ib ited at the Loan Exhib ition , inwhich b oth en ds of the strmg were con n ected with e lastic b ars kept invib ration b y mean s of electro-mag n ets . The period and p lan e o f the wavemotion could b e regulated .

In deed the n ote emitted affords a good measu re of the varying velocity of

gl

a

n d-waves in different materials. Se e the paragraph on velocity in solidLes.

l l l l l

14 ON SOUND . [CHAR

S o n om e t er f o r L o n gitu d in al V ib rat io n s .-A string to pro

du ce pu re lon g itu din al vibration s mu st b e stre tched be tw e enheavy and firm ly

-fixed te rm in al s. At e ach end of the l on gwooden re son an ce -box are clamps of solidmetal faced with lead .

A scale,u su ally of 15metre or more in leng th, is placed be low

the string , which can b e stretched e ithe r by wre st-pin s or bya we ight. A leaden clamp trave l s along the string . It is the ,

in e rtia of this we ight an d that of the te rm in als which de te rm in e s the n odal poin ts, an d n ot the ir rig idity as in the caseo f tran sverse oscillation s. The vibration s them se lve s con

sisting ,of alte rn ate rarefaction an d con den sation o f the

elastic mate rial close ly re semble those which take place inan organ -pipe

,or in the atmosphe re at larg e when conveying

sou nd .

The string may b e excited by m ean s of the thumb an d

fing e r du sted with powdered re sin , an d moved len g thwise ,or be tte r with the poin t of a violin -bow acting in the same

dire ction .

The sou n ds thu s e l icited are ve ry pu re,an d always mu ch

sharper re lative ly to the len gth of the strin g than those g ivenby tran sve rse vibration s: They have the same mu tu al re lation s as the se latter and the ir vibration -n umbe rs are inverse lyproportion al to the wave -lengths .To rs io n al V ib ratio n s o f S tr in g s .

—In depen den tly of the

two modes of vibration g iven above,every strin g performs a

third oscillation in separable f rom the othe rs. This can easilyb e

' demon strate d by hookin g lightly to the m iddle of the

string a small double rin g o f fin e w ire,in the shape of the

figu re 8, carryin g a l ittle paper flye r. When the string i stran sv ersely excited , whe ther by plu ckin g or with a bow

,the

rin g and flyer will beg in to tu rn rou n d with g reat rapidity,alte rn ating freqiren tly in the dire ction of rotation . Thisoccu rs whethe r the fu n damen tal ton e or an u ppe r partial b ee licited . It is eviden t that the strin g has a torsion al motion

,

which it commu n icate s with alte rn ate dire ction to the en ve l oping ring . The torsion al vibration s of strin g s are of littlepractical importan ce . B u t they u n expectedly in te rven ed inthe w rite r’s experimen ts with low n ote s on the dou ble -bass asdetailed e lsewhere . When eve r it was attempted to produ ceg rave ton e s by en larg ing the se ction al area o f the string , alimit was fou n d beyon d which the diameter of the strin gcou ld n ot b e in creased

,from the predom inan ce of the se

tprsion al ton e s. The bow,acting at the circumferen ce of the

string , had powe r en ough to rotate it in stead of commu n icating pu re ly tran sverse oscillation s. It was con sequ en tly

"1 MODES OF PRODUCTION OF SOUND.

15

n ece ssary to“

adopt, string s of smaller d iameter and g reaterspecific gravity.

'

It will thu s b e se en that a String vibratin g in the tran sversed irection has impre ssed u pon it at least fou r distin ct motion s ,two in the orig in al dire ction , on e . lon g itu din al , an d on e

torsion al . “ By acciden tal circum stan ce s the n umber may b e

indefin ite ly in creased .1

Summary of Strin g Vibration .

I. Trans verse — Alon e u sed in mu sic.

Velo city

1P eriod n

5L.

t_m

The harmon ies a complete series, as in open p ip es .

11. Longitu d inal

Unail‘

ected b y ten sion . Inversely as length .

Their pitch higher than in No . I. Vary in pitch with the material o f

the string.

Harmonics a complete series .

Afford ameasure of sou n d-velocity,

III. Torsiona lComplicated in theory. Not u sed mu sically.

V ib ratio n s o f B ars o r R o d s are the n ext in simplicity tothose of string s . They are of three kin ds

,longitudinal,

torsional,an d lateral. Of the se the last are the m ost im

po rtan t. Althou gh the thre e classe s of vibration s are qu itedistin ct in the ory , yet in actu al experimen ts it i s oftenfou n d impossib le to excite long itu din al or torsion al vibration swithou t the accompan imen t of some m easu re of lateralm otion . In bars o f ordin ary dimen sion s the grave st late ralmotion i s far g rave r than the g rave st long itu dinal or torsion alm otion .

Rods or bars with on e en d fixed can also b e made to vibratel ongitu din ally, the pitch be ing inve rse ly proportion al to thelen gth of the rod . The time of a complete vibration i s thatrequ ired for the son orou s pu lse to run twice to an d fro overthe rod. The first u pper partial of su ch a rod produ ce s an ode at on e -third from its fre e en d ; the secon d has twon ode s

,the highe r at on e-fifth o f the len gth from the free en d

,

the lower bise cting the remain de r of the rod . The orderOf the ton e s is that of ’ the odd

‘

n umbe rs 1,3,5,& c . ,

thu s

16 ON SOUND. [CHAR

re semblin g those of a stopped diapason pipe which will b ede scribed fu rthe r on .

The on ly in strumen t fou nded on this property o f rodsis b u t little kn own an d rare ly u sed

,be in g m ore an acou stical

cu riosity than anything .e lse . It con sists of a n umbe r of

deal rods , abou t twen ty or more,standin g u p v ertically

,

like the strin g s of a harp,f rom a sou n d-board obliqu e ly

5.-Longitu dinal vib rations of rods .

placed be low , in to which the ir lowe r extrem itie s are firmlyfixed . They are excited by ve rtical friction w ith the rosin edfing ers. A sim ilar in strume n t f u rn ished w ith g lass tubesin stead of wooden rods is occasion ally to b e heard in thestre ets of Lon don .

To r s io n al V ib rat io n s are even o f. le ss acou stical importan cethan those n am ed above . They w e re first

,like them

,investi

g ate d by Chladn i , b u t they have hithe rto con trib u ted n o

in strumen t to mu sic,an d are chiefly of a mathematical and

the oretical in tere st.Late ral V ib rat io n s o f Elas t ic R o d s , on the othe r han d , are

of larg e service b oth the oretically an d practically. They difier

mate rially accordin g to whe the r the rod is firm ly fixed at on e

or both en ds,or fre e at the two en ds

,an d supported at some

othe r poin t. A rod fixed at b oth e n ds behav e s exactly like a

string . It may vibrate in on e,or in two

,thre e

,or more

segmen ts. B u t the rapidity o f the vibration s,

and the

con sequ en t pitch of the n ote produ ced,diffe r e n tire ly from

those of a strin g . Whe reas the vib ration s of t he string risein a simple arithe rn etical se rie s , those of the rod rise as the

square s of the odd n umbers . When a string divide s in to two

18 ON SOUND.

se ems the simplest. It is said to have origin ated by acciden tfrom the n ote gi ven by a common n ail in se rted in the wain scotwhen a we ight was hu ng to it by a string . In this a b ow is

F ig . 7 .

—Vib ration s of a metal rod .

drawn ov e r a gradu ated se rie s of n ails,or rods

,fasten ed by

on e end to a block o f wood , thu s settin g them in vibration .

In the “Jew ’

s harp ” or Gu imbarde,which in variou s form s .

appears in many parts of the world as a popu lar in strumen t,

the f undamen tal ton e of the spring is modified by alteration s

L] MODES OF PRODUCTION OF SOUND.19

in the cav ity of the mou th to which it is applied . A cu riou sin strume n t o f the sprin g class is the Z anze , from We ste rnAfrica

,shown in the late Loan Exhibition at Sou th Ken sing ton .

F ig . 8 ,

— Na11-fidd le .

It con sists of a carved w ooden box, hav ing at the top a

n umber of iron tongu e s , which the performe r sets in v ibrationby mean s of his thumbs. This approache s n early to the

mu sical-boar, in which a“ comb of ste e l vibrators , we ighted

in the lowe r octave s with lead , are plu cked by pin s in se rted ina revolvin g barre l . The “

gangs”

of American an d othe rclocks are rods coiled in to a flat Spiral , attached to a heavymass o f me tal at the fixe d end , an d firmly screwed to the

wood of the case . An in strumen t te rmed the “B ell P iano,

”

with sing le sprin g s stru ck by hamme rs actu ated from a keyboard

,is also made . In tuning

-f orks , the n e ce ssity for firm lyfixing the base of the rod or sprin g is obv iated by attachingit to a secon d rod or spring , which vibrate s in opposition , an dkeeps the whole mass in equ ilib rium .

R o d s or B ar s fr e e at b o th e n d s may vibrate in sev e ral ways.

In the simple st case the rod has two n ode s,and thre e vibratin g

seg men ts the cen tral segmen t i s the lon g e st, the distan ce o f

each n ode from the e n d be in g abou t on e -fou rth of the distan cebetwe en them . In the se con d form the rod has three n ode s ,and fou r vibratin g parts

,the cen tral segmen t dividing in to

two . The fu ndamen tal n ote of rods thu s arran g ed is highe rthan that of a rod fixed at on e end in the ratio of 4 to 25.

The u ppe r partials are n ot mu ch re qu ired,the first mode of

vibration be in g that common ly u ti lized.

20 ON SOUND. [CHAR

These vib ration s of‘ rods rise to the ir highe st mu sical

characte r in in strum en ts common ly called Harmon icon s,which

Fig. 9.

— Zan ze .

may b e made of wood, g las s , stee l , or even of compactcrystallin e ston e . Many orien tal specimen s are made of the

first material,u sually of the silice ou s ou te r laye r of the

bamboo , an d are remarkable for the pre sen ce o f re sonatorsre in forcing the n ote , which will b e adve rted to farthe r on . Theton e is aston ishing ly large an d pu re. Su ch an in strum en t ofpin e wood

,bearing the n ame of Xylop hon e , has re cen tly be en

produ ced at many Lon don con ce rts . An othe r,made of com a

pact slate in bars,the Rock Harmonicon , was exhibited a few

L J MODES OF PRODUCTION OF SOUND. 21

years back . Mozart writes for a Similar in strumen t,probab ly

made of stee l,in his opera of the F lau to illagz

'

co,whe re it is

in ten ded to imitate the sou n ds of the classical Sistrum.

Fig-Mariu i ba.

By far the m ost importan t case of rods or bars free at bothen ds 18 that of the ordinary tu n ing-fork , n amed aboveTh e Tu n in g —fo rk , as an acou stical in strumen t of paramou n t

in te re st,requ ire s to b e de scribed in detail. It may b e looked

u pon as an e lastic b ar,free at both ends, and supported

fin the

22 ON SOUND. [cm-w .

middle whe re the stem is in serted , or as two mu tuallyan tagon istic bars , v ibrating in opposite phase s

,so that the

g en e ral cen tre of in e rtia i s u n distu rbed . It i s u sually madeof stee l , b u t is equ ally efficien t if f ormed of hard brass , or ofthe compoun d of tin and coppe r

,called be ll~ or gu n -m etal. I In

this case,the b ar, in stead of remain ing straight, i s ben t u n til

the two vibrating bran che s stan d paral le l to on e an othe r. Its

pitch be come s somewhat flatte r afte r this chan g e , and then odal poin ts approximate to on e an other. The stem or han dlei s u sually in se rted in to the con v exity of the ben d , and, in thebest con stru cted forks , i s Spread in to a solid block of me talcon tin u ou s with the f ork ; in deed the write r has fou n d thesoun d em itted by a f ork in which the cu rved part i s ex

ten ded in to a triangu lar prolongation fu ller an d more pu rethan in the ordin ary con stru ction . The su pport in this casestan ds at right an g le s to the plan e of the f ork’s vibration ,be ing in se rted in to a hole drilled at the cen tre of the triangu lararea.

It will b e seen that part of the motion given to the ordinarytu n in g -f ork is

,by its shape

,tran smitted in the dire ction o f

the stem'or han dle,which has an up

—and-down oscillation at

right an g le s to that of the pron g s themse lve s . This com

pon en t can b e farther tran smitted to a re son an t body, an d theton e mate rially au gm en ted by con sonan ce .

The sou n d of a tu n ing -f ork,when stru ck alon e , con tain s ,

be side s the fun damen tal n ote , n um e rou s u pper partial ton e sb u t the in te rval between them an d the lower sou n d isin fin ite ly greate r than in the case of string s. In thoseexam in ed by He lmholtz

,the n umb er of vibration s of the

first harmon ic varied from 5 8 to 66 time s that of the fu n dame ntal the rate s of the whole se rie s be ing as the squ are s ofthe odd n umbers 3

,5,7,9, &c. The re su lt of this is that the

u ppe r partial s are sin gu larly evan escen t, an d soon leave thefu ndamen tal practically pu re and u n complicated . This importan t acou stical property ismate rially in creased bym ou n tingthe fork on a re sonan ce chambe r

,which re in forces the groun d

ton e at the expen se of the others,as will b e explain e d in a

later chapte r.Be side s simplicity an d pu rity of ton e

,the tu n ing -fork

I In consequ en ce of the difference of rate in the tran smission of sou nd

throu gh differen t media, the srze o f b rass tu n ing ~forks is mu ch less than that

o i steel on es. The ve locity of so un d in stee l is at a maximum , amou n ting tometres per secon d . For b rass the velocity woul d b a

‘

less in ab ou t the

ratio 1“

5 : 1 . So that a tu ning-fork made o f b rass wou ld b e ab ou t a fifth lowerin p itch than if the material were steel.

” —Lord Rayleigh , Theory of Sou n d

p . 220.

L] MODES or PRODUCTION OF SOUND. 23

posse sses an othe r prope rty which is extreme ly valu able fortheore tical in v e stigation s that n ame ly of be ing on ly Slight]y

Fig . l l .— A tu n ing-fork mou n ted on a so u n ding-b ox.

affected by diff eren ce s Of temperatu re . Like all othe rm etall icb odies , it expands accordin g to a defin ite law , the coe fficien tOf which expan sion for each me tal w ith g ive n in cremen ts

o f heat is easily Ob tain ed ; the m odu lu s of e lasticity o f

the mate rial is d imin ished to a m in u te amou n t by the same

21 ON SOUND. Len in ).

cau se . Hen ce all forks flatten somewhat w ith warmth . It isn ot ve ry difficu lt , however , to apply a proper corre ction tothis e rror

,an d they th en become far the most tru stworthy

standards of pitch w e are acquain ted with.

x

On the othe r han d the ir g reate st disadvan tag e i s du e to therapid falling Ofi of v ibration from in tern al f riction an d

resistan ce o f the air. Nor is it easy to fin d a mean s of

exciting them which Shall e licit a su stain ed ton e . The u sualexcitan t is a blow with a heavy body

,covered with leather or

flan n e l. Fren ch f orks are made with a slight con v e rg en cebetwe en the in n e r Side s of the pron g s , an d are excited bydrawin g a pin som ewhat larg e r than the apertu re throu gh itfrom be low. The write r has su cce eded in ke eping larg e forksOf comparative ly slow motion excited

,by strikin g on e pron g

g en tly b u t repeatedly with a smal l hamme r attached to themechan ism o f an e le ctric “trembler ” be ll . B u t b y far the

be st m ethod is that larg e ly an d su cce ssfu lly employed b yHe lmholtz in the in ve stigation of vowe l sou n ds

,and in his

“Vibration Microscope ,” that n ame ly of cau sing the fork itse lf ,or on e in harmon ic re lation with it

,to b e come the con tact

breaker in a galvan ic circu it, in clu ding an e le ctro-magn et,whichke eps up a se rie s of syn chron ou s impu lse s on the ste e l prong sof the tu n in g -fork itse lf . It has be en common to place thee lectro-magn e t on on e side of the fork

,b u t this i s l iab le to

draw down the fork in to firm adhe sion to its pole . He lmholtzplace s the two p ole s of a rathe r wide horseshoe -magn e tou tside the two prong s of the fork

,so as simu ltan e ou sly to

draw them apart , b u t a still simple r an d more e ffe ctivem ethod was illu strated in the Loan Collection by some Fren chin strumen ts

,and is fig u red diagrammatically by Lord

Rayle igh 2 in which a sing le Short straight magn et, with w irewou n d in on e coil arou n d a core of bob b in Shape is in terposedbe twe en the two prong s , thu s ten ding to d raw them closertogethe r at eve ry con tact withou t exe rting any strain u ponthe su pporting stem of the fork .

I Tun ing-forks are among the in strumen ts the u se of which has exten ded

from sou n d in to other b ranches o f physics , after a p leasan t fash ion o f reci

procity. They have b een employed as measu rers of small in tervals Of time ;their p en du lar vib ration s are so regu lar

,so accu rate , an d so easily adju sted to

any on e period o f vib ration , that they fu rn ish an admirab le m ean s toward

th is en d . . A b eau tiful in strumen t of this n atu re was con trib u ted to the LoanExhib ition at Sou th Ken singto n b y the Fren ch Con servatoire des Arts et

M i’lters.

2 Theory of Sou nd , p . 56 , where it is state d that su ch interruptors are within

the capacity o f a village b lacksmith . The on ly drawb ack is the n eed o f

sp ecial wide forks ; tho se u sually made n ot leaving su fficien t room for the

in terp osed e lectro -magn et .

26 ON SOUND. [CHAR

Chladn i’s method of inve stigation con sisted in su pportingplate s of g lass or metal

,e ithe r squ are

,circu lar

,or o f some

othe r regu lar ou tlin e , by mean s o f a kin d of clamp , an d b ow ing

the edg e in differen t poin ts w ith an ordinary rosin ed how .

The y ib ration s thu s excited we re an alysed by mean s Of san dpre viou sly strewn on the plate it left the vib rating segmen ts

MODES OF PRODUCTION OF SOUND. 27

to heap itself on the qu ie scen t n ode s . A‘

vast variety of

beau tifu l fig u re s was thu s Obtain ed , each corre spon ding to a

particu lar n ote,an d to a special mode of vibration . The

more complicated forms w e re obtain ed by a combination o f

bowin g and damping ; the latte r be ing accomplished byto u ching the edg e of the plate in variou s place s with the tipsof the fing ers, and thu s hin de ring the motion of the spotstou ched . The rate of v ibration in a disc was foun d to b eproportional to its thickn e ss and inverse ly proportional to thesqu are of its diamete r.The se expe rimen ts were con tin u ed by Faraday among

othe rs,who modified them by addin g a light powde r su ch as

the spore s of lc podium m ixed with the sand . Thisin stead

, like the sand,of seeking the n odal poin ts of re st,

colle cted at the place s of most violen t vibration , a phen om en onu ltimate ly explain ed by the cu rren ts of air su rrou n ding the

vibratin g plate , an d n ot occu rrin g in cacao.

The laws which regu late the vib ration s Of plate s hav e be enmu ch discu ssed by mathematician s ; and in deed fu rn ish prohlems of con side rable complexity. B ut as n o practical application of this m ethod of e l icitin g sou n d occu rs

,it will b e

su fficien t to refer the reader to the works in qu e stion fo r

f u rther details . 1B e l ls may b e con sidered to hold the same re lation to plate s

in the ir mode of vib ration that the tu n ing -fork doe s to thelin ear b ar or rod . Indeed g ong s an d some varietie s of orien talb

plls are ve ry little modified from the orig inal shape of the

p ate .

The g ong is u su ally a circu lar Sheet of hard me tal,ren dered

more e lastic by hamme rin g , streng then ed at the edg e by a

de ep flan ge of the same n atu re . The vibration s here are

v e ry complex an d irregu lar, approximatin g som ewhat to thoseo f a ten se membran e , an d the n ote

,if n ote it can b e called ,

is a fortu itou s combin ation of seve ral discordan t ton e s. Notvery dissim ilar from these is the cymbal ; a n oise-produ cin gmachin e , in ten ded , like the drum ,

rathe r to mark rhythm an daccen t than to em it a defin ite n ote . It diff ers from th e g on gin Speaking from the treble regi ster, and in inten tion ally re

in forcin g the high clashin g harmon ics. For this e nd the twocymb als are sharply stru ck toge ther, tou ching in a few limitedpoin ts ; whereas the g ong ,

like the strin g in the pian oforte ,is g en tly b u t rapidly stru ck with a large soft beater ov e r a

1 Thomson an d Tait’

s Natu ral Philoso h Lord Ra lei The o SowndP

‘2 93 ; Donkin ; Kirchoff ,

p y , y gh, ory f i

28 ON SOUND. [CHAR

circle ab ou t on e-third le ss in diam ete r than the in strumen titse lf . NO dou bt this Situ ation f or the bl ows in dicate s thering of g reate st v ibration discove red empirical ly. The sam e

position , it will b e fou n d , produ ces the be st qu ality of ton ef rom the ke ttledrum .

B e l l s prope r are u su ally in the f orm of vessels,e ither o f

hemisphe rical Shape , as in c lock-b e lls or Of a ve ry complexou tlin e , as in chu rch an d hou se be lls . They giv e a n ote richin u pper partial ton e s

,or harmon ics

,u sually discordan t

to the fou n dation ton e , as will b e se en farthe r on vibratingalong the f ree edge , e ithe r in fou r

,Six, or mo re se ction s.

The de epe st n ote i s giv en by the div ision Of the edg e1n to fou r segmen ts , in which case the b e ll itse lf is m o

men tarily disfigu red by the blow of the hammer; in to an

e lliptical figu re ; retu rn ing by its e lasticity in to on e withits maj or axis at right ang le s to that orig inally formed . Other

Fig . l 4 .— Nodes an d segmen ts o f a vib rating b ell.

segm en tation s Of the vibratin g . edg e or soun d-b ow can b e

produ ced,as in the plate

,always pre se rving an even n umbe r

of 4, 6 , 8, 10 , or more vibrating parts. The sou n d o f the

hem ispherical hell is far more pu re an d u n iform than that ofthe ordin ary chu rch be ll. Hen ce this shape is common ly u sedfor clock chime s , an d for carillon s , to which also its compactou tlin e , gradu ated size

,an d con ven ien t facility of stowag e in

a lim ited space ren der it Spe cially fitted . A larg e n umbe r of

the se hemisphe re s can b e arran g ed on a Sin g le axis,w ith ju st

so mu ch of the free edg e projecting as is n eeded for the hammer

L] MODES OF PRODUCTION .OF SOUND. 29

to strike upon . On the other han d , the clock-be ll can n ot b eswu ng from gu dg eon s , an d doe s n ot, from the Shortn e ss of

its ax1s o f fig u re , adm it of an in tern al sw ing in g clapper. It

i s therefore u nfit f or ringin g in a peal.The m ore e lastic the material o f which a hell is composed,

the highe r will b e its n ote .

1 The n e ce ssary size and w e ightto e l icit a g iven n ote are main ly matters of trial an d experi

en ce,in con sequ en ce Of which m ost Of the olde r peal s of

be lls are grossly o u t of tu n e . The comm on shape of chu rchbe lls is that of a tru n cated con oid or paraboloid

,closed at

the apical en d b y a dom e -shaped roof,to which the su s

pen din g lu gg s are attached . Assum ing the d iam ete r at the

b ase as 15,an d the he ight as 12, with a cu rvatu re bel ow ,

in

se ction , of rad iu s 8 f or the lowe r half,an d for the u ppe r o f

radiu s 30,Dr. Hau ghton has g iv en a tab le o f w e ights an d

pitch . On e octave of the se de serv e qu otation ,in the hope

that they may some day b e con s tru cted .

NOTE. Diameter. Sou n d-b ow. Weight . Clapp er.

He lmholtz state s that the ton e s vary with the greate r orle ss thickn e ss Of the wall of the be ll towards the marg in , andthat it appears to b e an e sse n tial poin t in the art of castingbe lls to make the de eper proper ton e s mu tu ally harm on ic by

1 B ells have o f late b een « rnade of steel , b u t the common composition is ofcoppe r an d tin , hen ce termed

“b e ll or

“gu n -metal ,

”in the approximate

propo rtion s of six atoms o f th e former to on e o f the latter. This is equ ivalen tto a p ercentage b y weight of copper 76 5, an d tin 23

's. The exact atomic ratiose ems to pro du ce too hard an d b rittle an alloy, which de fect can b e redu ced .

b y slightly in creasing the qu an tity of copper. A common mixtu re is 13 copperb y we igh t to 4 tin . or b y van admixtu re o f zin c , an d p o ssib ly o f Silver

, althoughthe traditionary stories as to the effect of the latter metal seem to a greatexten t imaginary .

30 ON SOUND. [CHAR

g iv ing a certain empirical form . Gle itz , the organ ist, in hisHistorical Note s on the G reat Be ll an d the othe r Be l ls inErfu rt Cathedral ,

” state s that the first-n amed g ive s the follow ing se rie s o f n ote s : 6 , g#, b , e

’

,b

’

, It was castin 147 7 . Hem ou y of Zu tphen , in the seven te en th cen tu ry,requ ired a g ood be ll to have thre e octave s , two fif ths

,on e

maj or an d on e min or third . By the kin dn e ss of Sir F . G .

Ou seley an d Dr. Stain er, the write r is en abled to reprodu ce,

in mu sical n otation , the compou n d n ote s of sev e ral we llkn own bells . When played on the pian oforte the dampersshou ld b e raised .

HEREFORD CATHEDRAL. GREATTOM OF OXFORD.

5Hou r Be ll .Ten or Be ll .

B ro B EN OF WESTMINSTER .

PAUL’s .

— Larg e Hou r Be l l .

In the se n otation s,which can on ly b e con side red approx1

mate , m in im s are employed for lou d sou n ds an d crotchets forthose which are le ss promin en t. All the n ote s shou ld b e stru cksimu ltan e ou sly, b u t the m in ims loude r than the crotchets.In the larg e hou r be ll o f St. Pau l

’s, the u pper n ote i s mu ch

lou de r than the A flat, which ought to b e the tru e n ote of

the b e l‘ The wavy lin e is to show that the pitch of the

u ppe r n ote doe s n ot remain station ary. In B ig B en the Eis n ot a pu re n ote , b u t is combin ed w ith so many sou n dsthat it is impossible to g ive a n eare r analysis of the ton e of

the bell.It is foun d that the n umber of v ibration s made in a g ive n

time i s proportional to_the square of the n umber of segmen tsin to which the bell divide s itse lf. Thu s, if 2m b e the n umbe r:

L] MODES OF PRODUCTION OF SOUND. 3 1'

o f v ib rating segm en ts , we find,sin ce it varie s as W

,by

making m su cce ssive ly equ al to 2,3 , 4 , &c. ,

for the n ote s Of

the b e ll.4 9 16 25 36 &c.

1 g 4 9

(31 D, 0 3 egg, D,

corre sponding to its division in to the following segme n ts

4 6 10 12 850 .

Non e of the se se con dary n ote s are harmon ic s to C , exceptC . They are

,howeve r

,very variou sly audible in diffe ren t

be lls .The figu re o f the u su al chu rch he ll is irregu lar, more or le ss

con ical or cylin drical ln ou tl in e , an d on ly symmetrical arou n dits v e rtical axis . The shape o f be lls varie s mu ch in diffe ren tepochs an d cou n trie s. They are almost in variably made of

cast m etal , there in differin g from the g ong . Near the Ope nextremity is a thicken ed ring of m etal te rmed the sou n d-b ow ,

again st which the clappe r, ordinarily made o f soft wrou ghtiron

,strikes. After a time the be l l and clappe r g radually

m o u ld them se lve s to on e an othe r, an d the ton e be come s fu lle r,

w ith fewe r of the v e ry acu te u ppe r partials . B u t f rom ao

ciden tal irregu larities of shape , added to the varying spe cificg rax1t) o f cast m etal , n o chu rch be ll speaks with a sing lefou n dation sou n d

,or w ith on ly a sing le series of harm on ics .

The re is a comm on ten de n cy to the afte r-produ ction of n ew

sou n ds , which en croach u pon those orig inally emitted . Hen cethe we ird eifect of the se g rand sou n d produ ce rs, the difficu ltyof fixing the ir exact n ote , an d the mass of throbbin g beatsdu e to in te rfe ren ce which can always b e heard in the n eighb ou rho od of a large be ll . They are to a ce rtain exten t tu n ableb y chippin g away more or less metal at the sou n d-b ow

,b u t

the main pitch has to b e se cu red empirically throu gh the

carefu l im itation of a pattern kn own by expe rie n ce to produ cea ce rtain n ote .

Bells may b e excited in other ways than by strikin g .

Small spe cimen s,whethe r of g lass or me tal

,an swe r we l l

to the fiddle -bow,and the ir sou n d may also b e d eve loped by

other forms of friction . In these case s the sou n d is swee tand con tin u ou s . This fon n of excitation has been broughtto its highe st poin t 1n the in strumen t kn own as the M u s ical

G las s es , forme rly 1n g reat repu te , an d recen tly revived. They

32 ON SOUND.

were a collection o f g lass ve sse ls, vibrating on the prin cipleof be lls , se lected so as to form an approximate scale

, an dfarthe r tu n ed by pou ring in water. The ir n ote was excitedby rubbing the f ree edg e e ither w ith the m oiste n ed finger or

with acl oth wetted with acid . An ingen iou s modification Of ,

this arrang em en t exists at the Sou th Ken sing ton . Mu seum, ;

whe re in the bell s are fixed to a rotatin g spin dle , an d tou chedwhile in rotation with a wetted excitor. The soun ds thu s

34 ON SOUND.

The ir theory is in con sequ en ce somew hat complicated.

‘ Thethe ore tical m embran e i s a pe rfectly flexible an d infin ite ly thinlam in a of solid matte r, of u n iform mate rial an d thickn e ss ,w hich is stre tched in all dire ction s by a ten sion so g reat as toremain sen sibly u n alte red du rin g vibration . Its vi b ration shav e be en inve stigated by Bou rg e t in the same way as

Chladn i demon strated those Of plate s . He excited them bymean s of organ -pipe s , ren derin g the m otion visible by m ean s

o f san d scatte red on the mem b ran e . The prin cipal re su ltstve t e

Summary of Vibrations in M embranes.

1 . A circu lar memb ran e can n ot vib rate in u n ison with every sou nd . It canon ly p lace itse lf in

'

u n ison with sou n ds more acu te than that heard when thememb ran e is tapp ed .

2. The sou n ds are separated b y smaller in tervals the high er th ey b ecome .

3 . Nodal lin es are o n ly formed d istin ctly in respon se to certain d efin itesou nds . A little ab ove or b elow

,con fu sion en su es

,an d wh en the p itch o f the

p ip e is decidedly altered the memb ran e remain s u nmoved .

4 . Th e n odal lin es are circles or d iameters , an d comb in ation s of th ese . Whenthe n umb er o f diam eters exceeds two , the san d ten ds to heap itself con fu sedlytowards the midd le of the memb ran e .

As a m atter of practice,membran e s are b u t little u tilized

in mu sic as sou n d-produ ce rs . Of the thre e kin ds of drum s,the

side drum ,the bass drum

,an d the kettle drum

,on ly the last

is tu n ed to a defin ite n ote,an d that often ve ry impe rfe ctly.

The side drum an d bass drum are m ostly u sed in m ilitary ban ds,

an d serve chiefly the pu rpose of marking rhythm an d accen tfo r marchin g . According , howev e r, to the arran g em en t of

m odern orche stras , the kettle drum stan ds alon e in possessingtwo , or at m ost three , n ote s of the sixte en -fe et octave .

Z am b om b a .

— A cu riou s combin ation o f a m embran e witha rod vib ratin g lon g itu din ally is u sed in Spain an d Portugalu n de r the abov e n ame . To the cen tre of a membran e likethat of a tamb ou rin e is attache d a small sm ooth rod of can eor stron g straw . If this b e g en tly ru bbed be twe en the we ttedfin g e rs the l ong itu din al vib ration s thu s excited in it are

con veyed to the cen tral poin t of the membran e,an d a pe cu liar

drum-l ike sou n d is produ ced , posse ssin g a ce rtain rhythmaccordin g to the rapidity with which the fing ers are movedu pwards an d downwards .

Fu rther remarks will b e g iven on this su bje ct in the chaptero n spe cial application s of acou stics to practical mu sic .

V ib rat io n s o f R e e d s .—Re eds form the n atu ral l ink between

solid an d aeriform vibrators. Although they, as e lastic tongu e s1 See Lord Rayleigh, op. cit. p . 290 for farther details .

L] MODES OF PRODUCTION OF SOUND. 35

of wood or m e tal , have the ir prope r laws , following g en erallythose of rods or bars fixed at on e extrem ity an d f re e at the

other,they rare ly if eve r con form strictly to them , b u t are to

a g reater or less exten t coe rced by the e lastic flu id which sets

Fig. l 7 .— Strikin g Reed . Fig . 18.

-Free Reed .

them in motion , and from the con son an ce of which theymain ly de rive the ir powe r and qu ality of ton e . When

,as in

the harm on ium,they vibrate in to the g en eral mass of air

,they

are obviou sly le ss coerced than 'when they are adapted tocylin drical or con ical pipe s which posse ss the ir own n ote and

vibration n umbe r. B u t even in the latte r case , reeds , if of

sufficien t strength , are able to assert the ir in depen den ce ; asi s seen from the fact that the clarin et

,a reed in strumen t of

le ss than two fee t in len gth , is able to reach the lowest n ote sof the fou r fee t octave . This g reat exten sion of its compasscan on ly b e du e to the coercive power Of the slowly vibratingreed on the comparative ly short column of air.

36 ON SOUND. [0mm

Reeds are u sually divided in to two kin ds,the beating , an d

the fre e reed . The f or me r i s of con siderable an tiqu ity ; thelatte r appears to b e a m ode rn inven tion . Both form s posse ssthe e lastic ton g u e b u t the beatin g re ed diffe rs from the freein that the tongu e is made to cover the orifice throu gh whichthe air passe s an d slightly to ove rlap its edg e s ; whereas inthe latter it passes free ly

,as the n am e implie s , thou gh with

b u t little room to Spare , past the edg e s o f the slit in which iti s adapted . Orche stral reed in strumen ts and all the Olde rorgan -re eds are of the beat in g kin d .

I

The b e ating reed is farthe r subdivided in to sing le an d

double varietie s , the f orme r represen ted by the clarin et,an d

by the u su al organ re eds ; the latte r appearin g in the Oboeand bassoon . The se will b e de scribed in Chapte r VIII.

It has often be en poin ted ou t that the type of all reeds isthe little pipe of Wheaten straw made by children in the

fie lds . The hollow stem of a tall g rass cu t Off ju st above a

kn ot fu rn ishe s the stopped tube of the in strume n t,which is

made to speak by slitting the straw u pwards towards the kn otfor abou t an in ch , thu s leaving a tongu e stan ding o u t and in

a position to vibrate . The on ly in strumen t in which thisprim itiv e arrang emen t su rvive s is the bagpipe , the “dron e s ”

of which , e spe cially in Old spe cimen s,con tain a larg e reed

thu s made from the woody tubu lar stem of some peren n iallan t.pV ib ration s o f C o lum n s o f A ir .

—Next in importan ce tostring s as sou n d-gen erators may b e ranked pipes an d column s

of air. A ve ry simple expe rimen t will illu strate this. If

a piece of stou t g lass or metal tubing from a f oot to two fe etlong , an d an in ch or m ore in diame ter, b e taken , its en dssmoothed an d rou n ded to a blun t edg e , it will fu rn ish the wholeapparatu s requ ired . Holding it horizon tally in o n e han d an d

strikin g the Ope n e nd smartly with the palm of the otherhan d , su fficien t vibration will b e excite d in the con tain ed air toprodu ce a distin ct mu sical n ote . which Often lasts a se con d orm ore long en ou gh , at any rate , for its pitch to b e heard an d

de termin ed . If,afte r striking , the han d he qu ickly removed ,

a se con d n ote i s heard to follow the first at the in terval of an

octave above . In the former case the pipe vibrate s as what iste rme d a stopped pipe , with on e en d closed.in the latte r case as

an Open pipe . All the variou s form s of pipe u sed in the organand e lsewhere on ly diffe r f rom the se ru dimen tary forms in

I In Dr. Tyn dall’

s otherwise excellen t an d accu rate work on Sou nd,to which

the writer is mu ch in de b ted , the organ re ed is repre sen ted an d describ ed as a

free reed. Free re eds have b een found to fail in this position .

MODES OF PRODUCTION OF SOUND.

having a more complex me chan ism for orig inating and

main tain in g the mu sical vibration .

When both en ds Of the tu be are Open,a pu lse trave ll in g

backwards an d forwards within it is c omple te ly re sto red toits orig in al state afte r trave rsin g twice the len gth o f the tu be ,su ffe ring in the proce ss two refle ction s b u t when on e en d is

closed,a dou ble passag e is n ot suflicien t to close the cycle of

chan g e s. The orig in al state can n ot b e re cov ered u n til afte rtwo refle ction s f rom the open en d, and the pu lse trave ls ov e rfou r time s the leng th of the tu be .

To make the u n stopped tu be in the above expe rimen t yie ldthe same n ote as the stoppe d

,it w ou ld the re fore b e n e ce ssary

\ to g ive it dou b le the length. This law is u n iversal , and may\e asily b e explain ed .

B ut vibration may b e set u p in the column of air othe rwisethan by the blow above de scribed. If a g en tle stream of

b reath from the lips b e sen t obliqu e ly across the Open en d of

e ithe r an Open or a stepped tube , an au dible n ote re su lts ;in deed a common b u t simple in strum en t u sually n amed thepan dean pipe s acts on this prin ciple . A se rie s of small reed '

tube s of g radu ated len gths are arran g ed side by side withthe ir orifice s u pwards in a horizon tal lin e

,an d the ir closed

extrem ities downwards. They are fasten e d tog ethe r by stripso f wood . If the m ou th b e passed alon g the u ppe r row of

hole s an d the breath g en t ly u rged in to them , a scale more orle ss corre ct according to the accu racy w ith which the ir re lativelengths have been adju sted will re su lt. Hen ce it is Oftente rmed the mou th organ . If a vibrating b ody su ch as a

tu n ing -fork b e brou ght Opposite the orifice,an d the length

o f tube b e in a prope r ratio to it,the n ote will immediate ly

start ou t in to prom in en ce . This phen omen on will b e fu rthe rexplain ed u nde r the subje ct of con sonan ce . In organ -pipe s .flu te s and flageole ts , a thin she et o f compressed air is made

to impin g e again st a sharp edge . This edg e may b e the

sharpen ed extremity o f the Open tube itse lf,as i s we l l se en

in a remarkable example brought from Egypt by a frien d of

the write r’

s

Fig . 19.—Egyptian flu te .

We have he re e ssen tially an Open pipe blown on the prin cipleI F . Girdleston e , Esq . of the Charterhouse

,Godalmin g

.

38 ON SOUND. [CHAR

of the pan dean closed pipe, b u t fu rn ished with six late ral

hole s , by mean s of which the len gth Of the v ib rating column

may b e altered,an d the n ote correspon ding ly raised . It

aff ords an in stru ctive link in the history of sou n d-produ ction

Fig . 20 .— Prismatic son orou s pipes . Fig . 21.

-Cylindrical son orou s p ip es .

which has n ot to the write r’s kn owledg e be en pu blishedbefore . In the ordin ary flu te

,the u ppe r orifice is closed w ith

a cork,an d the stream of air i s passed ove r a hole w ith

thin n ed edge s pe rforated at the side n ear the top , the lowe rse rie s of hole s remain ing u nalte red.

L ] MODES OF PRODUCTION OF SOUND. 39

In the organ pipe , a'

more complicated arran g emen t i spre sen t which is be st seen by mean s of figu re s.From the win d-che st a tube leads in to a cavity the on ly ou tlet

of which is a lin ear crack form ing the foot of the pipe .

Ju st ove r this fissu re the w ood or m etal i s cu t away so as toleave a feather-edg ed portion , commu n icatin g w ith the ih

terior o f the pipe , exactly splitting the stream of w in d . An

explan ation has of late been ten dered as to the action hereset u p . The flat plate of compre ssed air blown throu gh the

sl it i s compared to the e lastic material of a vibrating reed .

In passin g across the orifice it mom en tarily produ ce s a sligh texhau stive or su ction al efiect, ten din g to rare fy the air con

tain ed in the lowe r part of the pipe . This b y the e lasticityof the air soon sets u p a corre spon din g compre ssion ,

and the

two allied state s react u pon the orig inal lam in a o f air i ssu in gfrom the bellows , cau sin g it to vibrate and to commu n icate itsmotion to that within the pipe . In deed

,the cou rse of the air

cu rren ts can to a ce rtain exten t b e demon strated by fe edin g thepipe with tobacco smoke or some othe r sem i-opaqu e vapou r.M o t io n o f A ir in P ip e s .

— Schn eeb eli drove air ren de redOpaqu e by smoke through a m oveable slit. When it passeden tire ly ou tside the pipe n o sou n d was produ ced

,b u t appeared

when the issu in g she e twas g e n tly blown on at right an g le s , contin n in g when on ce started u n til a coun te r-cu rre n twas produ cedby blow in g down the u ppe r orifice of the pipe . Little or n osmoke pen etrated in to the pipe . If the sheet of air passedin to the pipe en tire ly there was also n o sou n d

,b u t on blowin g

in to the u ppe r e nd sou n d was produ ced . He con clu de s thatthe L bf t-Lamelle, or aerial lam in a; acts a part an alog ou s tothat of the re ed in re ed-pipe s . He rman n Sm ith has come byin depen den t observation s to a sim ilar con clu sion

,term ing the

she et of air an“aeroplastic re ed .

” Schn eeb eli con siders itse ffe ct to b e con den satory ; Herman n

‘

Smith,with far g reate r

probability,holds it to b e exhau stiv e , an d similar to the com

mon spray-produ cin g apparatu s . The ton e s of the air re e dan d pipe he be lieve s to b e distin ct

,that of the forme r be in g

far more acu te than the latter,an d sometim e s capable of

coercing it. The re can b e n o doubt that this con dition of

thing s exists in the case of the re ed fitte d with a con son an tpipe , as in the case of the clarin et g iven above .

In all the above case s , the air,like the strin g or the rod

,

may assume several mode s of u n du lation . In the o p e n p ip e ,the embou chu re at which the win d en te rs is obviou sly a placeof greatest motion , corre spon ding to the ven tral segmen t ofa string . So also will b e the u ppe r open extremity. Half

40 ON SOUND. [CHAR

way between the se , at the poin t whe re the two Opposite an dcorre lative motion s mee t an d n eu traliz e on e an other, will b e a

n ode or poin t of re st. In this in stan ce the pipe will g ive itslowe st or fu n damen tal n ote . If the force of the cu rren t b ein creased , a shorte r wave may b e set in action

, a n ode be inge stablished at on e-fou rth of the Whole leng th from the emb onchu re , an d an othe r at the same distan ce f rom the top . Thepipe then speaks its first harm on ic

,the octave of the fu n da

men tal. By a fu rthe r win d-pre ssu re three n ode s may f orm ,

the first Of which is on e -sixth from the emb ou chu re, the third

at a similar distan ce from the top,an d the se con d halfway

between the othe r two the pipe g iving its se con d harm on ic a

twe lfth above the f oun dation . As the len gths of the wave sare in the proportion is, i , it is obviou s the time s of vibration will b e 3

, 2, 1 , or corre spon ding to the se rie s of n atu raln umbers.

Midway betwe en each con se cu tive pair Of n ode s the re is aloop , or place of n o pre ssu re-variation . At any of the sel oops a commu n ication may b e e stablished with the exte rn alair

,withou t cau sing any distu rban ce of the motion . The

Mom are place s of maximum ve locity,and the n ode s those of

maxrmum pre ssu re-variation .

In s to p p e d p ip e s a diffe ren t law obtain s ; for the wave shave clearly to traverse tw ice in stead o f on ce the le ngth

'

Of

the tu be , be in g re tu rn ed b y the closed extrem ity. This factalso in flu en ce s the position of the n ode s . When the fu n dam en tal n ote i s stru ck the column i s u nbroken

, the on ly n odebe in g at the stopped en d . In sou n ding the first harm on ican other n ode i s set up at on e -third Of len gth from the Openend. With the se con d harm on ic , a n ode form s at on e -fifth of

le ng th f rom the Open en d,the secon d dividing the lowe r

f ou r-fifths in to two equ al parts . In any case , the stoppede n d mu st b e a n ode , so that the secon d form of vibration of

the open pipe , an d all othe rs which wou ld rende r it the cen treo f a ven tral segmen t

,are exclu ded. Hen ce t h e harm on ic s o f

a stopped pipe follow the se ries of the odd n umbe rs 1,3,

5,&c. The se relation s betwe en the f u n dam en tal n ote of a tube

an d its ove rton e s we re d iscov e red by Dan ie l Bern ou ll i and are

g en eral ly kn own as the Law s o f B ern o u l l i . When the

len gth of a tube exce eds its d iame te r con siderably,the n ote

is in depen den t o f the latte r, an d varie s with the length alon e .

In both stopped and open pipe s, the distan ce f rom an openen d to the n eare st n ode i s a qu arte r-wave leng th of the n oteem itted. In the Open pipe the re is n o fu rthe r limitation b u t

in the case of the stopped pipe, the n eare st n ode to the mou th

42 ON SOUND.

and it is n ot difficu lt to see that the impu lse i s of a kin d toe n courag e the motion , an d to produ ce sou n d.

Fig . 23 .—Trevelyan

’

s in strumen t. Cau se of vib ratory movements .

Sondhau ss’

s Exp erimen t.

When a bu lb abou t 43 o f an in ch in diame te r is blown at

the en d of a n arrow tu b e 5 or 6 in che s in len g th, a sou n d issome time s heard proce eding from the heated g lass . It was

prov ed by Son dhau ss that vib ration of the g lass is n o e ssen tialpart o f the phen omen on . An explan ation of the produ ctiono f sou n d has been g iv en by Lord Rayle igh, which will b e f u llyde taile d in Chapte r VIII.

S in g in g P 1am e s .—Althou gh the serie s of small explosion s

by which the combu stion of gas is marked have con tribu tedsom e brillian t expe rimen ts to physicists , they can hardlyas yet b e said to b e a practical sou rce of mu sical soun d.

The flame of hydrogen has l on g been kn own as a mean s oforig inating a n ote an d o f late a secon d form o f the expe rim en thas u tilized the ordin ary flam e of coal-gas as a ve ry de licatecon son ator and te st for sou n ds produ ced extran eou sly in itsn e ighbou rhood. The former are te rmed sing ing , the latte rsen sitive

,flame s . The sen sitive flame s may be st b e con sidered

e lsewhere b u t a short accou n t of the sing in g -flame i s requ iredto complete the serie s of sou n d-produ cers .It i s easy, when eve r a jet of hydrog en is in flamed , or even

when coal-gas issu e s from a bu rn e r with some force,to hear

an u nmu sical hissin g or roaring accompanying the proce ss .Even in this case

,the n oise often pu ts on a m ore or less

defin ite form of vibration,an d an impu re n ote of coarse

qu ality make s itse lf man ife st. B u t if the jet b e su rrou n dedb y a re son atin g tube

,this has the powe r of redu cin g the

irregu lar vibration s to a g reate r u n iformity,an d Of se le cting

those which syn chron ise with its own vibration -period. Then ote in this case i s Often very pu re an d powerfu l . Theg en e ral type Of the proce ss may b e studied in the ordin aryBu n sen bu rn e r. In this very conven ien t laboratory applian ce .

ordin ary coal-gas is allowed to issu e by a small orifice in to a

large r tube , pe rforated at its lowe r extremity with several


hole s admitting more or le ss atmospheric air; the admissiono f. which is regu lated by a slide . An explosive mixtu re Ofgas and air i s thu s made

,which is preven ted from com

mun icating w ith the sou rce of g as by the cooling efiect ofthe su rrou n ding tu b e , ju st as occu rs in the wire-gau ze en

ve lope of a Davy safe ty-lamp . If on ly a small amou n t of

Fig . 24 .— Philosophical lamp or chemical harmon icon .

air is adm itted the m ixtu re bu rn s With a semi-lum in ou sflame an d sile n tly. B u t as the proportion of air is allowedto in crease by open ing the slide r, the flame loses its lum in osityand at length beg in s to roar. The combu stion g

raduallybe come s discon tin u ou s , an d i s inde ed composed of a serie s Ofshort su cce ssive explosion s . At leng th the m ixtu re be come stoo explosive even for this

,it lights throughou t its whole

44 ON SOUND. [can

leng th with emission of a sin g le report, an d carrie s the comb u stion down the tube to the small gas-jet itself. If theBu n sen bu rn e r, free ly Supplied with air

, b e in trodu ced in to a

tall ve rtical tube , the roaring i s ton ed to con son an ce an d

g ive s a pu re n ote in stead of an irreg u lar roar. It i s n ot

always easy with qu ietly bu rn ing coal-gas in a tube to Obtainany sou n d at all. B u t if the flame b e redu ced in size

,an d

m oved u p and down the con sonatin g tube , it may b e observedto be come tremu lou s at ce rtain spots , an d if left there , sud

den ly , by in crease of the pu lsation s , b ursts in to sou n d . Thisaction may b e dete rmin ed by the m ethod first n amed as a

m ean s o f excitin g mu sical oscillation in a pipe,n ame ly

,by

striking a slight b low with the palm of the han d on the u pperorifice of the tu be . The wave thu s sen t downwards is instan tly taken u p by the flame

,an d the n ote starts ou t

,some

time s with su ch vehemen ce as to extin gu ish it altogethe r.it can al so b e set g oing by the voice . Many o f the Olde rworks on chemistry and physics g ive the simple expe rimen twith a hydrog en flam e an d a g lass tu be . Dr. Higg in s n ame sit in 1777 ; b u t Wheatston e was the first who attemptedsu cce ssfu lly to produ ce a defin ite scale o f n ote s by thism ethod . His in strum en t i s n ow pre served in the Mu se um o f

King’s College , andwas re cen tly shown at the Loan Exhibition

at Sou th Ken sing ton . A serie s of g lass tube s , with m etalsliders for the pu rpose of tu n in g , are arran g ed in a row likethe pipe s Of an organ . Within each of the se i s a fin e con icaltu b e pie rced above with a capillary orifice . The l owe r en d

slide s air-tight in a se con d tube con n e cted with a su pply of

gas. In fron t is a short keyboard like that of a pian o . Eachkey , on be ing depre ssed , lift s the small gas-jet from itsposition at the bottom of the tube to abou t the ju n ctiono f the l owe r and two u ppe r thirds , which , be in g a sen sitive poin t

,immediate ly orig inate s the fu n damen tal n ote o f

the tu be .

Th e Pe p h o n e .— M. Kastn er has en deavou red to u tilize

this prin ciple in a mu sical in strumen t,b u t on a slightly

diffe ren t system . He says,

“If two flame s of a certain sizeb e in trodu ced in to a tu b e made of g lass, an d if they b eso disposed that they reach the third part of the tube ’s he ight,measu red from the bottom ,

the flame s will vibrate in u n ison .

The phen omen on con tin u e s as l on g as the flame s remainapart, b u t as soon as they are u n ited

,the sou n d cease s .” By

m ean s of fin ger-keys the flam e s are u n ited an d separated so

that a me lody can b e played . The re is some u n ce rtain tyabout the in strumen t, depen ding o n the pressu re of gas ; so


that although an n ou n ced for performan ce in Paris , it has n othithe rto been u sed .

A cu riou s acciden tal sou rce of sou n d appears to have beenseve ral time s discovered. It posse sse s n o practical importan ce, b u t affords an apt illu stration of the the ory of harm on ics.If a pie ce Of the ordin ary v ulcan ite tubin g , su ch as i s u sed

for con veying gas , an d which is preven ted fr om collapsin gby a spiral wire coiled rou n d its in te rn al su rface , b e cu t to

a leng th of abou t 18 in che s,an d g en tly blown in to

,a sof t

mu sical n ote of fe eble b u t re edy qu ality is produ ced . On

pre ssing the force of win d,it rise s su cce ssive ly to highe r

n ote s,which w il l b e fou n d on exam in ation to follow rou ghly

the orde r of the common chord to the fou n dation ton e . It isobviou s that the wire coiled in side the tu be produ ce s a serie sof equ idistan t obstacle s

,competen t to throw the air in to

regu lar vibration ; the rapidity of which vibration,and the

con sequ en t pitch of the n ote produ ced,vary with the speed

at which the air is blown in to the cal ibre o f the tu be .

The on ly remain ing sou rce of sou n d rs the human voice ;b u t this is of so mu ch importan ce that it will b e con sideredseparate ly in a late r chapte r.

46 ON SOUND. [CHAR

CHAPTER II.

MODES OF PROPAGATION OF SOUND. VELOCITY. WAVE-MOTION .

REFLECTION . REFRACTION

Th e Pro p ag at io n o f S o u n d appears to take place to some

exten t throu gh all bodie s,b u t in ve ry difieren t amou n ts an d

with varyin g degre e s of velocity. This factor has beenfou n d to vary dire ctly as the squ are root of the bodie s’

e lasticity , an d inverse ly as the root of its den sity. The f ormu la

therefore se rve s for all form s Of matte r. Solids,howeve r

,

be in g l iable to many kin ds of strain,an d flu ids

,whe the r

liqu ids or gase s , to on ly o n e , w e may have difieren t valu e s ofE

,and differen t ve loc itie s o f tran sm ission for the same solid .

In a perfe ctly fre e solid this valu e o f E is iden tical withYou n g

’ s modu lu s . The g reat maj ority of solids howeve rtran sm it sou n d m ore rapidly in on e dire ction than in othe rs.In solids

,more ove r, the the rmal corre ction

,to b e spoken o f

pre sen tly,is very small

,as it is also in flu ids

,whereas in air

it is large .

B y th e B ar th — There i s distin ct eviden ce o f its tran smission through the solid mass o f the earth itse lf f or longd istan ce s . Humboldt says, “At Caracas , in the plain s of

Calabozo,

and on the borders of Rio Apu re,on e of the

afflu en ts of the Oron oko ; that is to say, over an exten t of

squ are kilome tre s,

on e hears a f rightfu l report ,withou t experien cing any shock , at the momen t when a torren t of lava flows f rom the volcan o St. Vin cen t, situ ated inthe An tille s

,at a distan ce o f kilometre s ; At the time

o f the g reat e ru ption of Cotopaxi in 1744 , the subterran eanreports were heard at Hon da

,on the borders of Magdale na ;

yet the distan ce between the se two poin ts is 810 kilometres

MODES OF PROPAGATION OF SOUND.

the ir diff eren ce of leve l is metre s , and they are separatedb y the colossal mou n tain ou s masse s of Qu ito Pasto an d

Popayan , an d by n umberle ss ravin e s an d valleys. The sou n dwas ev iden tly n ot tran sm itted by the air b u t by the earth , andat a g reat depth . At the time of the e arthqu ake of New Granadain Febru ary, 1835, the sam e phen omen a we re reprodu ced inPopa~yan , at Bog ota, at San ta Maria, and in the Caracas , wherethe n oise con tin u ed for seven hou rs withou t shocks ; also at

Haiti in Jamaica, and on the borders of Nicaragu a.

”

NO bette r illu stration of the con v eyan ce of sou n d by solidm edia can b e fou n d , than that which occu rred in the re cen tcollie ry acciden t Coal is an exce llen t con du ctor of

sou n d , be ing both light an d e lastic . It was possible f rom the

v e ry first of the n ob le attempts to re scu e the five imprison edmin ers , to commu n icate with them throu gh a long barrier of

in te rven in g coal , by kn ocking on the extern al su rface of the

seam in which they we re in carce rated . In the same way theywe re able , as it we re , to te leg raph back the fact of the irexisten ce to the ir re scu ers.Mon s . Biot experimen te d on a cast-iron pipe 951 metres in

len gth, an d fou n d that sou n d is propagated throu gh thismetalw ith a mean ve locity o f metre s a se con d

,or more than

95 time s that through air o f the same temperatu re . The pipeu sed was of rathe r he terog en e ou s material

,a fact which ren de rs

the qu an titative determ in ation somewhat doubtfu l .An ingeniou s application of the prin ciple of propagation

through solids occu rs in Wheatston e ’s Telep hone, exhibitedat the Polytechn ic In stitu tion many years ag o . A ban d Of

pe rformers , with violin , clarin et, pian o , and othe r in strum en ts, we re placed in a basemen t room

,through the ce iling

o f which rods of fir-wood we re passed in to a con ce rt-roomabove . Each rod was attached by its lowe r extremity to on e

of the in strumen ts, at its u ppe r en d it was con n ecte d with a

con son ator su ch as will b e de scribed later. When t he

in strum en ts were played , n ot on ly the actu al n ote s, b u t eventhe qu ality and character of each we re distin ctly au dible toany n umber of listen e rs in the con ce rt-room . It will b e seen

from the Table that fir-wood tran smits sou n d at the en ormou sv e locity of metre s per secon d , or more than e ighte entime s that of its tran sm ission in air.

A cleve r toy has been late ly sold by which the tran smissionof sou n d through solids may b e simply demon strated . Itcon sists of two tin cylin de rs

,each closed at on e end

,an d

1 Qu oted in Guillemin’

s Forces of Nature.

48 ON SOUND. [CHAR

j oin ed tog ether by m ean s of a wire,or even an e lastic string ,

of several yards len g th. A sen ten ce gen tly spoken in to on e

cylin de r can b e distin ctly heard by applyin g the ear to theorifice of the other

,when it is qu ite in au dible from distan ce

throu gh the air.

The Te lephon e of Graham Be ll acts on a totally differen tprin ciple , conve rting the vibration s of a metallic plate in tomagn eto-e le ctric cu rren ts in a coil of wire su rrou n din g a

small magn et. By an exactly sim ilar apparatu s at the othe rend of the con du ctin g lin e , the u n du latory cu rren ts thu s produ ced are re conve rted in to mu sical ton e s .

The Microphon e of Prof . Hughe s really substitu te s for a

fe eble sou n d,on e mu ch lou de r produ ced b y varyin g resistan ce

between Oppose d con du ctors. It is the refore e ssen tially a re lay.

V e l o city in A ir .—The v e locity of sou n d in air has been the

subject of many expe rimen ts sin ce the time of Newt on . Thoseof Goldingham ,

pu b lished in the P hilosophical Transactionsfor 1823 , of Arag o in 1825, of Myrb ach an d Stampfer at

Vien n a, of M0 11 an d Van Beek in Hollan d,an d o f Greg ory,

seem the m ost tru stworthy. The observation s have g en e rallybeen these of the flash an d the report of a distan t can n on .

The sam e Obse rve r n ote s both phen omen a with the same

watch , an d if the distan ce of the gu n b e several m ile s , the rei s ample tim e to write down the obse rvation of the flash , beforepreparing for observation of the sou n d.

Tab le of Experimen tal Determinations

1 . Academie des Scien ces , 1738 .

2. B en zen b erg ,1811 (mean ) .

3 . Goldingham ,1821

4 . Bu reau d es Lon gitu des , 1822M0 11 an d Van Be ekStamp fer an d MyrbachBravais an d Martin , 1844

WertheimSton e

,1871

Le Roux

Regnau ltwp

p

qs

1

1

It has, however, been poin ted ou t by Airy ‘ that there is aphysiolog ical circumstan ce , the effe cts of which have hithe rtoe scaped n otice , b u t which probably produ ce s a sen sible e rrorit i s that two differen t sen se s, sight an d hearin g , are employed

On Sound and Atmospheric Vibrations, with the Mathematical Elemen ts 0

of Sou nd-Velocity.

Metres at00 cen t .

332

333

330 6

3322

3324

3324

331 6

332 4

330 7

50 ON SOUND. [CHAPJ

same, the v ol ume of a mass of air is in ve rse ly as the pressu re

it su pports ,”

show s that this is tru e of all gase s withince rtain lim its . Hen ce the v e locity

,on high mou n tain s , or

e ve n at the bottom of min es,doe s n ot vary if the temperatu re

is con stan t. With an in crease of tempe ratu re sou n d trav elsfaste r

,with a de crease

,slower. The rate of tran smission is

in cre ased ab ou t two fe et for each degre e Cen tig rade , or 1 1 4f e et f or each degree Fahren he it . At the tempe ratu re of 60

degre e s Fahren he it we may re ckon the v e locity o f sou n d at

ab ou t fe et per secon d , or 125m ile s per m in u te .

Velocity in

Carb on ic Acid 262metres p er secon d.

Oxygen 3 17

Air 33 1

Hydrogen 1270

By thi s mean s the distan ce of a son orou s body may b e

rou ghly m easu red , the ve locity be in g ab ou t a m ile in 4?se con ds at m edium tempe ratu re s.

The depth of the well in Carisbrook Castle has thu s beende termin ed

,by droppin g in a ston e

,an d w atchin g for the

sou n d in the wate r,allow ing of cou rse a corre ction f or the

time of fall . The clock of the Hou se s of Parliam en t strike sthe first blow of the hou r within a se con d of Gre e nw ich tim e

b u t five or six se con d s hav e to b e al low ed f or tran smissmn of

the sou n d to even moderately distan t station s .V e l o c ity in L iqu id s

— The ve locity w ith which sou n d ispropagated in liqu ids was admirably dem on strated by the

classical experim en ts of Co lladon an d Stu rm ,made in the

Lake of G en e va. The observers were station ed in two boatso n opposite side s of the lake . The sou n d was emitted from a

b e l l stru ck by a hamm er u n de r the wate r, an d re ce ived by a

l on g speakin g -tu be with a v ibratin g plate cove ring its larg e rorifice

,which was su n k ve rtical ly in the lake at the othe r

station , the ear of the l isten er be in g applied to its smalle r en d .

At the m omen t of strikin g the be ll some powde r was lightedb y a match fixed to the hammer

,an d the dete rm ination was

made by cou n tin g with a chron ome te r the time e lapsin gb etwe en the flash an d the sou n d. The station s we re de

termin ed to b e m etres apart ; the in terval was 9se con ds thu s g iv ing m etre s for the v e locity per seconin wate r at 8

°Cen tigrade . . It appears

,f rom su bsequ en t

experime n ts,that tempe ratu re cau se s con siderable variation

in the rate of tran smi ssion even in flu ids , thou gh far le sst han in gases ; the ve locity in the Se in e at 15

°Cen tigrade

IL] MODES OF PROPAGATION OF SOUND 51

having bee n metre s,in sea-water at 20° and at

23°

Ether at 0° cen tig . 1159 metres p er secon d.

Fresh Water at 15° con tig. 1437

SeaWater at 20° cen tig. 1453

Ditto at 23° contig . 1460

V e l o city in S o l id B o d ie s .— It is ow ing to the high e lasticity

o f solid bodie s su ch as g lass an d ste e l,that the ve locity of

sou n d-tran smission in them is so g reat, in spite of the irin creased den sity.

The simple st m ode of demon stratin g this v e locity is bym ean s of the Son omete r for lon g itu din al vibration s of w ire salready n amed .

If two Wire s b e stre tched side by side in this apparatu s , ofequ al len g th an d thickn e ss

,b u t of diffe ren tmaterial

,the n ote s

,

which hav e be en already stated to b e in depen den t of te n sion ,

w ill b e fou n d to diffe r con siderably. For in stan ce,if they

b e o f stee l an d brass,the f orme r will b e the sharpe r, ow in g

to the greate r ve locity in the m ore e lastic metal . With ironan d brass the ratio is that of 11 17 , repre sen ting an approximate ve locity of fe e t per se cond in the latter, ando f in the forme r .

Othe r m e thods o f de termin ation are g iven fu rther on .

Some of the prin cipal dete rm in ation s may he re b e summarized .

Tin 2498 metres p er secon d .

Silver 2684Platin um 2701Oak

, Walnu t 3440

Copp er 37 16Ste el and Iron 5030

Glass 5438

FirWood 5994

Table of Sou n d-P rop agation .

Occurs in all elastic b odies

I. In Solids .

1 . The most rapid of all forms ormatter.2. In free solids V You n g

’

s mod u lus of elasticity.

3 . Th e changes from heat very small.4. Solids n ot isotropic.

Compu tational determination follows from V l/ which applies tomatter gen erally.

52 ON SOUND. [CHAR

Experimental

1 . Strings , lon gitud inal vib ratio n s .

2. Ku ndt’

s exp erimen ts.

3 . Werthe im’

s exp erimen ts.

4 . Biot’

s measu remen t in cast-iron .

II. In Liqu ids .

Compu tational

In Water V 1/E 1/1033 x 931,

1)-0000457

14892 metre s p er secon d .

Experimen tal

1. Colladon and Stu rin .

2. Wertheim’

s experimen ts .

III. In Air an d Gases.

Compu tational

1 . Newton Principle . lSee b elow.

2. Laplace s correction.I

Exp erimen tal

1 . Tab le ab ove given .

2. Kun dt’

s exp erimen ts.

See b elow.

3 . B osscha’

s exp erimen ts.

W av e M o t io n — It mu st,how ev er , b e clearly u nde rstood

that the ve locity thu s spoken of doe s n ot imply the tran slationo f mate rial particle s from on e termin u s to the othe r. The reis n othin g re semblin g the flight of a rifle bu lle t betwe en a

sou rce o f sou n d and the observer’s ear. The proce ss ise ssen tially on e of wave m otion a condition in which , thoughe ach in dividu al particle passe s throu gh a very small distan cef rom its origin al pos ition of re st

,it propagate s the imparted

impu lse to its n e ighbou rs , an d each n e ighbou rin g particle tothose su cce ssiv e ly in con tact with it.It i s therefore n e ce ssary he re to advert briefly to wav e

m otion g en e rally, and to the the ory of u n du lation as appliedto sou n d.

“The theory of the tran smission of sou nd through the

air,

” says Profe ss or Airy,

1 “as we ll as throu gh othe r bodie s ,

is e spe cially fou nded u pon the con ception of the tran smissiono f wav e s

,in which the n atu re of the motion is su ch that the

movemen t o f ev e ry particle i s limited,while the law of

re lative mov emen t of n e ighbou rin g particle s is tran smittedto an u n limite d distan ce

,e ithe r withou t chang e , or with change

I Airy on Sou n d and Atmospheric Vibration s.

IL ] MODES OF PROPAGATION OF SOUND,53

f oll ow ing a defin ite law . In soun d w e have state s o f con

den sation an d state s o f rare faction , trave ll ing on con tin u allywithou t lim it in on e dire ction ; while the motion of ev e ryin dividual particle is extreme ly small

,an d is altern ate ly

b ackwards an d forwards. This is the con ception of a wav e

as depen ding on the m otion of particle s in t he sam e lin e as

that in which the wave trave ls . B u t the re are other kin ds o f

m ov emen t of particle s,which are e qu al ly in clu de d un de r the

con ception of wave . The m otion o f the particle s may b e

e n tire ly tran sve rse to the horizon tal lin e h ere it is n ot state s

o f con den sation an d rarefaction that trav e l con tin u ally in thesame dire ction

,b u t state s o f elevation and depre ssion that so

trave l. This is the kin d of wave which is re cogn ised as

applying to Polarized Light. B u t in all the se the re is on e

g en e ral character ; that a state of displacem en t trave ls o n

con tin u ally in on e direction w ithou t limit ; while the m otiono f each particle is

,or may b e , small and o f oscillatory

characte r. This is the g en e ral con ception o f a wave . Theidea appears to have been first en te rtain ed by New ton , an dwas ce rtain ly first dev e loped by him ,

for the p u rpose of ex

plain ing , what till the n was totally obscu re,the tran sm i ssion

o f sou n d through air it is w orked ou t in the third book o f

the P rin cip ia , and among the many won derfu l n ove ltie s o f

that won de rfu l w ork,it is n ot the least in te re stin g or the

least importan t. The mere con ception o f the motion o f

particle s in the way poin ted ou t is a v e ry smal l part o f

Newton ’s w ork the really import an t step is to show that theconden sation s an d rare faction s produ ced by the se m otion sw ill, by v irtu e o f the kn own prope rties of air

,produ ce su ch

m e chan ical pre ssu re s u pon ev e ry separate particle,that the

d ifferen t chan g e s of m otion which those pre ssu re s willprodu ce on each in dividual particle will b e su ch that theassum ed law s o f m ov emen t will n e ce ssarily b e main tain ed .

To re con cile his the oretical in fe ren ce for the ve locity of

sou nd w ith obse rved measu rem en ts,he su gg ested the idea

that the dim en sion s of air-particle s produ ced a se n sible effe ct.It has be en subsequ en tly discovered that the apparen t discrepan cy depen ds on chan g es of tempe ratu re deve lope d bythe u ltimate rarefaction s and conden sation s of which the

sou n d wave con sists.

Newton r origin ally compu ted the ve locity at 0°Ce n ti

g rade to b e 916 fe et per se con d . He took in to accou n t on lythe change of e lasticity resu lting from a change of den sity

,

1 Lee’

s Acoustics,Light, and Heat.

54 on SOUND. (am p .

b u t en tire ly overlooked the augmen tation of e lasticity re su ltingf rom a chan g e of tempe ratu re . Laplace was the first to showthe tru e cau se of this discrepan cy, and applied a correction to

Newton ’s in v e stigation which brin g s the theoretical in tocomplete accordan ce with the observed ve locity. This con sistsin mu ltiplying Newton ’s ve locity by the squ are root of the

ratio of the specific heat of air at con stan t pressu re ( 0 P) to

its spe cific heat at con stan t volum e Thu s,if V b e the

calcu lated ve locity, and V’the tru e ve locity, then

V'= V

The valu e of the ratio -g is Hen ce we have tru e

ve locity 916 X 10900 4.

W e rth e im ’s Exp e rim e n t s .

— If V b e the velocity of sou n din a particu lar gas , in fe e t per se con d ; 71 the wave -leng tho f. a g iven n ote in this gas, and n the v ibration n umb e r

,

then A.i s the distan ce trav e lled in Eof a se con d ; that

n

trav e l le d in a se con d V uh. n be ing con stan t f or all media,

and V varying dire ctly as A,the ve locitie s in two gase s may

b e compared b y ob serving the leng ths o f column s which g iv ethe same n ote . In column s of e qu al le n gth , the ve locitie s aredire ctly as the vibration -n umbers of the n otes emitted. There su lts in some gase s are g iven abov e .

Similar de te rmination s we re made by him by this method,

in l iqu ids an d solids . The v e locity in e the r an d alcohol was

fou n d to b e metre s per se con d, in solu tion o f chlorideof calcium metre s .

B o s s ch a ’s M e th o d .

— This method depen ds on the pre cisionw ith which the ear is able to de cide whe the r short sharpsou n ds are simu ltan e ou s or n o t . Two small e le ctro-magn e ticcou n te rs are con trolled by an in te rru pting fork , whose pe riod

146 se con d

, g iv ing syn chron ou s ticks . As o n e cou n te r isg ra

l

dually removed f rom the e ar,the two se rie s of ticks fall

asu n de r. When the distan ce 1s 34 metre s,coin ciden ce again

take s place , that be ing the d istan ce trav e lled by sou n d in Wo f a se con d .

W av e -l e n g th is the distan ce which sou n d trave ls i n anymed ium du ring the period corre sponding to the n ote sou nded .

Now the period which 1s the re ciprocal of the vibration n umber,or frequ en cy

,is lon g er the l owe r the pitch of the n ote , and

is a measu re of that pitch .

1L] MODES OF PROPAGATION OF SOUND. 55

VVav e -len gth may therefore b e taken as a m easu re of pitchif the m edium in which the sou n d trave ls b e kn own ; b u t itvarie s in passing from on e medium to an othe r. It varies alsoaccording to the temperatu re , be ing le ng the n ed when thisrise s

,ow ing to the decrease of den sity thu s produ ced.

O1i the othe r han d,the measu rem en t of wav e -le ng th is o f

importan ce in dete rm in in g the size of cavitie s u sed as

re son ators . For in stan ce,the ordin ary A of the Fren ch

n ormal diapason make s 435 double v ibration s in a. se con d .

Taking the ve locity of sou n d in air at as 1110 fee t

per se con d,the wav e - length .

15—0 2 ft. 66 02 ih . ; a

resonan ce b ox hav ing a qu arter of t lis wav e-le ng th or in .

will b e fou nd to produ ce the most complete con son an ce . Thesharp A of 456 vib ration s per secon d u sed in En g lish orche strashas a wave -leng th o f on ly two ft. 5 2 in . g iving 7 3 in . for

its re son an ce b ox.

K u n d t ’s d e t e rm in at io n o f th e V e l o city o f S o u n d i s su s

ceptib le of con side rable accu racy, and re qu ires on ly simpleapparatu s. A g lass tube abou t two metre s long and two

in che s in diamete r is closed at on e e nd by a stoppe r,the

other be ing fitted w ith a cork perf orated by a smaller rod o f

g lass or othe r elastic mate rial bearin g at its in n e r extremitya piston fitting smoothly in to the large r tube . By rubbingthe proje cting e nd of the rod it i s set in to long itudin al vib ration which is commu n icated to the air in the section of the

tube betw een the s toppe r and the piston . Some light powde rsu ch as the spore s of Lycopodium is con tain ed in the tube ;they are set in to active v ibration arranging them se lve s in smalllin ear patche s or heaps repre sen ting the n odes. The meandistan ce be twe en the se is equal to half a wav e -leng th in air.

If the rod b e g rasped at its middle by the cork , the wavelength o f the sou nd it emits 1s twice its length . As the v e locityof sou n d in any b ody 1s equal to the wave -leng th in that bodymu ltiplied by the n umbe r o f vibration s in a se con d

,if this

latte r b e the same in both case s,the v e locity of sou n d in the

rod is to that in air as the len g th of the rod is to the distan ceb etwe en the heaps. If

,fo r in stan ce

,the rod b e o f g las s , and

clamped in its middle,and the distan ce be tween the cork and

piston b e of the same length , the n umbe r o f he aps will b ef ou n d to b e 8, corre spon ding to a ve locity o f sou n d in g lass16 time s that in air. The method can also b e employed io rmeasu ring v elocity in other gase s than air by in trodu cingthem in to the tube and comparin g the distan ce s of the heapsw ith the len g ths of the v ib rating rod . By varying the

material of the rod,the ve locitie s o f sou n d in variou s e las tic

56 ON SOUND. [a r'

.

s olids may b e also obtain ed. Brass was thu s fou n d to g ivea v e locity of 10 87

,ste e l carbon ic acid

0 8,coal-gas an d hydrog en air be in g

1 °

O,or n early as the in verse squ are s of the ir re

spective den sitie s.In a dou ble apparatu s dev ised by the same experi

men te r,the sou n ding tu b e was cau sed to v ib rate

in its secon d mode by f riction applied n ear the

m iddle,an d thu s n ode s we re formed at poin ts

d istan t f rom the en ds by o n e -fou rth of the le n g thof the tu b e . At each o f the se poin ts con n e ctionwas made w ith an in depen den t wave -tu be

,pro

v ided w ith an adju stab le stoppe r,an d w ith bran ch

tu b e s an d stop-cocks su itable f or adm itting variou sgase s to b e experim en te d on . The du st-fig u re sin e ithe r tu be thu s correspon d rig orou sly to the

sam e pitch,an d therefore comparison of the in te r

vals of the ir re cu rre n ce g ive s a corre ct de te rm in ation of the v e locity of propagation for the two

gase s w ith which the tu be s are filled .

A few of the re su lts arrive d at w e re1 . Ve l ocity o f sou n d in a tu be d im in ishe s w ith

diame ter, b u t ab ov e a ce rtain diameter the chang eis n o t pe rceptib le .

2. Dimin u tion of ve locity in crease s w ith the

wav e -le ng th .

3 . The pre sen ce of powder, or rou ghen in g the

in te rior o f the tu b e,dim in i she s the v e l ocity

,

e spe cially in small tu b e s .

4 . In wide tu be s the ve locity is in depen den t o fpre ssu re

,b u t in small tu be s in crease s with it .

5. All chan g e s of v e locity are d u e to frict1on,

and to exchan ge of heat be twe en the air an d the

s ide s of the tube .

6. The v e l ocity of sou n d at 100° ag re e s exactlywith that g iven by the ory.

R e fl e ct io n an d R e fract io n o f S ou n d . Whensou n d-wav e s m eet a fixed obstacle , they are re

fle cted,ju st as occu rs in the case of light. The

two sets are propagated as if starting fromseparate sou rce s . In the case o f a flat su rfacethe refle cted u n du lation s se em to d iv e rg e f rom a

sou rce situ ated behin d it, which corre spon ds to the

v irtu al imag e o f a plan e m irror. The an g le s o f

in ciden ce an d reflection,as in the case of light, are equal.

58 ON SOUND. [0mm

Ech o e s .— The on ly importan t practical illu stration of these

physical facts occu rs in the case o f e choes. It appe arsthat the sen sation of sou n d occ upies ab ou t the te n th o fa secon d , du ring which time the sou nd-wave trav e ls ab ou t 34

me tre s . If the distan ce of the refle cting su rface exce eds hal fthis distan ce , we are able to hear separate ly the re tu rn ing vibration . If the re b e parallel refle cting su rface s at a distan ce

,

IL] MODES OF PROPAGATION OF SOUND. 59

the echo become s mu ltiple in stead o f sing le . Bu ilding s ,rocks

,clumps o f tre es

,eve n clouds can produ ce su ch

refle ction,as in the case of thu nde r.

The followin g qu otation fairly sum s u p the b est kn own of

these phen om en a.

I In an cien t an d mode rn w orks a n umbe rof mu ltiple e choe s are m en tion ed , the su rprisin g effe cts o f

which may b e qu e stion ed , b u t which are all easily explain edb y su cce ssive refle ction .

“Su ch an echo is said'

to hav e existed at the tomb ofMete lla, the wife of Crispu s. which repeated a whole v erse o f

the E n ez'

d as many asv eight times. Addison speaks of an

e cho which repeated the n oise of a pistol-shot fif ty-six time s

,

like that of Sim on etta in Italy. The e cho of Ve rdu n , formed

by two larg e tow ers ab ou t fifty-two m etre s apart , repeats the

same word twe lve or thirtee n time s . The g reat Pyram id o f

Egypt con tain s su b terran ean chambers con n e cted by lon gpassag es , in which words are repe ated ten tim e s . B arthiu s

speaks of an e cho situated n ear Cob len z , o n the borde rs o f

the Rhin e , which repeats the same syllab le seven te en time s ,with a v e ry pe cu liar e ffe ct

,the pe rson speaking be ing

scarcely heard , while the repetition s produ ced b y the echo arev e ry distin ct sou n ds . Among e choe s in Eng lan d may b e

n oted on e in Woodstock Park, which repeats seven teen

syllable s b y day , an d twen ty by n ight while in the Whispering Galle ry of St. Pau l ’s , the slighte st sou nd is an swe red fromon e side of the dome to the othe r.”R efl e ctio n fr om G as e s .

-Refle ction may als o take place fromlayers o f gase s posse ssing d iff eren t den sitie s , a fact whichhas be en stu d ied by Tyn dall . Sou n d from a high-pitchedre e d be ing con du cted throu gh a tu b e towards a se n sitiveflame serv ing as an indicator

,was cu t o ff b y the in te rposition

o f a coal-gas bu rn e r of the ordin ary “bat’ s-w ing kin d ;an d by holding the latte r at a su itable ang le , the sou n d cou ldb e refle cted from the flam e

,throu gh an othe r tube

,in su fficien t

qu an tity to excite a se con d sen sitive bu rn e r.On accou n t of the g reat diffe ren ce of de n sity refle ction i s

n early total at the bou n dary be tw e en air an d solid or l iqu idmatte r. Hen ce sou n d in air is n ot eas ily commu n icated to

wate r, and soun ds whose orig in i s u n de r wate r are heardwith diflicu lty in air.

S o u n d Sh ad ow s .—Whe n wave s of sou n d imping e u pon

an ob stacle,a portion o f the motion be in g thrown back as

an echo,the re is formed u nde r cove r of the obstacle a sort

I

l Gu illemiu’

s Forces of Natu re, p . 140 .

60 ON SOUND. [CHAR

o f sou n d-shadow . To produ ce this in anythin g like Opticalperfe ction , the d ime n sion s of the in te rv e n in g b ody mu st b econ side rab le . The stan dard is the wave -le n g th of the v ib ration , an d it requ ire s almost as extrem e con dition s to produ cerays in the case of sou n d

,as in optics to av oid produ cing

them . Still, sou n d-shadows thrown by hills or bu ildin g s, areoften tole rably complete .

‘

R e fract io n o f S o u n d b y th e A tm o s p h e r e is produ ced(1) by temperatu re , or (2) by w in d .

1 . The dev iation of son orou s rays from a re ctilin ear cou rse ,du e to hete rog en e ity of the atm osphere

,has practical in te re st.

The chang e o f pre ssu re at diffe re n t lev e ls does n ot g ive riseto ref raction

,sin ce the ve locity of sou n d is in depen den t o f

den sity ; b u t , as Reyn olds has poin te d ou t,the case is dif

feren t w ith variation s o f temperatu re as u su ally me t w ith .

The se are de te rm in ed chiefly by the rarefaction or cou de nsation which a portion of air mu st u n de rg o in its passag efrom on e leve l to an othe r. Thu s acou stical refraction depende n t on tempe ratu re has almost the '

same explan ation as thato f the optical phen omen on of m irag e . In the n ormal stateof the air a ray startin g horizon tally

,tu rn s g radu ally u p

wards,an d at a su fficien t distan ce passe s ov e r the head o f

an obse rv e r on the same lev e l as the sou rce . The sou n d ishe ard

,if at all

,b y diffraction . The obse rv e r may b e said

to b e situ ated in a sou n d-shadow,thou gh n o ob stacle may

in te rv en e .

The re fraction is in creased when the su n shin es,

an d

dim in ishe s du rin g rain fall .2. It has lon g b een kn own that sou n ds are g en erally b e tte r

heard to le eward than to w in dward o f the sou rce , b u t Profe ssorStoke s first showed that in creasin g v e loc ity of w in dmu st in te rfe re with the re ctilin ear propagation of sou n d-rays . When thew in d in crease s ov e rhead, a horizon tal ray trav e ll in g to w in dward is g radu al ly ben t u pwards ; rays trav e ll in g w ith the

w in d,on the othe r han d , are ben t downwards

,so that an

obse rve r to le eward hears b y mean s o f a ray which startsw ith a sl ight u pward in cl in ation , an d which has the advan tag eof be in g ou t of the way of obstru ction s for the g reate r partof the cou rse .

2

The re su lts of Reyn olds’

s expern n en ts w e re1 . When the re is n o w in d

,sou n d proce eding ove r a rou gh

su rface is more in ten se abov e than be low.

2. If the w in d-v e l ocity b e g re ate r above , sou n d is lifted tow in dward . an d n ot destroyed .

1 Rayle igh ,u . p . 107 .

2 Ib id,10 0 . cit.

1L] MODES OF PROPAGATION OF SOUND. 6 1'

3 .Un de r the same con dition s it is brou gh t down to le e

ward,an d its ran g e is exte n ded at the surface o f the g round .

Atmosphe ric re fraction has be en mu ch stu died w ith t e

f eren ce to f og signals at sea. Tyn dal l has m ore ov e r shownthat sou n d may b e in terce pted by alte rnate laye rs of gase s ofd iffe ren t de n sity. It is probab le that both cau ses are con

ce rn ed in the capriciou s behav iou r of the se warn in g s . LordRayle igh , m ore ove r, poin ts ou t that there is a diffe ren ce inb ehav iou r b etwe en lon g an d short sou n ds . This ag re e s w ithTyn dal l’ s ob servation that in som e state s of the we athe r a.

how itze r firin g a 3 lb . charg e comman ded a larg er ran g e thanthe whistles , trumpets , or sire n , while o n othe r days thein fe riority of the g un to the siren was demon strated in the

cleare st man n er.In fl u e n c e o f P o g .

— It has g en erally been believed, on the

au thority o f De rham ,that the in flu e n ce Of fog was pre

j udicial to the dispersion o f sou n d . Tyn dal l prove d that thisOpin ion is e rron e ou s, an d that its passag e i s fav ou red by thehomog en e ou s con dition o f the atmosphe re which accompan ie sfog . When the air is satu rated w ith moistu re

, the fall o f

temperatu re with e levation is m u ch le ss rapid than in dryair

,on accou n t of the con den sation of vapou r which ae com

pan ies expan sion . From a calcu lation of Thomson ’ s 1 it

appears that in warm fog the e ff e ct of e vaporation an d con

de n sation w ou ld b e to d imin ish the fall of temperatu re byo n e -half . The acou stical re fraction du e to temperatu re w ou ldthu s b e le ssen ed

,an d in othe r re spects the condition of the

air w ou ld b e fav ou rable to the propagation of sou n d,pro

v ided n o Obstacle we re off e red by the su spen ded particle sthemse lve s .

3 Manchester Memo irs . 1861-2.

62 ON SOUND. [CHAR

CHAPTER III.

INTENSITY,CONSONANCE

,INTERFERENCE.

In te n s ity o f S ou n d .—The wave s of rare faction an d

con den sation issu ing f rom a son orou s b ody in a homog en eou smedium

,like the “ rays ”

of light proce edin g f rom a can dle ,mu st n ot b e regarded as m oving m ere ly in a lin eardire ction ? It is tru e that both in the case Of soun d

, an d in

that of light, the commu n ication be tween the produ ce r an d

the re cipien t take s a lin ear form ; b u t the real con stitu ti onof the u n confin ed sou n d-wave is sphe rical. There be in gn othin g to impede the oscillation Of the u ltimate particle s

,

each impu lse spreads in an en larg in g an d con cen tric she ll,

the qu an tity o f matte r set in m otion au gm en ting as the

square o f the distan ce from the sou rce . The in tensity , orlou dn e ss , mu st therefore dimin ish in the same ratio . This istermed the law Of In ve rse Squ are s , an d is tru e also for light.The small Space throu gh which each particle v ibrate s backward an d forward is termed the amp litu de Of its u n du lation

,

an d the in ten sity of sou n d is proportion al to the squ are of

this amplitu de .

If the son orou s wave b e con fin ed in a tube,of cou rse

its prog re ssiv e extin ction by tran sferen ce o f m otion to

rapidly in creasing masses Of matte r doe s n ot take place,and

it may b e conv eyed for long distan ces with on ly v ery slighten fe eb lem en t. On this prin ciple are con stru cted the ordin aryspeaking-tu b e s . M. Biot, in the experimen ts by which b e

de te rm in ed the v e l ocity o f sou nd in solid bodie s,proved the

fact that sou n d tran sm itted by the air in the wate rpipes

of Paris was n ot sen sibly e n fe ebled at the distan ce of n earlya kilometre . TWO person s speaking in whispers cou ldeasily hold a conversation through these pipe s. “There is

m .] INTENSITY, CONSONANCE,INTERFERENCE. 63

on ly on e way n ot tob e heard,says M. Biot ; “

n ot to speakat all.

” 1

The re is,however, an importan t differen ce betwe en the

propagation Of sou n d in a u n iform tu b e an d in an open

Fig 28 .— Propagation of a son orou s Wave through an u nlimited medium.

space . In the forme r case,the laye rs o f air corre spon din g to

su cce ssive wave -len g ths are of e qu al mass,an d the ir move

Quoted in Gu illemin’

s Forces of Natu re. Regnau lt fou n d the report of

l

a

pis

ttlol in a p ip e of 1 1 0111. to b e au dib le at a d istan ce equ ivalen t to

me es.

64 ON SOUND. [CHAR

m en ts are precise ly alike,except in so far as they are in te r

f ered w ith by friction . Regn au lt f ou n d that in a con du it of' 108 of a m etre in diam eter

,the report of a pistol charged

w ith a gramm e of powde r ceased to b e heard at the distan ceOf m etre s. In a con du it of ‘3m . the d istan ce was 3

,810m .

In the g reat St. Michel sewe r Of 1 '

l0m . the sou n d was made,

b y su cce ssive refle ction s , to trav erse a distan ce o f

me tre s w ithou t be com in g in au dible . In an Open Space,each

su cce ssive layer has to impart its own en e rgy to a larg e rlaye r hen ce the re is con tin u al dim in u tion Of amplitu de in thev ib ration s as the d istan ce from the sou rce in crease s . An u n

du latio n involv e s the onward tran sfe ren ce of“

e n ergy ; an d

the amou n t of e n e rgy w hich trav e rses,in u n it tim e

,any

closed su rface de scribed abou t the sou rce , m u st b e equ al tothat which the sou rce em its in u n it time . The in ten sitytherefore follow s the same law as that o f radian t heat, an d

o f light, as state d above . The e n e rgy of a particle ex

e cu tin g s imple v ibration s in obedie n ce to e lasticity,has be en

said to vary as the squ are of the amplitu de of its v ib ration s fo r

the amplitu de bein g redou bled , the distan ce w orked through ,and the m ean w orkin g force are both dou b led

,so that the

w ork don e is qu adru pled . At the extrem e position s all is

poten tial en e rgy ; in the m iddle all is kin e tic en e rgy ; atin te rm ediate poin ts it is partly in on e form an d partly in theothe r. If we sum up the poten tial and als o the kin e tice n e rg ie s of all the particle s con stitu ting a wave

,we shal l find

the re su lts to b e e qu al . 1This assumption is n ot absolu te ly tru e ; sin ce vibration

implie s friction , an d friction implie s the g ene ration of heat .

So n orou s en ergy there fore d im in ishe s more rapidly thanac cordin g to the law of inv erse square s, an d

,in be coming

extin ct,is con verte d in to heat .

Maye r has dev ised a plan by which the in ten sitie s of two

sou nds of the sam e pitch may b e dire ctly compared. Thetwo sou n ds are separated by an imperv iou s diaphragm ,

and

in fron t o f e ach is a re son ator accu rate ly tu n ed to them .

Each re son ator is attache d by caou tchou c tu be s Of e qu alle ng th to a U-tu be , in the m iddle of which is a bran ch leadingto a man ome tric capsu le .

If the re son ators are at the sam e distan ce from the sou n din g bodie s , an d on e b e excited, the attache d flame v ibrate s .If both are produ ced in the same phase an d in ten sity theyin te rfe re comple te ly in the tu be

,an d the flame is station ary.

l Everett’

s Deschan el , p . 799.

66 ON SOUND. [CHAR

the sou rce of sou n d,its e ffe ct is immediate ly tran smitted to

the particle s in con tact, an d with an amou n t of f orce whichat first se em s disproportion ate to its in he ren t en ergy. Foralthou gh t he third law of Newton re spe cting the equ ality of

action an d reaction mu st Obviou sly b e fu lfilled , the e lasticityo f most bodie s en able s them to take u p tran sm itted vibrationin a ve ry high deg re e . Those which posse ss this prope rty inthe m ost marked man n e r are calle d son orou s , an d the ir re

spon sive vibration is te rm ed con son an ce . Withou t con son an cethe effect of mu sical sou nd wou ld b e slightly, if at all

,

appre ciable , for it is by this mean s that its chief propagationan d dispe rsion i s e ffe cted. In the first ran k as con son ators

stan d the produ ce rs them se lves. A tu n in g -fork is set

in to sympathetic vib ration by an other vib rating in u n isonwith it. A string will pe rform the same Office

,an d an organ

pipe in stan tly re inf orce s the sou n d of a correspon din gtu n ing -fork he ld n ear its open extrem ity. Even a jar orbottle

,the cavity Of which bears some defin ite ratio to the

wave -lengths Of the sou n ding body,an sw ers a similar

pu rpose .

The we ight an d den sity Of the con son an t body do n ot

n e ce ssarily preven t its actin g as a propagator Of sou n d if itsmodu lu s of e lasticity b e high . Lead or clay for in stan cedeaden sou n d by the ir in e rtn e ss, while ste e l and g lass convey

29.—M. Helmholtz

'

s resonan ce glob e .

it with the u tmost facility. B u t bodie s Of lighte r characte rand le ss den se molecu lar con stru ction , su ch as the softe rw oods , are obviou sly the fitte st for this fu n ction . It i s to the

UL] INTENSITY,CONSONANCE, INTERFERENCE. 67

highly re son an t stru ctu re of pin e -wood that the predom in an tton e of the violin fam ily e spe cially is du e .

Gen erally speaking , re in forcem en t in soun d is corre lativewith the powe r Of produ cing it. All sou n ding bodie s re in force ,b u t some hav e be en div ided Off in to what Cle rk-Maxw e llte rm s distrib u tors . Othe rs have the powe r of sin g lin g ou t

particu lar sou nds f or re in forcemen t. If , for example , thedampers b e lifted Off a pian o an d the voice b e u sed in itsn e ighbou rhood , it will b e heard to sin g ou t lou dly with a

humm in g ton e the n ote s which hav e be en spoken on . Thesame effe ct occu rs with drums an d tu n in g -f orks : even the

flat crown Of a hat re spon ds by v ibration sen sible to the

tou ch when lou d n oise s occu r in proximity to it.This powe r of sing ling ou t sou n ds has b e en u tilized by

He lmholtz f or the an alysis Of mu sical n ote s in makin gR esonators.

1 They origin ally had external membran es,b u t he

afte rwards fou n d that the tympan ic m embran e , or drum Of

the ear itse lf , cou ld b e u sed f or the sam e pu rpose,b y makin g

the re sonan t cavity Of a particu lar size,su ch that

,itse lf

speaking a ce rtain n ote,it will sing le ou t that n ote from all

o the rs, an d re in force it vig orou sly.

The simple st m ethod,however

,of demon strating re son an ce

is to take a tall jar or tube and hold over it a sou n din g body.

su ch as a tu n ing -fork . As long as the f ork an d the cavity of

the jar are in n o defin ite relation to on e an othe r,the sou n d is

u n altered . B u t if,by g radu ally pou ring in wate r, we alte r

the depth o f the cavity,a poin t is su dden ly reached at which

the n ote starts ou t with exce eding clearn e ss . It will then b ef ou n d that the leng th of the column of air in the tu be be arsan exact proportion to the wave-len gth of the v ibration sem itted by the f ork

,u su ally that of on e to fou r. The reason

Of this is obviou s. At each vibration of th e f ork , a wav e of

con den sation trave ls down the tube,i s reflected from the

bottom ,an d retu rn s to fin d it in the same phase as when it

started . It thu s su perposes its own motion u pon that of thefork

,and by a su ccession of su ch action s re in force s the sou n d.

In so doing , howeve r, the fork has the addition al labou rimposed u pon it of setting in motion the con tain ed particle sOf air as we ll as its own

,and therefore come s soon e r to re st

than when vib ratin g indepen den tly. An effe ctive expe rimen tis produ ced by combin ing a son orou s be ll with a re son an tcavity Of variable dimen sion s. A sou rce Of sou n d mayalso act u pon a tu n in g -fork by con son an ce . If two forks in

1 See Chapter V .

68 ON SOUND. [CHAR

accu rate u n ison b e placed at some distan ce from on e an othe r,

an d on e b e excited , the othe r imm ediate ly beg in s to sou n dw ith vig ou r, an d if the first b e damped

,it may b e again set

in motion by the con tin u an ce of de riv ed vibration e stablishedin the se con d. Tu n in g -forks and othe r son orou s bodie s

,su ch

as g lass or metal vase s , ofte n commen ce sou n din g spontan e ou sly when a mu sical in strume n t

,an organ or harmon ium ,

is played on in the sam e room even the g lass win dow s o f a

chu rch are apt to take u p the n ote of a particu lar pedal pipeto the exclu sion Of those in its immediate n e ighbou rhood .

He lmholtz has shown that a stretched string may b e madeto pe rform the Office Of a re son ator. 1 If a sou nding tun in gf ork have its stem placed o n a strin g , and it b e m ove d so n earthe bridg e that on e Of the proper ton e s of the se ction of

string lying between the fork an d the bridge is the same as

that of the tu n in g -fork , the string beg in s to vibrate strongly ,an d con du cts the ton e of the tu n ing -fork with g reat powe r tothe sou n ding -board an d su rrou n din g air whereas the ton e isscarce ly if at all heard as lon g as the se ction is n ot in u n isonwith the ton e of the fork .

A simple apparatu s was u sed by Savart to show the

in flu en ce Of jars or boxe s in stren gthen in g sou n d . Close toa sou rce of sou n d , su ch as a be ll or tu n ing-fork , was placeda hollow cylin der

,closed at its farthe r en d by a mov eable

bottom,by m ean s o f which its capacity cou ld b e in creased

or dimin ished . This was supported on a slidin g re st, so thatits open en d cou ld b e brou ght n ear or remov ed f rom the

v ibratin g body. The hell or f ork be ing excited by a rosin edb ow

,the cav ity o f the re son ator was alte red u n til it coin

cided in pitch w ith it, an d immediate ly the sou n d, Orig in ally

fe eble , an d all b u t in au dible , b e came distin ct an d lou d . Thelou dn e ss cou ld then b e varied to any exten t by moving the

open e nd of the con sonating cav ity in to closer proxim ity tothe sou rce of sou n d . He lmholtz has u til ized the latterphen om en on in his syn thetical reprodu ction o f compou ndvowe l-qu alitie s . Koen ig has improv ed on the orig in al form of

re sonator by in trodu cin g a slide su ch as that n amed ab ove ,by which the same in strumen t may b e made to re in forceseveral n ote s .

Th e o ry o f R e s on at o rs .- In a pipe cl osed at on e end

,we

have a mass of air vibratin g in certain defin ite pe riods pe cu liarto itse lf , in more or le ss comple te in depe n de n ce o f the

extern al atmosphere . If the air b eyon d the open en d we re

I Helmh oltz, Sensation s of Ton e, Ellis’

s tran slation, p 88.

UL]'

INTENSITY,CONSONANCE,

INTERFERENCE. 60

en tire ly w ithou t mass, . the m otion within the pipe wou ld hav e

n o ten den cy to e scape , b u t in actu al expe rim en t the in e rtiaof the exte rn al air can n ot b e g o t rid Of . When the d iamete rOf the pipe is small

,the e ff e ct of this is small , an d vibration s

on ce excited hav e a ce rtain degree of persisten ce . Then arrowe r the chan n e l of commu n ication betwe en the in te riorcav ity an d ou tside , the greate r doe s the indepen de n ce be come .

In the pre sen ce o f an exte rn al sou rce of sou n d , the con

tain ed air v ibrate s in u n iso n , w ith an amplitu de depen den ton the re lative mag n itu de s of the n atu ral an d con strain edperiods , risin g to g reat in ten sity in the case of approximate

isochron ism . When the orig inal cau se of sou n d cease s, there son ator yields back the v ib ration s stored up within it, thu sb e com ing itse lf for a short tim e a se con dary sou rce . A ve sse lcon tain ing air, o f which the capacity is su fficie n tly large , commu n icatin g w ith the extern al atmosphere by a n arrow n e ckrepre sen ts a state of thin g s in which . the kin etic e n e rgy mayb e n eg le cted except in the

.

n e ighbou rhood of the ape rtu re ;the air m ovin g approximate ly as an in compre ssible flu idwo u ld do u n de r like circumstan ce s

,su fficien tly so for a

calcu lation Of the pitch .

‘

S o u n d -b o ards g e n e ral ly as C o n s o n at o rs .—It is obviou s

that the vib ration s of strin g s , tu n in g -f orks,reeds

,an d othe r

g en erators o f sou n d,can n ot

,u n aided

,impre ss any larg e

amou n t of m otion on the su rrou n din g air con sequ en tly theton e s they produ ce alon e are v ery fe eb le . It has been shownhow the se may b e in creased for in div idu al n ote s . The re ishoweve r a larg e class of applian ce s with which m ost mu sicalin strume nts are fu rn ished , te rmed sou n d-boards . The se don ot so mu ch te n d to in ten sify particular rate s Of vibration

,as

to impart son ority to all. The ir f un ction has be en we l l explain ed by Stoke s whateve r the ir ab solu te size

,they pre sen t

a far larg e r area to the air than the string or fork itse lf . B ythu s laying hold , as it w e re

, Of the rarer body, they prev e n tthe dissipation Of the alte rn ate rarefaction s and con den sation sof which a sou nd-wave con sists . Sou n d-boards u su ally take thef orm of large su rface s or boxes of some e lastic mate rial

,su ch as

pin e -w ood . The Olde r pian os had a flat she e t Of varn i sheddeal laid immediate ly ab ov e , b u t n ot tou chin g , the strin gs .

The most remarkab le example , howev e r, occu rs in the body ofall in strumen ts o f the v iol or v iolin fam ily. We hav e here a

flatten ed box Of a ve ry complicated shape , the fron t or be llyof which is made of fin e e ven -

g rain ed pin e -w ood ; the back

I Rayleigh , On Sou nd , II. 156.

70 ON SOUND. [earn

u sual ly of a harder and m ore compact material . The se twoare u n ited by a b ar o f sim ilar wood termed a sou n d-post

, an dthe f orme r is pierced by two Open ing s , u sually in the shape ofthe lette rf . Upon the be lly stan d the fe et Of the bridg e , andclose to on e Of the se fe et is the sou n d-post. No be tter combin ation cou ld have been adopted for se cu rin g the larg e stpossib le amou n t of reson an ce . The wood itse lf has been shownto b e the m ost rapid and pe rfect of all sou n d-tran smitte rs itss urface i s larg e proportionally to the siz e of the in strumen t ;the two face s are rig idly con n ected b y a con du ctor

, an d a

con side rable mass of air i s in closed,commu n icating by the

two orifice s above n am ed w ith the su rrou n ding air. This ispe rhaps the most pe rfe ct in stan ce of the fact n am ed in thein trodu ctory remarks

,of a machin e which has grown u p for

tu itou sly , the re su lt en tire ly of arti stic experien ce,b u t which

prove s to con form in the m ost perfect way to the orie s su b sequ en tly deve loped .

In t e rf e r en ce o f S o u n d may b e looked u pon as the log icalcorre lativ e and con trary Of con son an ce . It has alreadyb e e n shown how two wave s of similar pe riod an d phasere in force on e an othe r and it n atu rally follows that if they b eo f u n equ al pe riod or Opposite phase they mu st b e m u tu allydestru ctiv e . If the phase al on e diffe r, the period of the tw o

be ing equ al , the in te rfe ren ce is con stan t ; if , however, theperiods are slightly u n equal, alte rn ate re in forcemen t and in terf e re n ce takes place

, g iv in g rise to the phen omen on of Beats .

As in the case of con son an ce , this action can b e followedu p from the me chan ical departmen t in which pen du lums

,

and even wave s in wate r are type s , throu gh so u n d,in to

the far subtle r an d more rapid u ndu lation s which characte rizelight. If two wav e s pass simu ltan eou sly through the sam e

m edium, the actu al m otion of each particle is the resu lt

of the two comb in edf This w ill b e the sum o f them if

they b e in the same dire ction,the diffe ren ce , if in Opposite

dire ction s , an d n othing if they b e equal and Opposite . Thisstatem en t is tru e whe ther the u n du lation s b e on the su rfaceo f wate r

,in air

,or in the a the r which is believed to tran sm it

light. Redu cing it to the te rm in ology of sou n d,two u n du

lation s of equ al wav e -len gth an d amplitu de mu st e ithe r b e inthe same phase or always diffe r by the same amou n t if the

sam e,the re su lt is a doubling of the sou n d , and i s a case o f

con son an ce : if Opposite , that is if on e b e exactly half a

wave -len gth in advan ce of the othe r, the two motion s will b eopposite

,an d silen ce w il l resu lt from the con j oin t action o f

two sou n ds. This is the case of complete in terfe ren ce ; a

INTENSITY, CONSONANCE, INTERFERENCE. 7 1

condition which can , howeve r, b e on ly partially realized inpractice . The in te rfe ren ce s can al so b e produ ced betwe en a

d irect an d a refle cted wave , and were carefu lly expe rimen tedon by Savart and Se ebeck , u sing a distan t wall for the

reflector.

Fig . 30.— B eats of imperfect un ison . Ordinates of ten waves .

Fig. 31. -Ordin ates of eleven waves tran smitted in the same time .

The simple st illu stration Of in te rferen ce is afforded b y an

o rdin ary tun ing-fork. If it b e set in vibration an d he ld tothe ear, its n ote will as a ru le b e easily heard ; b u t if it b eslowly rotated on its stem

,it will b e fou n d that the re are

fou r position s in each rotation in which the n ote i s distin ctlyau dible , an d fou r othe rs in which it disappears altog ethe r. Itis heard when eve r its two pron g s are he ld paralle l to the ear,and also when they stand at rig ht ang le s to that plan e . Betwe enthe se are fou r obliqu e situation s almost at an an g le of 45

°

w ith those previou sly n amed,which are position s Of silen ce .

The sou n d of a tu n in g-fork is at all time s mu ch dim in i shedby the Opposite motion of the two prong s , which tends tocan ce l the v ibration Of the su rrou ndin g air. This may b e

shown by passing a smal l tube ove r on e pron g , withou ttou ching it, an d thu s shie ldin g the othe r from its an tag on isticinflu en ce the sou n d immediately b ecome s materially stronge r.When on the other han d , the fork is he ld e dg ewise to the ear

,

the prong n ext it, be in g in the more favou rable position Of

the two , exe rcise s a predom inan t in flu en ce,an d enable s the

sou n d to b e heard. B u t when the Obliqu e position i s Obtain ed,

a more complete an tag on ism re su lts and the n ote i s en tire lyextingu ished. This expe rimen t may b e made more distin ctb y rotatin g the f ork ove r the mou th Of a con sonating jar ortube .

If two pe rfe ctly similar stopped organ -pipe s b e set on the

same win d-che st,an d e ithe r b e sing ly excited , a n ote is

produ ced in the ordin ary way. B u t if the two are blown

7 2 ON SOUND. [CHAR

togethe r, in te rferen ce take s place betwe en them ,and n othing

is heard beyon d the ru shing Of the win d .

“

If a circu lar disc b e excited by mean s Of a bow,so

as to produ ce six v ibrating se ctors,an d the palm of the han d

b e brought ov e r on e of the se , the sou n d is at on ce in ten sified. This doe s n ot occu r if the two han ds are placedove r two con se cu tive portion s

, b u t i s more marked whenalte rn ate se ctors are so treated . The han ds

,acting as

dampers , che ck the spread of the vibration s from those partsof the plate immediate ly ben eath them

,an d all ow the oppo

site m otion o f the in terv en in g plate to deve lop , comparative lyfree f rom in terferen ce

,in to sou n d . The experimen t is even

more striking if , in stead Of the han d , a pie ce Of pasteboardwith three alte rn ate se ctors cu t ou t o f it b e employed as the

dampe r. It has be en also varied by fixin g a tu be bifu rcatedat its lowe r extrem ity

,an d fu rn ished at its u pper with a

vibrating membran e ove r the plate . San d is scatte red on the

membran e when the two orifice s of the bifu rcation are he ldove r adjacen t sectors

,little or n o motion Of the m embran e is

seen,b u t when ove r alte rn ate se ctors

,the san d is immediate ly

thrown “Off ; in the first case there is in te rferen ce

,in the

se con d coin ciden ce of v ibration .

Pe rhaps the m ost beau tifu l illu stration is on e proposedorig inally by Sir John He rsche l . A brass tu be d iv ide s in totwo b ran che s , which reu n ite fu rthe r on in the ir cou rse . On e

bran ch can b e drawn o u t by m ean s Of a slide to a greate rlength , so that the wave s from a sou n din g body can b e made

to trave l ov e r difieren t len gths of tube . If a vib ratin gtu n ing -fork b e brou ght n ear to on e orifice , an d the ear appliedto the othe r, as lon g as the two tube s are Of the same len g ththe n ote i s clearly heard . B u t on draw ing o ut the slide , apoin t is reached at which the sou n d van ishes . This take splace when the extended bran ch is half a wav e -leng th long e rthan the othe r ; indeed it affords a rough method of measu r

ing wave -len gths.B e ats .

— The most u su al form in which in terfe ren ce i s met

with is that of beats . The se gen e rally orig inate in the

simu ltan e ou s sou n ding of two n ote s n ot qu ite in u n ison .

They will b e shown later to b e ou r most subtle an d tru stworthy method Of se cu ring an d te stin g u n ison or perfe ctcon cords. They are easily explain ed on the prin ciple o f

in te rfe ren ce . The wav e -lengths of tw o n ote s b e in g slightlydifferen t while the ve locity o f propagation is the same

,

the phase w ill alte rn ate ly ag ree and disagre e in the ir cou rse .

When the phase s Of the two wav e s are coin ciden t they

74 ON SOUND. [CHAR

lou dn e ss is elicited . B u t as the clockwork lifts an d drops itsw e ight at regu lar periods, before the first v ibration is ex

tingu ished , two con dition s may occu r . e ithe r the hamme rmaymee t the be ll in the same phase as its own , in which case an

extreme ly lou d toll resu lts,or it may fall on it in the Opposite

phase , when g reat part of the momen tum is employed incan ce llin g the in terfe rin g vib ration s already set u p and a

very fe eble sou n d 1s g iven ou t. The se difieren ces of in ten sityin the su cce ssive stroke s can b e plain ly heard at 11 O

’clock ormidn ight when the air is still.

1v. ] PITCH.— ITS MEASUREMENT, ETC. 75

CHAPTER IV.

PITCH.-ITS MEASUREMENT, LIMITS, VARIATION, STANDARDS, AND

TONOMETRY.

IT has been already stated that all sound con sists of threee lemen ts , n ame ly, in t e n s ity , p itch , an d qu al ity . The first ofthe se depen ds en tire ly on the amplitu de o f the vibration sthe last has been shown by He lmholtz to b e con n e cted withce rtain secon dary and affiliated oscillation s termed harmon ics ,from which few mu sical ton e s are en tire ly f re e . This will b eadve rted to in a su bsequ en t chapte r. The remain ing con

stitu en t , n am e ly,pitch

,has been the subje ct of mu ch importan t

re search . It depen ds en tire ly on the n umbe r of vibration sin a g iv en time .

Lim its o f A u d ib l e S o u n d .— Savart showed that the facu lty

of pe rce ivin g sou nds depen ds rathe r on the ir in ten sity than on

the ir acu te n es s , an d by in creasing the diame te r of his toothedwhee l, carried it u p to vibration s per se cond.

For deep sou n ds b e su bstitu ted for the toothed whee l a b arabou t two fee t lon g , rev olv ing on an axis between two thinw ooden plate s abou t 0 0 8 Of an in ch distan t f rom it. A grave ,con tin u ou s , very deafen ing sou n d was thu s produ ced

,w ith

7 to 8 vibration s in a se cond . The se re su lts are dispu ted byDe spre tz and He lmholtz , the f orme r placing the limit at 16,the latte r at 30 vibration s , the defin ite mu sical characte raccord ing to the latter Obse rve r be ing on ly Obtain ed at 40

v ibration s per se con d. The se discordan t re su lts are n o dou b tdu e to the difie ren t capacitie s of difie ren t Obse rvers for theperce ption of sou n d , in de ed the extreme u ppe r lim it ofau dib le sou nd appears to vary mate rially with the in dividu al.M. Despretz had a diaton ic octave of smal l forks fromto vib ration s in the secon d, tu n ed by M. Marloye , who

declared that with practice he cou ld tu n e still highe r scales .

7 6 ON SOUND. [CHAR

He did tu n e an octave f ork to that last n amed,which wou ld

g ive vibration s. He state s that in the proce ss o ftu n in g he wen t twe lve time s over the whole octav e . On thetw o first attempts he heard n othin g . At the third attempt hewas able to distingu ish the in te rvals in the follow ing orde r :Fou rth , Fifth , Min or Sixth , Min or Third , Maj or Sixth

, MajorThird , Min or Sev en th , Major Seven th

,an d at last with g reat

difficulty , the Maj or Se con d . Upon this he make s the su gg estiv e remark that if there b e a n atu ral scale for the ear

,

the se obse rvation s wou ld poin t to its be ing min or rathe r thanmajor.Appu n n has made thirty-on e tu n in g -forks

,in tru e maj or

scale s from vibration s u p to which m ost earscan distin guish, although they Often produ ce a v ery pain fu lsen sation .

Captain Dou g las Galton has shown a method of produ cin geven highe r ton e s

,b y mean s of small whistle s. Man y of

the se n ote s appear to b e more au dibleto the smaller mammalia,

e spe cially to cats , than they are to the human e ar.

Preyer has made expe rim en ts of con side rable pre cision bymean s of which he fixe s the min imum limit f or the av erag ee ar be tween 16 an d 24 sing le vib ration s p er se con d ,-themaximum at many person s of fair hearin g powersbe ing howeve r deaf to sou n d of or e ven fewe rv ibration s .

It appears,from experimen ts made by the write r

,that the

mu sical characte r Of l ow ton e s depen ds mate rially u pon thepre sen ce or absen ce of a su fficien t con sonan t body for the irre in forcemen t an d co -ordin ation . Sixteen -foot C can b e oh

tain ed on the dou ble bass,w ith distin ct mu sical character

,by

spe cial treatm en t of the re son an t body of the in strumen t.

The size Of the room ,m oreover, mu st b e con siderable for the

larg e wave s thu s orig inated to spread withou t damping an d

in te rfe ren ce . The 32-foot octav e of pedals in the organ at

the Albe rt Hall is pe rf e ctly mu sical in effe ct. Probab ly theexperimen ts o f Helmholtz

,which w ere made w ith fe eble

sou rce s of sou n d su ch as me tal string s , w e re deficien t in therequ isite s ju st n amed . The l owe st and gran de st n ote to b eheard occu rs when a train passe s in to a short tu n n e l . Thesu cce ssive explosion s of issu in g s team

,varying in rapidity

f rom 8 to 20 in a se con d , can,if the spe ed b e gradu ally

in creasing , b e clearly heard to coale sce in to a prof ou n dhumm in g n ote of steadily rising pitch du e to the con son an ceOf the g igan tic re son ator fu rn ished by the bore of the tu n n e litse lf .

iv ] PITCH.

— ITS MEASUREMENT, ETC . 77

The dete rm in ation Of the n umbe r Of v ib ration s in a g ivenpe riod corre spon ding to a particu lar mu sical n ote may b e made

b y the variou s methods of TONOMETRY, as this bran ch of

acou stic s has b e e n te rm ed . The se may b e g iv en , in tabu larform

,as follow s

I. Mechan ical me thods .

II. Optical methods .

III. Photog raphicm e thods .

Prof . Blake ’s expe rimen ts .

IV . Ele ctrical methods .

V . Compu tative methods .

I. M e ch an ical M e th o d s .

—The simple st me chan ical me thodin the ab ove list Of con trivan ce s is f ou n ded on the fact thatslight su cce ssive n oise s cau sed by collision , which in div iduallypre sen t n o mu sical character whateve r

, g radu ally coalesce in toa defin ite ton e if the ir in te rvals b e regu lar an d if the ir su ccession b e su fficien tly rapid . A comm on watchman

’s rattle

,an d

ev en a stick passed rapidly across the bars of a g ratin g , mayb e u sed as a popu lar illu stration of the fact ; Dr. Hau ghtonhas fu rn ished an exce llen t example

,n amed in the in trodu ctary

remarks .

“The gran ite pavemen t of Lon don is fou r in che s in width,

and cabs driv ing ove r this at e ight miles an hou r,cau se a

su cce ssion of n oises at the rate of thirty-five in the secon d,

Savart’s toothed whe e l .Cagn iard de Latou r

’s siren .

Perron etThompson ’s we ightedmon ochord .

Duhame l’s vibroscope .

Le on Scott’s phon au tog raph .

Edison ’ s phon ograph .

1. Lissaj ou s’ method .

He lmholtz’s v ib ration m icroscope .

Koen ig’s man om etric flam e s .

McLe od an d Clarke ’s cycloscope .

Meye r’s e le ctrical ton omete r.Lord Rayleigh

’s pen du lum

expe rimen t.

Chladn i’ s rod ton omete r.Sche ible r’s . Tonmesser w ithtu n in g -forks .

Appu n n’

s ton ome te r with f re ere eds .

78 ON SOUND. [CHAR

which corre sponds to a we ll kn own mu sical n ote,that has

been re cogn ised by many competen t obse rve rs ; an d ye t

n othing can b e imagin ed m ore pu re ly a noise,or le ss mu sical

,

than the j olt of the rim of a cab -whee l again st a projectingston e ; yet if a regu larly repeated su ccession of jolts takeplace

,the re su lt 18 a soft, de ep , mu sical sou nd

,that will well

iv. ] PITCH.

— ITS MEASUREMENT, ETC. 79

b ear comparison with n ote s de rived from more sen time ntalsou rce s.” 1

( 1) To o th e d W h e e ls .-Even as e arly as the time of Galile o ,

that philosophe r produ ced a mu sical sou n d by the passage of

a kn ife ove r the se rrated edg e o f a piastre an d in fe rre d fromthis that the pitch depen de d on the rapidity o f the impu lse s .

“On Ju ly 27 , 1681, Mr. Hooke showed an expe rimen t o f

making mu sical and othe r sou n ds by the he lp of tee th of

brass whe e ls , which teeth we re made o f e qu al bign e ss formu sical sou n ds

,b u t of u n equal for vocal sounds .

” 2

Savart first redu ced the system to accu racy by mou n tingthe toothed or se rrated whee l on an axis con n e cted withmachin e ry competen t to g ive it rapid rotation , an d attachin gto it a cou n te r or in dicator showing the n umbe r of rev olu

tion s in a g iven pe riod . Thu s, g iving the whe e l 60 0te e th , an d rotating it forty time s in a se con d , he cou ldobtain collision s in each se cond

,which corre spon d to

an extreme ly high n ote . A pie ce of card or a metall ic platewas applied to the passing te eth , which of cou rse re ce ived ina se con d as many blows as the produ ct of the tee th in to therotation s .(2) Th e S iren ,

inven ted by Cagn iard-Latou r in 1819, su bstitu te s for the su cce ssiv e collision s a se rie s of small pu ff s o f

air. In its simple st form,as made by Seebeck , it con sists o f

a rotating disc , perforated w ith circu lar ring s of orifice s , toe ach or any of which can b e adapted a n ozzle deliveringwin d at high-pre ssu re from powe rfu l be llows. In this f ormhoweve r the sou n d is fe eble . The Siren , in spite of the

qu ain t in accu racy of its n ame,3 was a con s ide rable advan ce

u pon Savart’s whe e l . The te e th of the latte r are here replace dby coin ciden t open in g s in two sim ilar circu lar plate s , the on e

fixed, the othe r rotatin g above it,w ith b u t slight friction ,

u pon an axis. In the act of rotation sim ilar su pe rposed ring sof hole s alte rn ate ly open and close a passag e for the w in dissu in g in a steady stream throu gh the lowe r fixed plate .

The isolated “pu ff s ” soon u n ite in to a con tin u ou s n ote .

In Cagn iard-Latou r’

s orig in al in strum en t,a more compli

cated arran gemen t exists. The rotatin g disc tu rn s on a

v e rtical axis above a re ceptacle , the u ppe r su rface o f which,

1n close approximation to the u n de r su rface o f the disc,is

pierced , n ot with a sing le hole , b u twith a rin g of hole s equ al

1 Natu ral Philosophy Popu larly Exp lain ed. p . 157.

2 B irch ’

s Hwtory of the Roual Society. Qu oted in Tyn dall On Sou nd .

3.

It is said to have derived this name from its power of singing u nder water,a gift With Wll lCh Homer’

s za pfiveg were n ot endowed .

80 ON SOUND. [CHAR

in n umbe r an d similar in position to those of the disc abov ethem . In stead , howeve r, of the hole s in the tw o rin g s b emg .

pie rced vertically both are in clin ed ob lique ly , the lowe r row

Fig . Seeb eck'

s sire n .

in on e dire ction,the u ppe r in an Opposite sen se . The ob je ct

of this is to fu rn ish m otive powe r to the rotating disc fromthe horizon tal e lemen t of the Win d-pressu re . It wou ld havebeen bette r to om it this device , an d rotate the disc by exte rn alf orce , as w ill b e p re sen tly de scribed . The u pper en d o f the

Spin dle carryin g the disc is fu rn ished with an en dle ss screw,

which works in to a small registe ring train of whe e ls . The secan b e thrown in an d ou t of ge ar at will so that the rotation s in any g iven n umb e r of secon ds can b e approximatelyin dicated on dials ou tside . The in dication is on ly approxi

mate , sin ce the added f riction of the train,howeve r small

,

82 '

ON SOUND.

Fig. 37 .—Helmholtz

’

s dou b le siren .

[CHAP .

xv.] PITCH.


The siren , althou gh the ore tically a pe rfect in strument , failssom ewhat in practice , chiefly in con sequ en ce of the dlfiicu ltyexpe rien ced in keepin g its n ote steady. The characte r of then ote itse lf i s harsh and scream ing , so that beats with softe rsou rce s of sou n d are all b u t inaudible . As it is u sually mademore ove r, the re is n o way Of preven tin g a steady acce lerationof the rotation , or the corre spon ding rise in pitch . The blastof wind be ing made to accomplish two pu rpose s

,as a drivin g

pow e r as w e ll as a sou rce of soun d,can n ot b e mate rial ly

alte red w ithou t at on ce redu cing the impe llin g force and the

ton e . Mr. Ellis n ote s that “as each revolu tion of the dis c

re ckon s as twe lve vibration s , an error Of on e revolu tion in a

se con d , which is easily made,v itiate s the re su lts by twe lve

vibration s or '4 of a sem iton e at the pitch of O, which is a

large amou n t. Practically a siren can n ot b e depen ded on

within ten vibration s.”He lmholtz

,in whose han ds the siren was made to g ive v e ry

fair re su lts,employed an e le ctro-magn etic driving machin e

to _actu ate it. It is con n ected with the discs by a thindrivin g-b an d. The siren doe s n ot then requ ire to b e blown .

In stead of blowin g , he place s on the disc a small tu rbin econ stru cte d of stiff pape r, which drive s the air through theOpen in g s when eve r they coin cide w ith those in the che st.This arran g emen t

,

” he state s , “gav e me extreme ly con stan t

ton e s on the siren,rivallin g t hose on the best con stru cted

organ -pipe s .

”

Err o r o f S ir en — An other sou rce of e rror in the in dication sof the siren does n ot hitherto seem to have be en n oticed.

This is du e to the amoun t of compre ssion to which the air

i s subjected . For properly d1iving the disc at high speedsvery con siderable force is n ece ssary

, on accou n t o f its in ertiaan d friction .

,

The Win d in the che st shou ld su pport a columno f water from 12 to 24 in che s in he ight, a pressu re e qu ivalen tto f rom 5 lb . to on e pou n d per squ are in ch . In passin gthrou gh the perf oration s of the siren it is therefore alterin gmaterially in volume

,and stil l m ore perceptib ly in

’

heat . Boththe se e lem en ts exercise a powe rfu l in flu en ce on the

'

tOn e

emitted by wind in strum en ts of all kin ds,as will b e shown

in g reater detail in a later chapter,an d cann ot b e n eg le cted

in this in stan ce with impu n ity.

It s R e al V alu e .—The real practical u se of the siren i s for

demon stratin g the formation of the scale,an d the vibration

ratios wh ich distin gu ish the‘ principal con cords an d dis

son an ce s . The se remain pe rfe ct an d u n distu rbed in spite ofvariation s in the absolu te n ote u pon which they are fou n ded .

G 2

84 ON SOUND. [CHAR

( 3) De termin atio n b y th e M o n o ch or d .— On e of the earlie st

su cce ssfu l attempts at accu rate dete rmin ation of pitch was

made by Perron et Thompson . For this end he rev ised an d

perfe cted the an cien t in strum en t of Eu clid an d Pythagoras ,the m on ochord . According to his con stru ction it was fivefe e t long , ten in che s b road

,an d six de ep ; the w ire was of

Stee l the twen tieth of an in ch in dlam eter,con tain ing 145

f e e t to the~

pou n d avoirdu pois,breakin g with a we ight of

300 lbs . The load requ ired to produ ce ten or C of the pian of orte was from 240 to 250 lb s. The sou n d was brou ght ou t

by the application of a we l l-rosin ed bow,an d had the stren gth

o f a Violon ce llo . The m ethod of u sing the abov e apparatu sfor the enharm on ic tu n in g of an organ , will b e de scrib edin a later chapter. Here it will b e sufficien t to n ote the

dire ct physical method of m easuremen t w ith su ch an in stru

men t . A string is tu n ed to a g iven n ote,and its vibration s

are determin ed by kn owing the stre tchin g we ight , the w e ightof the wire as stretched

,and the vib ratin g length of the

string .

1 The following is the f ormu la u su ally adopted,as

'

ven by Mr. Ellis in his excel len t commu n ication to theociety

‘of Arts .

Let V Pitch,or n umber of double vibration s in on e

se con d .

W Number of grain s in the stre tching we ight .S Number of grain s in on e in ch length of stretched

string .

L Number of in che s in vibrating string .

Hen ce SL We ight of vibrating string ; which, cu t Off ,we ighed and measu red , g ive s L , 8L ,

an d S.

Then

P be ing the length of the se con ds pen du lum at

G reenwich , and rr the con stan t 3 1 4159.

The string is brou ght in to sen sible u n ison w ith the g ivenn ote by shorten in g or len g then in g the wire , and cu t to the

corre ct leng th. It is carefu lly m easu red for L,and we ighed

for 8L. The we ight with its attachmen ts is we ighed for W.

In this way Dr. Smith , in the year 1755, in the mon th of

September, tu n ed a

' wire to g ive a n ote two octave s be low the

D pipe of the organ in Trin ity Colleg e , Cambridg e

J ou rn al of Society of Arts, May 25, 1877 , On the Measu remen t and Settlemen t o f Mu sical P itch ,

”b y Alexan der J . Ellis, are.

86‘

ON SOUND. [cam

they may b e typified by attaching a smal l poin t or style to theprong of a tu n ing -fork,an d allow in g this to trace its move

men ts u pon a pie ce of smoked paper or g lass al lowed to travel

steadily before it. If the fork b e n ot sou nding , the poin t

wil l de scribe a straight lin e . B u t if it b e firs t se t in

yib ration , the attached point will con stan tly move backwards

m] PITCH.-ITS 'MEASUREMENT, ETC. 87

and forwards,and the wave-lin e remain s as a perman en t

image of the motion performed by the f ork du ring its mu sical

oscillation s . In practice it is b e st to wrap the paper rou n da rotating cylinde r ; rotatory movemen ts be ing , as a rule ,more easy to regu late , an d steadie r than those in a straightlin e . The cylinder may , moreove r, b e made to mov e in a

spiral,in stead of perform ing a simple axial rotation , and

thu s the l in e may b e in defin ite ly exten ded within a l im itedspace .

The cu rve obtain ed from any simple harmon ic motion is on eof those den om in ated cu rve s of Sin e s ; an d it may , as

He lmholtz remarks,b e made to reprodu ce the m otion of the

v ibrating poin ts, by cu tting a n arrbw ve rtical Slit in a pie ce of

pape r,an d placing it ove r the cu rv e -tracing : if this latte r b e

drawn u n iform ly u n de r the slit from right to left , the poin t,se en throu gh the s l it

,will appear to m ove backwards and

f orwards pre cise ly in the same mann er as the orig in al tracerattached to the fork.

This m ethod is su sceptible of very con siderable accu racyand in de ed has been employed for the exact dete rm in ation of

both its factors. Chron ographs, in which a steadily vibratingtu n ing -fork is the stan dard , tracing its o scillation s on a she e tof blacken ed pape r carried past it

,have be en con stru cted

a fin e example was exhibited at the Loan Exhibition Of

Scien tific Apparatu s. A se con d style,be side that on the fork ,

i s u su ally made to n ote,by an ab ru pt m otion on the same

strip of pape r,the exact in stan t of any astron om ical or other

phen omen on , su ch as the flight Of a proje ctile,which it is

desired to determ in e . On the other han d , the Phon au tog rapho f Scott and the Vibrosc ope of Duhame l

,which is practical ly

the apparatu s de scribed abov e,aim at measu ring the pen

du lar vibration s of a sou n ding body in term s of the cylin der’

s

rotation .

( 5) Th e Ph o n au to g rap h .— This in strumen t replace s the tu n

ing-fork shown ab ove

,by a hollow barre l abou t 18 in che s long

and a foot in d iamete r. On e en d is open,the othe r is closed

except by a tu be carrying on it a stre tche d memb ran e . Uponthe membran e is fixed a . bristle , which move s with the membran e , and acts as a style . In orde r that this shou ld n ot b e ata n ode , the membran e is tou ched by a m oveable pie ce

,which

,

be ing made to tou ch the membran e first at on e p oin t an d

then at an other,enables the Observer toalterthe arran gemen t

of the n odal po in ts . ”The bristle is thu s made to coin cidew ith a loop , that is a poin t at which the vib ration s of themembran e are at a maximum .

88 I” “

ON'

SOUND.

When a soun d is produ ced, the air in the cavity o f the

barre l and the membran e vibrate in u n ison with it,and the

style is made to trace on a rotating blacken ed su rface the

Fig. 39.—Comb ination of two parallel vib ratory movemen ts.

form of the se vibrations . Each u n du lation corre sp on ds to adou ble vibration of -the style . The f ork u sed in the vibroSCOpe is retain ed , and trace s bes ide the tracing of the style

90 ON SOUND. [CHAR

shee t of tinfoil,as will b e more fully

'

de scrib ed in a laterchapte r.II. ( 1) Op t ical M e th o d s .

— A ve ry material improvemen t onthe graphic me thods ju st n amed has be en made by substitu tin g

a ray o f light for the style . This me thod was orig in allydu e to M. Lissaj ou s , who in trodu ced a beam of light in to a

dark chamber, an d, focu ssing it by mean s of a len s u pon “

a

mirror attached _to on e of the prong s of a_vibrating “tun ing ;

lv.] PITCH— ITS MEASUREMENT,ETC. 91

f ork , threw the image thu s obtain ed u pon a'

screen . The Spotof light orig in ally produ ced by the qu ie scen t m irror is e longated on the fork be ing set in to vibration

,and form s a lin e of

con tin u ou s illum in ation .

If e ithe r the tun ing-fork carryin g the first m irror, or a

se con d mirror from which the beam is su bse qu en tly refle cted ,b e m oved throu gh a small horizon talang le , the imag e will also m ove , an d

the lin e will b e expan ded in to a sin u ou sfigu re in every re spe ct reprodu cin g the

cu rve of Sin e s above de scribed. Theadvan tag e of this method Obviou sly isthat the w e ight and friction of the styleare en tire ly dispen sed with . A se condtu n ing-fork vib ratin g at right an gle s tothe first can b e made to carry the se con dm irror

, an d thu s the composition of the

two harmon ic m otion s may b e cau sedto produ ce the beau tifu l fig u re s u suallyn amed after the ir origin al produ cer.An in g en iou s in strumen t of the same

n atu re,b u t depen ding on the vibration

of reeds in stead of tu n in g -forks, wasexhib ited at the Loan Exhibition byM. Pichle r. It con sists of a win d-chestw ith mean s of blowing , above whichare two re eds ; on e fixed in a ve rtical ,the othe r in a horizon tal dire ction ; byshiftin g the bearing of on e re ed it mayb e made to in crease its length , so thatthe vibration s of the two shall b e toe ach other in an y ratio from that of

u n ison to that of an octave . They can

pass throu gh all in termediate fig u re s.On each reed is placed a smal l m irror

,

an d a beam from a strong sou rce of

light fall s first on the mirror Of'

the

u ppe r reed,when ce it retu rn s and is

refle cted on the se con d , an d then ce it isthrown on the screen . While the mirrorsare motion le ss , the spot o f light remain s still wh en they are se tin vibration the on e in a ve rtical position g ive s a ve rtical lin eo f light when the other in a horizon tal ‘plan e i s added theycomb in e two harmon ic motion s , g iving the cu rve s alreadynamed. The circle denote s_unison .

,

the varied figures are

92 ON SOUND. [CHAR

produ ced by varying .phase s and v e locitie s of the two ree ds.The ear hears the diffe ren t in te rvals at the same in stan t.Con cord is thu s den oted simu ltan e ou sly by the absen ce of

beats, and by the stilln e ss as we ll as the regu larity Of there su ltan t Optical figu re . Discord can b e pe rce ived by the

rough clashing of the in te rfering u ndu lation s,and b y the

94 ON SOUND. [CHAR

a silver head attached to the end of a' rod produced ,"when set

in vibration , t he patte rn s en grav ed above by virtu e . of themomen tary pe rsisten ce of retin al impression s . It is alsodemon strated by the Vibration Microscope o “He lmholtzmen tion ed e lsewhe re .

( 3) Koen ig’s M an om e tr ic H a rl em— This method con sists

in tran sm ittin g the movemen ts of son orou s wav e s through a thin membran eof caou tchou c to a smal l re se rvoir of

Ordin ary. gas con n e cted with a flame .

A capsu le of wood or metal is dividedin the m iddle by su ch a memb ran e

,

form ing two compartmen ts . On e of

the se i s con tin u ou s with the vibratingmass of air

,the other with the ordi~

n ary gas main s,and with a bu rn er

( Fig . The membran e thu s form in gpart o f the wall of the pipe

,yie lds to

the alte rn ate con den sation an d rarefaction of the air

,an d tran smits the se

alte ration s of pre ssu re to the streamof gas. The re su lt is that the flam e

fl icke rs u p an d down in coin ciden cew ith the vibration to b e obse rved (Fig .

To ren de r the flickerin g s distin ctfrom on e an othe r they are rece ived on

a rotatin g min or. While the flame

bu rn s steadily, there is thu s formed a

con tin u ou s ban d o f light. B u t if the

capsu le is con n e cted with a tu be sou n ding its fou n dation n ote s the flame take sthe form Shown in the first figu re(Fig . If the octave b e sou n ded itassume s that in the se con d .

If two tu be s sim u ltan e ou sly g ive thefun damen tal an d its octave , we Obtainthe f ollow in g appearan ce (Fig . If

the in terval b e that of a third , theflam e take s the more complicated shapeshown be low (Fig .

C lark e an d M acLe o d ’s M e th o d .

—

dA n ew method for in ve stigatin g an

dete rmin in g ve locitie s of rotation has

re cen tly been published in a paperread before the Royal Society in April 1877 , which be sides its

1v.} PITCH— ITS MEASUREMENT, ETC. 95

o ther'

u sefu l"

applications , aff ords an exce llen t,probably the ‘

be st, method of determimng the pe riod of tu nmg -forks. It

Fig . 46 .—Apparatus for the comparison of the vib ratory movemen ts of

two sonorou s tub es .

96 ON SOUND. [on AP.

.

If ’ a circle of dots equ ally spaced rotate s in f ron t o f a

tumng-fork provided with a len s or m irror then if the fork

1s so arranged that it imparts to the image of the dot a

movemen t at right an g le s to the m otion of the latte r,this

double movemen t aided by the“con tin u ity o f vision will

p rodu ce on the eye the impre ssion of a wavy line . . The form

98 ON SOUND. [CHAR

Now if the exact ratios stated above obtain . the wav e s corre spon din g to them Wl ll appear to b e station ary. If how e v e rthe speed of rotation is a little too qu ick for the exact ratio

,

the wave will have a sl ow prog re ssive motion in the same dircotion as that of the dots , while if the Speed of rotation i s a l ittletoo slow

, the wave will move slowly,

in the Opposite direction .

” 1

1 Extracted from a pamphlet b y the inven tors. See also Proceed in gs ofthe Royal Society for April, 1877.

PITCH — ITS MEASUREMENT, ETC.

It i s n ot n ece ssary to en te r in to all the de tails o f thisvaluable con trivan ce b u t it i s obvi ou s that as with a tun ingfork vibrating at a stan dard rate

,ve locitie s o f rotation can

b e accu rate ly dete rm in ed,so with a steady standard o f

rotation , the e rror of a tu n ing -fork from its the ore ticalvibration -n umber can b e immediate ly de tected .

III. (1) Ph o to g rap h ic M e th o ds .—Proilessor Blake of Bron e r

Un ive rsity’

has re cen tly con tribu ted to the American J ournalof Science an ingen iou s method of photog raphing vibration s .

H 2

100 ON SOUND. [CHAR

From the cen tre of the vibratin g disc, made of thin iron as

in the te lephon e , a wire proj e cts which is con n e cte d b y a

short arm with the back o f a smal l ste e l m irror capable of

rotatin g in a vertical dire ction betwe en two ste e l poin ts. Therefle ctin g su rface of the m irror is firmly fixed pe rpen dicu larto the vibratin g disc. A he l iostat sen ds a beam o f su n lighthorizon tally in t o a dark close t , an d at a distan ce of seve ralf e et fall s u pon the mirror, which is in cl in ed 45° to the horizon .

The rays refle cted ve rtically downwards pass throu gh a le n s,

at the focu s of which they form a lum in ou s image o f the

open in g of en try. A carriag e movin g smoothly on fou rwhee ls trave ls ben eath the len s at su ch a distan ce that a

sen sitized plate laid u pon it is at the focu s for actin ic rays .

Un iform ve locity is g iven to the carriag e an d is measu red bya tu n ing-fork Of 512 v ibration s fitted with a style . If the

carriage b e set in motion alon e a straight lin e is photog raphed.

B u t on cau sing the disc to vibrate,each Of its m ov em en ts

carrie s the refle cted beam from the o scillating m irror throughtwice the ang le of the mirror’ s motion . Cu rve s are thu sre corded on the photographic plate , the abscissa o f whichare measu red by the kn own ve locity of the plate an d carriag e ,an d which se rve to de term in e the pitch the ord in ate s represen tin g the amplitu de of vibration of the cen tre of the d iscmagn ified in this case 200 time s . With the voice , speakin gin an ordin ary ton e an amplitu de of n early an in ch is Obtain ed .

This con trivan ce has be en applied more to the an alysis of

vowe l- sou n ds than to determ in ation s of pitch ; thou gh it isobviou sly a fo rm Of g raphic determ in ation , an d thereforede se rvin g of re cord in this place .

The beau tifu l tran script on p. 99 was Obtain ed by thisme thod.

(2) S ch e ib l e r’s M e th o d .

—Tne first p erson who hit u pon a

practical method of Obtain in g exact measu remen t was

Sche ible r, of Crefe ld , who de scribe d it in a pamphlet publishedin 1834 . His syst em ,

as modified by He lmholtz’ s more re cen tre searche s , is thu s de scribed .

1 If we strike a tun ing -fork,

w ith an othe r an octave highe r, an d hold them both ov e r the irproper re son an ce chambe’rs

,we shal l b ear n o heat

, even if theyare ou t of tu n e . B ut if they are applied to a soun din g-board ,a beat may b e heard be tw e en the se con d partial of the lowe r ,an d the l owe st of the u ppe r fork . If both b e he ld ove r there son an ce chamber of the uppe r fork

, the beat is heard mored istin ctly . Cou n ting the n umbe r of beats in a secon d

,they

g iv e the diffe ren ce betwe en the n umbe r of vibration s of -theu pper fork, an d double the n umber of the l owe r. Suppose

1 J ou rnal of Society of Arts , May 1877 .

102 ON SOUND. [CHAR

( 3) Ap p u n n ’s R e e d To n om e te r .

— Appu n n has substitu tedf ree harmon ium re eds for Sche ible r’ s forks . The se

, althou ghsomewhat more affe cted by chan g e s o f temperatu re thantu n in g-forks, have the advan tag e of produ cing a l ou de r an dm ore coe rcive ton e . The reed qu ality is pecu liarly rich inu ppe r partial s , produ cing a strong , hard , and somewhat harshsou nd

,which is o f g reat se rvice scien tifically , be cau se it pe r

f e ctly discrim inate s all the con son an ces,allowin g a slight

error to b e imm ediate ly detected by dissiden t beats. Thearrang emen t of Appu n n

’s in strumen t is as f ollows z— Sixty

five re eds are arrang ed in a l ong re ctang u lar box, and excitedby a ste ady win d-pre ssu re . The re eds each act in a separatechambe r

,con trolled by a wire which Open s a valve fu lly

,or

to any smalle r amou n t. By pu shing in the valve , the n ote i sflatten ed u p to abou t 25 vibration s in a se con d . The reedsare so tu n ed that each beats exactly f ou r time s a secon d withe ithe r of the ad jacen t reeds . The l owe st is n umbe red 0 , andthe highest 64. Con sequ e n tly the highe st is fou r time s 64

,

or 256 vibration s sharper than the lowe st. The lowe st an d

highe st, sou nded tog ether , make a perfe ct octave . The differen ce betwe en the n umbers of vibration s be ing 256, it followsf rom what has be en shown abov e

,that the lowe st reed make s

256, and the highe st 512 vibration s in a second .

Unf ortu n ate ly this apparatu s is mate rially in flu en ced bythe powe r which the reeds , when vibratin g stron g ly , have o f

in flu en cing on e an other. Determ ination s made with it byMr. Ellis are dispu ted by M. Koen ig , an d Lord Rayle igh hasrecen tly added some exce llen t eviden ce to the same effe ct.IV. (1) M ay e r

’s El e ctr ical To n om e t e r is de scribed by Mr.

Ellis , from an u n pu b lished lette r Of the inven tor. The se condsp en du lum of a clock has a wedg e of platinum foil attachedto its lowe r extrem ity, which , at every sw ing , passe s throu gha g lobu le of m ercu ry

,placed vertically u n der it in the cup of

an iron bin din g—screw Con n ected with on e wire Of a smallbatte ry

,of which the othe r w ire i s con ne cted to the p1imary

coil of a larg e in du ctorium ,when ce a wire passe s to the top

o f the pen du lum . By two othe r wire s the secondary coil ofthe in du ctorium is con n e cted with a tu n ing -fork and a revolv

ing cylin de r. The tu n in g -fork carrie s a de licate pie ce of

platin um,which

,as the cylin de r revolve s , will mark

,a cu rv e

on its smoked su rface . Every tim e the pen du lum leave s them ercu ry g lobu le , a sing le spark is proje cted from the foil onthe fork, which pierce s the covering of the cylin der

,and

marks the begin n ing and en d of the se con d . As the mercu ryg lobul e may n ot b e tru ly un der the poin t of su spen sion , the

1v. ] PITCH — ITS MEASUREMENT, ETC. 103

len gth of every two se conds is u sed . The n umbe r of sin u ositie s of the cu rve between the spark hole s , div ided by tvxog ives the pitch of the fork. This me thod is ve ry accu rate ,b u t seem s to b e slightly in flu en ced by the we ight of the

platin um on the fork , an d also by the f riction on the cylin de r,as will b e n otice d fu rther on .

(2) Lo rd R ay l e ig h’s B xp e r ixn en t .

— 1A stan dard f ork byKoen ig which was supposed to g ive 128 vib ration s in a se con dwas excited by mean s of a b ow , and the obje ctwas to compareits frequ en cy w ith the se con ds pen du lum of a clock ke epin g good time . The remainder of the apparatu s con sisted of

an e lectrically main tain ed fork in terruptor with adju stablewe ights , makin g abou t 12 vibration s per secon d, and a depende n t for

'

k, the frequ en cy o which was abou t 125. The cu rren t

f rom a Grove c ell was ren de red in term itten t by the in te rru ption

,an d as in He lmholtz’s vowe l experimen ts excited the

vibration s of the se con d fork, the pe riod of which was '

asn early as p ossible an exact submu ltiple of its own . Whenthe apparatu s was in steady operation , the sou n d emittedf rom a re sonator ass ociated with the higher f ork had a

f requ en cy determin ed by that of the in te rru ptor an d n ot bythat o f the highe r fork itse lf ; n eve rthe le ss an accu rate tu n ingn ow n ece ssary in orde r to obtain vibration s of su fficien t inten sity. This tu n ing was efie cted by gprolon gin g as mu ch as

possib le the period of the beat heard when the depen den tfork starts from re st. The beat may b e regarded as du e toan in te rferen ce of the forced and n atu ral n ote s . By cou n tingthe beats du ring a m in u te of tim e it was e asy to compare thehighe r fork and the stan dard with the n e ce ssary accu racy ;all that remain ed be ing to compare the frequ en cies of the

in terruptor an d of the pen du lum . For this pu rpose the

prong s of the in terru ptor are provided with small plate s o f

tin so arrang ed as to afiord an in term itten t view of a smallsilvered bead carried by the pen du lum

,and su itably lighted .

Un de r the actu al circumstan ce s of the experim en t, the brightpoin t of light is visible in g en eral in twen ty-fiv e position s ,which wou ld remain fixed if the frequ en cy of the in te rru ptor were exactly twen ty-five time s that of the pen du lum .

In accordan ce , howe ve r, with a w e l l kn own prin ciple,the se

twen ty-tive position s are n ot easily obse rved when the p en

du lum is simply looked at for the motion then appears to b econ tin u ou s . The difficu lty is easily evaded by the in te rposition of a som ewhat n arrow vertical slit throu gh whichon ly on e of the twen ty-fiv e position s is visible . In practice

1 Natu re, Novemb er 1 , 1877.

104 ON SOUND. [CHAR

it is n ot n ecessary to adju st the slit to any particu lar position ,sin ce a slight departu re from exactn e ss in the ratio of fre

qu en cies bring s all the visible position s in to the fie ld of viewin tu rn .

In makin g an expe rimen t the in terru ptor was tu n ed at firstby sl iding the we ights , an d afte rwards with soft wax, u n tilthe in te rval betwe en su cce ssive appearan ce s of the brightspots is su fficien tly long to b e conven ien tly obse rved . Witha slow pen du lum the re is n o difficu lty in distin g u ishing inw hich direction it is vibrating at the momen t when the spotappears on the sl it

,an d it is be st to atten d on ly to those ap

pearan ces which co rre spo n d to on e dire ction of the pe n dulum ’s m otion . This will b e be st u n dersto od by cons ide rin gthe case of a con ical pen du lum

,whose m otion , real ly circular,

appears to an eye situ ated in the plan e of m otion , to b ere ctilin e ar. The re striction n am ed then amoun ts to su pposingthe b in der half of the circu lar path to b e invisible . On

this u nde rstan ding , the in terval betwe en su cce ssive appearan ce s is the time requ ired by the fork to gain or lose on e

complete vibration as compared with the pen du lum . Whe therthe differen ce is a loss or gain is easily determ in ed in an yparticu lar case by obse rvin g whethe r the apparen t m otion o f

the spot across the slit, which shou ld have a visible breadth ,is in the same or the opposite dire ction to that of the pen dulum ’s motion . The in terruptor gain ed on e vibration on the

clock in abou t 80 se con ds,so that the frequ en cy of the fork

was a thou san dth part greate r than 125 or 12 51. Thedepen den t f ork gave the n in th harmon ic

,with a frequ en cy o f

The beats betw e e n this fork an d the stan dard , thepitch of which was . the highe r, were 180 in sixty secon ds , sothat the frequ en cy of the stan dard was as n early as possible

agre e in g ve ry close ly with Koen ig’

s scale . The e rror ofthe dete rm in ation may have amou n ted to 1 b u t cou ld n o t

we ll exceed “

2. The approximate determ in ation of the fre

qu ency of the in terru ptor had to b e made in dependently, as

the obse rvation on the pen du lum doe s n ot de cide which mu l

tiple of if most n early coin cide s with the frequ en cy of the fork .

Also the re lation b e tween the two au xiliary forks was as

sumed ; b u t on this poin t there cou ld b e n o dou bt,u nl e ss it

b e su pposed that Koen ig's scale is in e rror to the exten t of

a whole ton e .

V. ( l ) C h lad n i’s R o d To n om e t er .

— It has already be enstated that the vibration s o f rods with o n e en d fixed are

in ve rsely proportion al to the squ are of the ir length in depe nden t of the ir area or cross-section

,and that they g ive a serie s

106 ON SOUND. [CHAR

de te rm in ation of the Fren ch n ormal pitch b y mean s of

Appu nn’s ton ome te r is still Open to som e dou b t

,on accou n t of

discrepan cie s du e to tempe ratu re and the mu tu al in flu en ceof the re eds on on e an other . He state s that in stead of

435 or 870 the n ormal A real ly has a vibration -n umbe r of439 or 878. This statemen t has been attacked byM . Ru dolphKoen ig . B u t the re is n o reason to su ppose that the re lativ epitche s of the variou s stan dards exam in ed su ffer f rom thisin itial difficu lty which may b e easily remedied by a smallcon stan t correction to b e af terwards applied.

Mr. Ellis make s five prin cipal g rou ps , as followsI. An cien t Low Pitch

,0 below 500.

-The first thing thatstrike s u s is the g reat flatn e ss of the olde r pitche s . Dr. R.

Smith’s D, 262, in 1755, taken an octav e higher, as D524 ,

g ive s n early the p re sen t Fren ch n ormal for C. Hen ce his pitchwas almost exactly awhole ton e flatter than the pres en t Fren ch0,and a ton e an d a qu arte r flatte r than B roadwood’

s presen thigh pitch , which we may take to repre sen t con ce rt pitch .

”

B u t re je ctin g this as in all probability wron g ly asce rtain ed,

we have the fork tun ed to Father Schm idt’s C pipe at HamptonCou rt before the organ was re con stru cted

,an d this is more

than a sem iton e flatte r than C 512, an d of a ton e flatte r than“ con cert pitch .

” It follows that vocal mu sic composed a

hu n dred years ago ou ght to b e tran sposed a whole ton e,if

su ng at the prese n t pitch,to produ ce its prope r effe ct

II. The Han del P itch, C 500 to 513 .— C 512 which was ih

sisted on so strong ly by Sir John Hersche l at the meetin g of

the Society of Arts,to con sider the Report of the Comm ittee

on P itch in 1860 , was in favou r 50 to 100 ye ars ago . VVie

pre cht g ive s a Be rlin pitch of that amou n t, b u t the measu rem en t may b e doubted . We find

,howeve r, the fork to which

Mr. Peppercorn tu n ed pian os for the Philharmon ic con certs in1815 was abou t 511

,an d reckon in g by the old tu n ing , a f ork

u sed at the Plymou th Theatre abou t 1800 , g iv e s n early thesame

,while Han de l’ s fork of 1751 gave 0 510 . We may

take then the dawn of mode rn pitch to b e C 512, which wou ldb e ful ly a sem iton e flatte r than the pre sen t con ce rt pitch .

Hen ce vocal mu sic of Han de l’s time shou ld b e tran sposed a

sem iton e lowe r than it is written when played at con cert pitch .

The same remark applie s to the mu sic of Mozart,and pro

bably of Haydn an d Be ethoven .

The valu e of C 512 which appears to have be en aimed at

abou t the pe riod of this g roup , is en tirely arithm etical . It

has n o othe r particu lar advan tage . Arithmetician s can dealwith any othe r C with equ al ease by mean s of decimals . In

1v.] PITCH.


measu ring pitch it is n ever n ece ssary to con sider more thantwo place s of d e cimals , an d ev en the last place is u sed on lyto preven t an accumu lation of e rror.III. F rench Normal P itch, C 514 to 527 .

—Abou t forty yearsago there was a Fren ch pitch in u se almost coin ciden t with thattheoretical ly established in Paris in 1859 ; on e fork from a

g ood maker measu red by Scheible r in 1834 actually gaveA 4349 or practically 0 517 . The pitch , howeve r, mu sthave risen rapidly to abou t A 452

,an d the obje ct of the

Fre n ch Comm ission was to regain this older pitch. Thismodern version of the older fork in the Paris Op era an d

Con servatoire was preceded in En glan d by the alm ost iden ticalb u t flatter pitch of Sir Ge org e Smart, an d B roadwood

’

s vocalpitch

,and also by the very slightly sharper pitch of Sche ible r

in Germany,which b e ing chosen by him as the mean pitch of

Vien n a g ran d pian ofortes , represen ts the Vien na pitch o f the

time . From having be en accepted by a con g re ss of Ge rman

physicists,who met at Stu ttgart in 1834

,it is c ommon ly

kn own as the Stu ttgart p itch . Altog e the r, this g rou p , whichis comprised within abou t a qu arte r of a ton e , repre sen ts thatmost in vogu e n ew on the C on tin en t

,an d con sequ en tly has the

g reate st claim s on ou r atten tion,althou gh its h ighe st forks are

13of a sem iton e be low ou r presen t high pitch .

IV. Medium Pi tch, C 520 to 536.

—The in te rval of abou tg of a ton e between the Fren ch n ormal and high pitch is n otwe ll marked . We h ave in deed within this g rou p a fork fromLe ipzig , pu rporting to b e the Dre sden low pitch ; on e fromVie n n a, measu red by Sche ible r

, b u t diffe rin g materially fromthe othe r V ien n a forks on e from the Lice o Mu sicale at

Bologn a, in 1869 the m edium pitch empirically adopted byMe ssrs . Broadwood and in the organ of St. Pau l’s . There werealso several fore ign forks in this g roup . The theore tical forkof the Society o f Arts which beg in s it, was n e ve r really made

,

and Griesb ach’

s A,like Hu llah

’

s C, were acciden tal e rrors .

It wou ld seem that the whole of this g rou p is n ot g en erallysatisfyin g ; it is both too sharp an d too flat

,an d can only b e

regarded as a n eu tral medium pitch.

V. M odern High P itch, C above 536.

—The highe st g roupcon tain s the moderate ly high pitch which the Fren ch Comm ission fou n d so excessive

,an d the still sharper Eng lish

con cert an d mil itary pitch of the pre sen t day , with the highpitch of Bru sse ls

,strong ly advocated in a report of a com

mitte e to the Belg ian Min ister in We find that the reI The p itches given shou ld b e corrected b y sub tracting 4 for the error which

Mr. Ellis attrib u tes to the French Normal.

108 ON SOUND. [CHAR

was a tu n ing -fork in Paris in 1826 g iv in g A 445 C 529 ‘

2

for the Fren ch Opera, an other g iv in g A449'

5 C 5346 fo r

the Italian Ope ra, and an othe r A 452 C 537 5 for the

Opera Comiqu e . The two first be lon g to the pre cedin g g rou p .

Man y Operas we re composed to the last pitch which was

afterwards raised to A 455 C 541. The report men tion sthat when the Fren ch Commission was appoin ted the Operapitch was A 453 C 5387 ,

an d that Lissajou s wished tolower it to A 4495 C 5346 ,

b u t that a con trary Opin ionprevailed. The Comm ittee say that to this high pitch be lon gthe tu n in g -fork of the Bru sse ls Con servatoire

,on e in u se at

Ghen t, an old f ork of the Paris Opera Com iqu e in 1820 , the

tun ing -fork of the Philharmon ic Society of Lon don , that of theBerlin Opera in 1861,

“an d lastly that of the Choral Society

of Colog n e .

We have thu s , according to Mr. Ellis’s observation s,a

rise for C from 467 to 546 , or 80 vibration s 26 sem iton e sin 130 years , or, if the e arly obse rvation s b e re je cted as

possibly e rron eou s,from the u ndoubtedly au then tic fork of

Han de l,which g ives 507 4 to the v ib ration n umber of 546 5,

at which the ban d of the Be lg ian Gu ide s were playingin 1859. The write r can state from his own carefu l oh

servation s made at the Han de l Fe stival of 1877 , du rin gthe performan ce of the Israe l in Egypt, on an extreme ly hotday in Ju n e

,the the rm ometer be in g n early 80° u n de r the

dome of the orche stra,that the pitch of A rose to 460

,

which is equ ivale n t to a C of 547 , an d is higher than anypreviou sly re corded.

C au s e s o f th e R is e in S tan dar d P itch .— It will b e see n

that the ten den cy of the stan dard pitch has always been torise

,except when au thoritative ly an d su dden ly lowe red

,as in

Fran ce,and m ore re cen tly

,though with little su cce ss

,in

En g lan d. It is also obviou s that the rise occu rs chiefly inorche stral pe rforman ce s . Mu ch of this is du e to temperatu re .

All win d in strum en ts rise with the warm breath of the playe r,

e spe cially the clarin et,which varie s almost a sem iton e .

Som e part is also du e to the fact that strin g ed in strumen tstun e to perfe ct fifths , which can b e shown to b e in com

m e n su rable , from the ir larg er in terval, with the octave .

Con siderable we ight mu st b e g iven to the fact that the ear

i s physiolog ically liable to se le ct the sharper of two n ote s forim itation ; b u t the chief cau se is an in stin ctive b u t vu lgarin clination in the players them se lv e s to give the ir own in strumen t an u n du e prom in en ce at the expen se of the others byslight sharpen ing .

110 ON SOUND. [curs

CHAPTER V.

NATURE OF MUSICAL TONE. QUALITY. HARMONICS.

RESULTANTTONES.

N atu r e o f M u s ical S ou n ds . Q u al ity .— It has been accepted

as an axiom that the sen sation of mu sical ton e i s du e toa rapid periodic motion of the son orou s body that of n oise

,

to n on -pe riodic m otion s an d it has be en shown that mu sicalton e s are distingu ished : 1 , B y the ir force or lou dn e ss .2, By the ir pitch or re lative height . 3 , B y the ir quality.

Q u al ity .— It i s to the third of the se con stitu en ts that atten

tion is n ow to b e dire cted. The quality of a ton e , which was

formerly den oted by the an g l ic ised° Fren ch term Timbre,i s

that pecu liarity which distingu ishe s the violin from the fl ute orthe clarin et , an d the se from the human voice

,when u ttering

sou n ds o f the same pitch or frequen cy. Un til the re searche so f He lmholtz this last characte ristic had remain ed u n explain ed.

He show ed in a con clu sive man n er that the observed diff eren ces depended on n o abstru se or recon dite prope rty

,b u t

simply on the co -existen ce w ith the prin cipal of otherse con dary an d affiliated vibration s

,which accompany and

modify the sen sation by alteration s which they produ ce u ponthe form of the sou n d-wave itse lf . He lmholtz aptly illu strate sthe possibility of differen ce even in pe riodic m otion s su ch as

are so slowly performed as to b e capable of be ing followedby the u n assisted eye . For in stan ce

,the motion of a p en

du lum or an ordin ary vib rating sprin g is on e which is rapidin the m iddle o f its path , an d slow at e ither extremity :

that of a hamme r moved by machin ery is marked by be in gslowly raised and fall ing su dden ly. A ball thrown up vertically an d caught on its de scen t b y a blow which sen ds it u pagain to the same he ight, occu pie s the same time in rising as

in fal l ing , b u t at the l owest poin t its motion i s su dden ly

v. ] NATURE or MUSICAL TONE, ETC.111

in terru pted ; whereas above it passe s throu gh'

graduallydim in ishin g sp eed of ascen t in to a g radu ally m oreasrng speedof de scen t. Non e of the se f orm s of

motion are similar to on e an othe r, n or

wou ld they,if tran slated in to sou n d , pro

du ce the same effe ct. The pendu lummay b e g raphically repre sen ted by the

dou b le cu rve of sin e s be fore n amed ,passing equ ally on e ithe r side of a

straight m iddle l in e ; the hamme r by a

se rie s of long in clin ed plan es terminatedby a short downward cu rve ; the bal l bya serie s o f arche s abru ptly refle cted fromon e side of a base l in e . It is u pon thisdiffe ren ce in the form of the vibrationthat qu ality of ton e depends. Ohm was

the first to de clare that there is on ly on e

form of vibration which wil l con tainn on e of the se secondary w ave s

,an d w ill

therefore con sist sole ly of the prime ton e .

This is the form pe cu liar to the pen du luman d to tun in g-forks , an d hen ce they are

called simp le or p endu lar vibration s .

P artial s .— The affiliated or secondary

wav e s , when occu rring in the same periodas the primary

,are term ed harmon ics

,

overton es, or most accu rate ly

,u er

partial ton es. The characte r g iven io a

particu lar n ote by the ir pre sen ce,by

an over-literal tran slation of a Ge rman

w ord,has been te rmed clan g -tin t, ” it

was form erly designated as timbre,b u t

i s be st repre sen ted by the fam iliarEn g lish word qu al ity.

When several resonan t bodies simu ltan eou sly excite dif eren t systems of wavesof sound

,the changes of density of the air , and the disp lacemen t,

and velocities of thep articles of air within the ear are each equalto the algebraical sum of the corresp onding changes of density ,disp lacemen ts, and velocities

,which each system of waves wou ld

have sep arately p rodu ced if it had acted indep enden tly.

‘

The mu ltiplicity of vibration al f orm s which can b e thu sprodu ced by the composition of simple pen du lar vibration s is

I Helmholtz,Sen sation s of Tom , p . 43 et seq.

1 12 ON SOUND. [CHAR

infin ite . The Fren ch mathematician,Fou rier

,has proved the

corre ctn ess of a mathematical law which may b e thu s en u n

ciated : “An y g iven periodic form of vibration can always b eprodu ced by the addition of simple vibration s

,having vib ra

tion al n umbers which are on ce,twice

,thrice

,fou r time s

,& c. ,

as g reat as the vibration al n umber of the g iven m otion .

”

P ou rie r ’s Th e o r em .

— For the pu rpose o f application to thetheory o f sou nd

,this law may b e expre ssed as f ollows

Any vibration al motion of the air in the au ral passages,corresp on ding to a mu sical ton e

,may be always, bu tfor each case

on ly in a sin gle way, exhibited as the sum of a n umber of simple

vibrational motions, correspon ding to the partia l ton es of thatmu sical ton e.

It was,howeve r

,in cumben t u pon He lmholtz as the proposer

o f su ch a hypothetical explan ation o f qu ality,to show by

experimen t that the observed facts were in this way fairlyr epre sen ted . This b e complete ly accomplished by the he lpo f the re son ators n am ed above . Starting with an an alysiso f sympathetic vibration , he en u n ciate s the law that “the

simple partial ton e s con tain ed in a composite mass o f

mu sical ton e s produ ce p ecu l iar me chan ical eff e cts in n atu realtog ether indepen den t o f the human ear and its sen sation s

, an d al so altogether indepen den t of mere ly theoreticalcon sideration s .

”

Commen cing with the phen omen a of re inforcemen t al readygiv en , sympathetic reson an ce is stu die d in a circu lar mem

b ran e strewed with san d,as in Chladn i’ s inve stigation s .

It is shown that it is easie st to set the m embran e in g en eralmotion by sou n ding its prime ton e . The first form in whichthe re son ator appeared was that of a pig

’s bladde r stre tchedov er the larg e r orifice o f a g lass re ce iv er , the mass of con

tain ed air in the latte r v ibrating sympathetically with the

membran e . It was f ou n d that su ch a membran e is n o t on lyset in vibration by mu sical ton e s of the same pitch as the

proper ton e o f the bottle,b u t also -b y su ch mu sical ton e s as

con tain the proper ton e of the membran e among the ir u ppe rpartial ton es.

R e s o n ato rs .— A great improvemen t, however, re su l ted f rom

the substitu tion of the tympan um or drum of the ear itse lffor the m embran e above de scribed. Hollow sphere s of g lassor metal we re made with two Open ing s , on e larg e with sharpedg e s, the othe r fu n n e l-shaped an d adapted for in sertion in tothe ear; The g lobu lar or cylin drical cavity of the re sonatoris e spe cially pron e to vibrate in un ison with its own primeton e , while it ten ds to damp all others. Hen ce any on e

,

“even

114 ON SOUND.

will b e absen t . For in stan ce,if it b e plu cked in the middle ,

the even partial ton e s will b e absen t, and the sou nd wil lhave a pe cu l iar hollow characte r. Or after it has been stru ckin the ordin ary way, a cam e l’s-hair pen cil may b e appliedso as to damp all the simple vibratio n s which have n o n odeat the poin t tou ched. A p ian o strin g plu cked in the cen tre ,an d then tou ched at the same poin t

,cease s en tire ly to sou n d

,

whereas, if plu cked e lsewhere,and tou ched in the m iddle

,the

secon d partial will b e heard .

’

The re sult of these an d sim ilar experimen ts is that thesensation of a mu sical ton e is compou n ded ou t of the sen sation s

of several simp le ton es. The prM e ton e i s g en eral ly lou derthan an y o f the u pper partial ton es

,an d hen ce it al on e dete r

m in es the pitch .

There are very few case s in which the soun d of a reson an tbody con sists o f a simple ton e

,the chief in stan ce s b e ing

tu n ing -forks m ou n te d on a con son an t box , an d larg e stoppedorgan -pipe s blown with a very g en tle stream of wind . On

the othe r han d , the u n ion of several comparatively simpleton e s in to on e compou nd o f g reate r power is artificial ly produ ced in the o rgan by the steps te rmed corn ets

,se squ ialteras ,

an d m ixtu re s. As a g en e ral ru le1. The u pper partial ton e s corre spon din g to the simple

vibration s o f a compou n d m otion o f the air are fe lt,even

when they are n ot always con sciou sly perce ived .

2. They can b e made obj e cts o f con sciou s pe rception withou t any othe r he lp than a proper dire ction of atten tion .

3 . Even in the case of the ir n ot be ing separate ly perce iv ed ,becau se they fu se in to the whole mass of mu sical sou n d

,the ir

existen ce in ou r sen sation i s e stablished by an al teration inthe qu al ity of ton e

,the impre ssion of the ir higher pitch be ing

characteristically marked by in creased brightn e ss of qu al ity,an d apparen tly g reater sharpn e ss of pitch .

A l ist of the first an d m ost importan t of the se u pper partial sas fou n ded on the bass C’ i s as follows

Th e Syn th e s is o f To n e s .-The corre spon ding operation to

the an alysis of mu sical qu ality requ ire s simple ton e s ‘

of greatpu rity which can have the ir force and phase exactly reg ulated .

These are obtain ed from tun in g-f orks , the lowe st ton e of

v. ] NATURE OF MUSICAL TONE, ETC.

which is re in forced by a re son an ce cavity , an d thu s comm u

n icated to the air. To se t them in motion they are placedbetween the pole s of an e le ctro-magn e t, the cu rren t in whichis regu lated by a separate apparatu s . To make the ir sou n dau dib le

,the re son an ce chamb er has to b e brou ght n ear them .

Its m ou th can b e closed by a l id attached to a leve r,by

partial open ing of which,the sou n d can b e regu lated to an y

am ou n t o f streng th . The string s acting on a n umber of su chlev ers“ are attached to a sort of key-board.

He lmholtz at first experimen ted with e ight forks of thiskin d , repre sen tin g B ") an d its first seven harmon ic partials ,n ame ly bb, f , b

’

b d", f a

"han d b

”

b. He af terwards had f orksmade of the pitch of f a

mb and b

"’

b.To set them in m otion

,in te rm itten t cu rren ts had to b e

passed throu gh them e qu al in n umber to the vibration so f the l owe st in a second

,n am e ly 120: The f un dam en tal

fork thu s re ce ived an impu lse at ev e ry vibration , the highe rm embers of the serie s at pe riods inv erse ly corre spon ding to

the ir rapidity .

The m oving apparatu s con sisted of a tu n in g-fork , to thetwo pron g s of which were fixed platin um wire s

,dipping in

cu ps filled half with me rcu ry an d half w ith alcohol . Thesem ade con tacts at every oscillation

,which

,sen din g an in

termitten t cu rren t through the e le ctro -magn e t, se cu red p ersiste n ce of the vibration . A l ittle stee l clamp was placed on

on e pron g of the f ork,by which its oscillation s might b e

brou ght in to exact u n ison with the fu n dam en tal ton e .

When the re sonan ce chambers w e re closed,and the whole

serie s of f orks was in vibration,n othin g was heard b u t a

g en tle humming sou n d,b u t on Open ing each of the se , its

corre spon ding ton e was heard,an d it becam e possible to

f orm d iffe ren t combin ation s of the prime ton e with on e orm ore u pper partials having differen t degree s of lou dn e ss

,and

thu s to produ ce ton e s of diffe ren t qu alitie s .The v owe l sou n ds of the human voice are marked by com

parativ ely Iow partial s,th o se of E an d I alon e som ewhat

exceeding the lim its of th e f orks u sed. U, 0 , the modifiedGe rman 0 and even A cou ld b e m ore or le ss im itated . Byfu rther addition s to the serie s of forks E,

an d in deed all b u tI we re procu red.

In the same way the qu ality of ton e produ ced by organpipes of difieren t stepswas reprodu ced

,also the reedy ton e of

the clarin e t, by u sin g a se rie s of u n even partials , an d the sof te rton e s of the Fren ch horn by the f u ll choru s of all the forks.From all the se experimen ts it was fin ally shown that t he

1 2


qu ality Of the mu sical portion of a compou n d ton e dependssole ly on the n umbe r and relative stren g th of its partials imple ton es

, an d in n o respe ct on the ir diffe ren ce of phase .

V ow e l To n e s .—If the lowe st ton e of the reson an ce

chambe r doe s n ot corre spon d with the prime ton e,b u t with

sOme of the u ppe r partials,the correspon din g u ppe r partial is

really more re inf orced than the prime,and con sequ en tly pre

dom in ate s . The qu ality thu s produ ced,m ore or less re sem b les

the v owe ls Of the human voice,which are really produ ced by

the v ocal chords,with the m ou th as a re son an ce chambe r

,

capable,by chan g e of shape an d v olum e

,of re in forcing

d iffe ren t partials Of the compou n d ton e to which it isapplied .

With the assistan ce of re son ators it is possible to re cogn isevery high partials, u p to the sixte en th when o n e Of the morebrillian t vowe ls is ' su n g by a bass voice . The ir lou dn e ssd iffe rs con siderab ly in diff eren t person s

,an d they are

g e n eral ly n ib re difficu lt to re cogn ise b y the u n aided ear thanin mu sical in strum en ts. The inve stigation of the re son an ceof the cavity of the m ou th is therefore Of con siderable importan ce. If tu n ing -forks Of diffe ren t pitche s are he ldb efore it while the shape Of the cavity is altere d for theseveral vowe ls it can b e dete rm in ed which of them is m ostre in forced . It can thu s . b e shown experim e n tally thatthe pitch Of strong e st re sonan ce depen ds sole ly on the vowe lfor pron ou n cing which the m ou th has be en arrang ed , there son an ce s be ing essen tial ly the same in men

,wom en

,an d

children .

The vowe l s are arran ged by He lmholtz afte r Du BoisReymon d in this serie s( 1) The broad A as in father is the common orig in of all

,

the cavity o f the mou th be ing fu n n e l-shaped , with u n iforme n larg em en t ou twards. A lowe r se rie s con sists of O, as in

more, an d U as in p oor, in which the lips are con tracted an d

tongu e depre ssed so as to form a bottle-shaped cavity.

(2) In the vowe l s A , E, I, and the m odified Ge rman A the

lips are drawn far apart, and a con traction is made betweenthe m iddle o f the tong u e and the hard palate . These have a

highe r an d a deepe r re sonan ce ton e .

(3) In the third serie s from A throu gh 0 and U,in addition

to the con traction of the ton gu e an d palate , we have also a

con traction of the lips in to a sort of tu be like that of a bottlewith a ve ry long n e ck .

The reson an ce of the cavity of the mou th for vowe ls mayb e thu s expre ssed in mu sical n otes

118 ON SOUND. [CHAP.

Ratio m 2m‘

3m 4m 5m 6m 7mNote 0 1 C

2G 2 O3 E3 G3 X

V ibration n umbe r 64 128 192 256 320 384 448

We have he re thre e C’s, two G

’s, on e E,

an d a n ote markedX lying betwe en A3 an d B 3 . If the octave abov e C b e

sou n ded w ith it, the harmon ic se rie s produ ce d by C2 will b e ,in the first three octave s,

Ratio . 2m 4m 6m 8m l om 12m 14m

Note c,

c G 0 4 E4 G XV ibration n umber 128 256 384 512 640 768 896

again g ivin g three C’s,two G ’

s,on e

'

E,and on e X an octave

ab ove the forme r.If the n the two n ote s Cl and C2 b e sou n de d tog e ther, the

first thre e Octave harm on ics o f C1 will b e compou n ded w iththe first two octave s of C3, as follows

C11st in ten s ity.

C ; C21st an d 2n d in ten sity .

G2

2 3rd in te n sity.

C 3 2n d and 4th in ten sity.

G3 G3 3rd an d 6th in ten sity .

E3 5th in ten sity .

X1 7th in ten sity.

It i s the refore eviden t that when tw o n ote s C1 an d C2 aresou n ded tog ethe r, they produ ce Overton es

,Upp er P artial

Ton es, or H armon ics, 1n which,be side s C’

s , the n ote G (thefifth) w ith its octave s , an d the n ote E ( the third) with itsoctav e s , are m ore n early re lated to C than any othe r n ote sare

1

He re in lie s the physical explan ation of the fact that anysou n d an d its octav e bear the same n ame , in accordan ce w ithn atu re , sin ce the two sou n ds so accord or tu n e tog ethe r thatthey se em to b e alm ost like on e sou n d .

It may b e remarked that the e ffe ct . of the combination of

n ote s differin g by an octav e i s to throw the n ote s X fu rthe rb ack , makin g them less au dib le , an d brin g ing ou t m ore clearlythe re lation ship of C ,

E,an d G . Now this g rou p of n otes

which are actu ally sou n de d whe n ever C, e ither alon e or w ith

1 For fu rth er d etails see Hau ghton’

s Natu ral Philosophy, p . 170 et seq ,

where this su b ject is treated with great lu cidity .

v. ] NATURE OF MUSICAL TONE,ETC. 1 19

any o f its octave s is - playe d— fortu s the Maj or: Triad , thefirst

,third

,an d fifth o f the n atu ral scale ; the 1r v1b rat1on

n umbers be ing in the simple ratio of the n atu ral n umbers4 5 , 6."The harmon ic serie s has an othe r m ost importan t bearing

on mu sic, which n ow may re ce ive its fu ll e lu cidation .

1 . In the reg u lar division of the sou n din g strin g ,2. By in creasin g the b last of w in d in an organ -pipe ,3 . Or by alteration of the embou chu re in brass in strumen ts ,

the same orde r an d sequ en ce of sou n ds is Obtain ed su cce ssiv e ly as has be en here shown to co -exist simu ltan e ou sly.

( 1) The produ ction of harmon ics in strin g ed in strumen tshas b e en already n oticed . (2) In organ -pipe s the u se of b ar

mon ic stops , the con son an t tu be s of which are made tw icethe ir prope r len g th , pe rforated w ith a smal l hole in the midd le ,and with this a high pressu re Of w in d

,i llu strate the same

prin ciple . (3) In the Fren ch horn an d sim ilar in strumen tsn early the whole harmon ic se rie s is u tilize d in the scale of

what are te rm ed “open n ote s,

”as follows

C G C E G X C D E

F G X C

The real fou n dation n ote is o f cou rse an octave lowe r thanthe lowe st he re g iven ; it is all b u t impossib le to produ cew ith the u su al m ou thpie ce , b u t can e as ily b e obtain ed byaffixing to the tu be some othe r sou rce of sou n d

,su ch as a

clarin e t re ed .

If the harmon ic serie s b e exten de d to the fu ll rang e of

ove r fiv e octav e s , the seven sou n ds o f the mu s ical scale can

b e deve loped ou t o f it in regu lar su cce ssion from the gradu alapproximation of the con stitu en t ratios . This has be en w e lldemon strated by Mr. Colin Brown

,Eu in g Le ctu re r on Mu sic

at the An derson ian Un ive rsity, Glasg ow,on the n ote F

,whe re

the twe n ty-f ou rth, tw e n ty-sev en th,thirtieth

,thirty-se cond

,

thirty-sixth,fortieth

,f orty-fif th

,an d forty

~ eighth harmon icsprodu ce a corre ct e n harm on ic scale of e ight con se cu tive n ote sfrom treble C to its octave . The harm on ics which do n ot

be lon g to the scale are marked w ith a cross on the approxi

mate lin e of the staff to which they be l ong .

122 ON SOUND. [CHAR

M u s ical To n e s o f S trin g s .— The in strumen ts of this class

which are excited by plu cking are the harp , g u itar, an d zither .In bowed in strume n ts the same efiect is produ ced and term edP izzicato . The the ory of the ir m otion u n de r the se circumstan ce s is complete . It i s the sam e for a string which hasbeen stru ck in on e poin t by a hard sharp edge . In the pian oforte , where soft hamme rs are u sed

,it is m ore complicated .

The f orce of the u ppe r partial s depen ds on— l,The n atu re of

the stroke . 2, The place stru ck . 3,The den sity

,rig idity, an d

e lasticity of the strin g .

(1) A sharp poin t produ ce s a shrille r ton e with a larg e rn umbe r of high u ppe r partials. When the hamme r 1s softand e lastic

,the motion has time to spread before the hamm e r

rebou n ds,an d greate r softn e ss of ton e re su lts from the cor

respon ding de crease Of the higher partials .(2) As regards the place stru ck , if it approach the en d of the

strin g , the u ppe r partials in crease,whereas if it b e n earer the

m iddle,the even partial s de crease and the sou n d is m ore

hol low . The poin t stru ck in pian os is from on e—seven th te lo n e n in th of the len gth from the end of the strin g , an d thenthe seven th an d n in th partial s becom e comparativ e ly weak .

The se are the first in the se ries which do n ot be long to themaj or chord of the prime or f un damen tal ton e

,an d the ir

exclu sion ren ders the qu ality m ore sm ooth and pu re .

(3) Rig id string s can n ot form ve ry high u ppe r partials , an dl ight strin g s of gu t, on the other hand, are fav ou rable tothem ; b u t the inf erior e lasticity of this latter material ten dsto damp them. Hen ce plu cked string s of catgu t , as in the

g u itar an d harp , are mu ch le ss tin kling in sou n d than those of

m etal .V ib rat io n M icr o s c o p e .

— The vibration s of bowed string swe re exam in ed by m ean s of the vibration -miscroscope abov en amed . They were fou n d for the m iddle strin g of a violinto b e e ssen tial ly diffe ren t from simple vibration s

,be ing an

g u lar an d su dden , l ike the m otion of the tilt hamm er n am edpreviou sly . They resemble the teeth of a saw

,the downward

lin e be in g often so rapidly exe cu ted as to be come invisible .

Du ring the greate r part of each vibration,the strin g cling s

to the bow an d 1s carried on b y it ; it then su dden ly detache sitse lf and rebou nds , be in g again se ized by other poin ts of thebow

,an d again carried forward.

Q u al ity o f Org an Pip e s .— In the flu e pipe s of organ s the

qu ality difiers materially with the d iam e te r an d len gth o f

the pipe itse lf . The tu b e which con stitu te s the air-chambe rof the pipe streng then s by re son an ce su ch ton es as corre spond

v. ] NATURE OF MUSICAL TONE,ETC. 123

with its proper ton e , an d make s them predom in ate over there st. It i s on ly in n arrow cylin drical pipe s that the higheruppe r partials of the tube exactly corre spon d with the harmon ic u pper partials of the prime ton e . By u sin g a re son ator

,He lmholtz finds that in n arrow pipe s partial ton e s may

b e heard clearly up to the sixth . Wide pipe s , on the con

trary , produ ce the prime ton e strong ly an d f u l ly,with mu ch

weaker se con darie s . The n arrow er stepped cylin drical pipeshav e proper ton es corre spon ding to the u n even partials of

the prime,the third partial or twe lfth , the fifth partial or high

maj or third,an d so 0 11. Wide stopped pipe s when g en tly

b lown, g iv e the prime ton e almost alon e .

R e s u l tan t o r C om b in atio n al To n e s .-TWO n ote s , when

sou n ded tog ether, produ ce , u n de r ce rtain circumstan ce s, othern ote s which are n ot actu al con stitu en ts of e ither ton e . The seare termed resu ltan t or combin ation al

,and are of two kin ds ,

d ifferen tial and summation al. The form er have a vibrationn umbe r e qu al to the difieren ce of the ir compon en ts , the latte ro n e which is the sum of the ir frequ en cie s. The sou nds comb in ing to produ ce this re su lt may b e e ither fu n dam en tal oru ppe r partials ; hen ce n ote s rich in harm on ics may yie ld a

larg e n umbe r of re su ltan t ton e s. They were observed in the

last cen tu ry by Serg e an d byTartin i , and were u n til late ly attribu ted to beats , the frequ en cy of be ats having b een shown tod epen d dire ctly on the vibration -frequ en cie s of the ton e sprodu cin g them ; b u t this explan ation fails to accou n t for thephen om en a

,and wou ld f orm an exception to the ru le which

appears to b e g en e ral , n am e ly,that eve ry simple ton e arise s

from a correspon ding simple vibration . It doe s n ot explainthe summation ton e s

,an d in certain case s the differe n ce-ton es

and the beats can b e heard tog ethe r. The pitch of a com

b in ation -ton e i s g en e rally differen t from that of e ither of

those produ cing it or from that of the ir u pper partials. Bothprimary an d upper partial ton es ma give rise to them . Thed iffe ren tial are g en erally stron ger than the summation ton e s.

As the prime ton e g en erally predom in ate s ev er the partials,

the differen tials Of the f orm er are m ore distin ctly heard . Tohear them

,two sou n ds f orm in g a ju st in terval shou ld b e

stron g ly he ld tog ethe r ; a w eak low ton e will b e heard . In

the fifth the ton e thu s produ ced is an octave be low the lowerg en erator ; in the f ou rth a twe lfth ; in the maj or-third twooctave s in the m in or third

, two octave s an d a maj or thirdin the maj or sixth

,a fifth in the m in or sixth

,a maj or sixth

,

accordin g to the following scheme

124 ON SOUND. [a r

Dif erence Tones.

MajorFou rth Third

The summation al ton e s are u su ally weaker than the differen tial. By the ir defin ition

,they are always sharper than

those produ cing them ,as f oll ows

Summation Tones .

Major MajorFou rth Sixth Third Min or Sixth

In the last two cases the summation ton e l ie s between thetwo n ote s g iven in the u pper stav e .

They are most easily heard on the polyphon ic siren an d o n

the harm on ium when tu n ed to tru e in ton ation, of which

inde ed they may b e u sed as a te st. ’ They com e ou t very distin ctly when two powe rf u l sopran o voice s sing in thirds , an doccasion ally produ ce a very painfu l effe ct in the fu l l choru sof a larg e organ .

B e ats o f Up p e r P artial To n e s occu r when the two g en erators are more than a m in or third apart

,an d are of con siderably

more practical importan ce for mu sical pu rpose s than the

beats of combin ation ton es . It w ill easily b e seen that beatsmay occu r when eve r any two u pper partial s lie n ear tog e ther,or when a prime of on e approache s an u ppe r partial of

the other. The n umber of beats is the diff eren ce o f the

vibration al n umbers of the partials,or of the u pper prim e

and the first partial of the lower. t is fr om the perfe ct

126 ON SOUND. [earn

the same g en erate them , an d the rapidity of the beats is thesame in both case s. Hen ce there i s on ly a slight in crease inthe strengt h of the beats. B u t with simple ton e s the case i sdifferen t, sin ce with them n o beats whatever cou ld b e produ ced u n le ss they we re n early in u n ison . S u ch beats don ev erthe less occu r, though mu ch weake r

,even with wide r

in te rval s. In the imperfe ct octave the first differen tial iscompeten t to produ ce beats , b u t in the fif th the beats are du eto a m ore complicate d re lation , which in crease s in complexityfor the fou rth , and in the imperf e ct maj or third is hardly tob e re cogn ised .

Dzfieren tz’

als and Upp er P artials of In tervals.

I

Octave F ifth Fou rth MajorThird2 1 3 2 4 3 5 4

Min orThird Major Sixth Min or Sixth6 5 5 3 8 5

The n ote s of the in te rval are repre sen ted as m in ims,the

diffe ren tial s as crotche ts ; the se con d diffe ren tial s to the leftof the larg e r n ote s.

I From Cu rwen’

s Mu sical Statics.

vi .j EFFECTS OF HEAT,Em 127

CHAPTER VI.

EFFECTS OF HEAT,ATMOSPHERIC PRESSURE, MOISTURE, DENSITY.

Effe ct s o f H e at .

— It has be en shown by carefu l Ob se rvation s that the ve l ocity Of Sou n d in air at 0

°Cen tigrad e or

fre ezin g poin t of w ate r is abou t 332 m etre s or 1090 f e e t p erse con d . This in crease s with rise of temperatu re , be in g proportion al to the squ are root t of ab solu te temperatu re .

I If t

b e the ordin ary cent ig rade temperatu re , a the coefficien t o f

expan sion, 0 0366 , the ve locity at any temperatu re may b e

f ou n d by the formu la.

1090 J l a t in f e e t per se con d .

The velocity mu st b e the oretically given by the f ormu la

where D IS the den sity and E the e lasticity of the air.

If P den ote the pre ssu re of the air in u n its of f orce peru n it of area

,and the temperatu re b e kept con stan t du ring com

pre ss ion,a smal l addition al pre ssu re 19 will , by Boyle ’s law

,

produ ce a compre ssion equ al to l ),an d the valu e of Ewill

P

b e simply P .

0 11 the other han d , if n o heat is al lowed to en te r or e scape ,Ab solu te Temp eratu re is the p oin t at which the motion of gaseou s molecu les

wou ld cease . A gas is in creased a, Of its volume for each degre e C en tigrade ,

an d at 273° is dou b led . If theESemp eratu re o f a given vo lume were loweredthrou gh the con traction wou ld b e equ al to the volume ; that is , thevolume wou ld n ot exist . In al l prob ab ility, b efore reaching this temp eraturethe gas wo uld u n dergo a change of state .

128 ON SOUND. [CHAR

the temperatu re will b e raised by compre ssion,an d additi on al

re sistan ce w ill take place . The compression will the refore b ep where 1 B is the ratio of the two spe cific heats

P ( 1 B )which f or air an d Simple gase s is The valu e of Ewillb e P (1 B ). Be tw e en the se lim its the ve locity in air willvary . Ob se rvation Shows it to b e close u pon the latte r of

these valu e s .

Effe ct s o f P re s s u r e .— Ve locity is in depen den t of the he ight

of the barom eter Sin ce pre ssu re an d den sity are afiected tothe sam e amou n t, an d in the sam e dire ction . At Qu ito, whe rethe mean pre ssu re i s on ly 21 8 in che s

,the ve locity is the

same as at the sea-leve l,provided the tempe ratu re b e the

same .

If 9 b e the force of g ravity, h the barometric he ightredu ced to ze ro , an d d the den sity Of me rcu ry at zero thenfor a gas at atmosphe ric pre ssu re E g h d, and Newton

’

s

formu la be come s

If tempe ratu re in crease s from 0° to t

° its volume willin crease f rom u n ity at ze ro to 1 a t at a be in g the

coefficien t Of expan sion o f the gas . B u t den sity varie sin ve rse ly as the volume , the ref ore d become s d ( 1 a t)hen ce

g h

7 (1 + a t).

Valu e s of V thu s obtain ed are how e ve r le ss than the

experime n tal valu e s,in con sequ en ce of the heat produ ced by

compre ssion , as already stated .

The coefficien t of expan sion Of air,In a fraction al form has

been g iven as5373 per deg re e Cen tig rade . Hen ce takin g the

ve locity at fre ezing poin t, as 1090 fe et per second,that at

a g iven temperatu re t° will b e

1090N/1

Con ve rtin g this in to degree s Fahren he it it g ive s abou t1110 ft. at 50

°

Fahren he it and 1 148 at 86° Fahren he it

,or

an in crease of velocity of abou t a foot per se con d for eachdeg ree Fahrenhe it.

130 ON SOUND. [CHAR

when the ten sion was produ ced by mean s of a we ight, an dthe string thu s allowed to len g then . there was still a n otablefall Of pitch

,du e , in all probability, to alteration of its

e lasticity.

String s of catgu t, be in g very hyg rom etric,are al so mate

rially affe cted b y m oisture , which swe ll s the mate rial laterallyand te n ds to shorten it. In a hot damp con cert-room violin svary rapidly an d somewhat irregu larly from this cau se .

On Tu n in g -fo rk s .— The effe ct of heat on tu n ing -forks was

n oticed an d rou ghly compu ted by Perron et Thompson . He

state s that they are made flatte r by heat at the rate of a hu ndredth of a comma for every thre e deg rees Cen tig rade ( or 54Fahrenhe it). His experimen t was con du cted as follow s. A'

ste e l tu n in g -fork sou nding treble C’was cooled to the freezingpoin t in sn ow an d the l oad n oted which brou ght the harmon icdou b le octave of his m on ochord in to u n ison w ith it ; this was240 lbs. The fork was then he ated in b oilin g water ; u ponwhich 2 lbs. had to b e taken Off the l oad to bring it againin to u n i son . The in crease Of len gth may b e e stimated at

0 147 of an in ch to the foot, which is competen t to flattenthe pitch by le ss than the ten th Of a comma. B u t the f orkwas f ou n d flatte r by the third of a comma : more than twothirds of the e ff e ct mu st therefore hav e proceeded f rom som e

othe r cau se. “For which n othing so readily pre sen ts itse lf as

a re laxation of the e lastic pow e r of the m etal at the Shou lde ro r ben d . In confirmati on of which it was observab le thatthe forks

,when heated to boiling -poin t

,l ost mu ch Of

the ir streng th of ton e, an d did n ot en tire ly re cove r it on

cooling .

”

Perron et Thompson ’s observation s g ive the coefficien t of

alte ration as 000023 . Sche ible r g ive s as the re su lts Of his

obse rvation s what is e qu ivalen t to 0 000573 at 440 vibration san d “

000 0505 at 220 vibration s as the alte ration of pitchper deg ree Fahrenhe it. Profe ssor MacLe od an d the write rmade the amou n t larg er , n ame ly ‘

000 125 for 1° Cen ti

g rade or‘

00007 for 1° Fahrenhe it. Profe ssor Maye r saysthat an n crease or dim in u tion of 1° Fahrenhe it leng then s orshorten s the fork by 1mm part

,thu s making the coefficien t

o f chan g e‘

00004545. The latter dete rm in ation was made byexposing a f ork in a room with an open win dow du ring f o u rdays of cold weathe r, n otin g the tempe ratu re , an d an othe rin a room heated to 70° by ste am pipe s. The se discrepan cie sare dou btless du e to the kin d Of stee l employed for makingthe forks.

1 On J ust In ton ation , p . 74 .

n ] EFFECTS OF HEAT, ETC. 131

Efie ct o r B arom e tric Pr e s s u r e .— When the barometer

rise s,the efiect on string s , wire s , and tu n ing -forks , i s to

in crease the flatten ing attribu table to the re sistan ce of the airby 3

1

3 for eve ry in ch of rise . From this cau se the harmon icoctave s are too sharp to the fu n damen tal ton e , an e rror whichmay b e referred to the efiects of the air’ s resistan ce on differen tlen gths of strin g . To take the in stan ce of the octave , thevibration s o f this will b e twice as many in a g iven time as

those of the Open string , b u t the exten t of each vibration mayb e e stimated as half ; SO that the whole Space trave lled ove rmay b e con sidered as un alte red . SO far, the refore , the re sistan ce Of the air may b e expected to produ ce the same

e ff ects on both . B u t there is an othe r e lemen t,which is that

the shorten ed lengths pre sen t proportion al ly le ss su rface tothe air. Hen ce the g reater len gths will b e more retarded thanthe smalle r

, and the in crease of retardation of the l on g e rstring s, which is the same as over-Sharpn e ss of the shorte r,may b e expe cted to b e as the diffe ren ce s in length. It isexperimen tally fou n d to b e so.

O n Fr e e R e ed s .— In the harmon ium reed the proce ss of

alteration by chang e Of temp eratu re i s far more complicated.

For whereas the reed itself is equ ivalen t to a vibratin g rodsu pported at on e extrem ity

,its vibration s slacken by its

expan sion,and by its dimin u tion of e lasticity ; on the other

han d,the air which it sets in m otion be come s less den se , an d

tran sm its the sou n d-wave with in creased vel ocity. The latte raction predom in ate s on the whole ; hen ce fre e re eds do n ot

flatten with heat as has been stated ; b u t they sharpen somu ch le ss than flu e -pipe s in an organ as to produ ce the sam e

e ffe ct. For this reason they are almost en tirely disu sed inthis combin ation . They are slightly in fe rior in this re spe ct totu n in g -forks, which , as above stated , alte r, according to on e of

the higher e stimate s, 0 125 per cen t. for 1

°

Ce‘

n tigrade , whereasfrom expe rimen ts recen tly made with a brass reed , it seemed tov ary abou t '0277 per cen t. for 1

°

Cen tigrade of temperatu re .

As stan dards Of pitch,however

,in spite o f this triflin g in fe ~

riority, they are far supe rior on accou n t of the ir more in cisiveton e , the abu n dan ce of the ir u ppe r partial ton es

,an d the

con sequ en t lou dn ess of the beats they produ ce .

On Or g an P ip e s .— The g en eral effe ct of heat on an organ

pipe is to Sharpen the n ote it em its . The compen satoryeffe ct n amed in the case of the reed occu rs in this in stan cealso , b u t to amu ch less exten t. For the pipe itself len gthen s ,e spe cially if it b e made of a very expan sile metal

,su ch as tin

or pewter. B u t the pitch of the flu e -pipe is far more dependen tK 2

132 ON SOUND. [OEArL

on the con tain ed column of air than on the en ve lope whichsu rrou nds it, and con sequ en tly the rarefaction of thepmediumtolls upon the n ote with an influ en ce qu ite predom in an t. In

wooden pipe s , the expan sion Of the tu b e IS SO very smal l as tob e en tire ly in appreciable . Small pipe s g row re lative ly sharpe rthan larg e on e s u n der the same in cremen t Of heat. Perron e t

Thompson state s that the m iddle 0 pipe Of the open diapason ,two fee t lon g , and two in che s In diam e te r

,sharpen s a comma

u n de r a rise of 10° Cen tigrade , or 18° Fahren he it,me te r be in g station ary.

Difieren t kinds Of pipes , su ch as stopped an d open diapason s,

vary n ot on ly In the amou n t of sharpen ing , b u t also in the

rate with which it take s place . Open pipe s sharpen morequ ickly than stopped pipe s , an d metal pipe s more rapidly thanw ooden on e s . The smaller pipe s are affe cted by change s of

heat soon e r than the large , as well as more .

In an organ of several stops , on su dden ly raisin g the heatfr om 10° to 15°Cen tigrade (50° to 59° Fahrenhe it), the smalle stpipe s of the metal open diapason g rew sen sibly too sharp forthe othe rs in the cou rse of half an hou r. This Sharpen ing con

tin n ed to in crease for fou r hou rs , when they we re too sharpfor the larg e st pipe s , by a qu arte r of a comma

,the in term e

diate pipe s in the meanwhile g rowin g sharper,the smalle st

first. In the stopped diapason Of wood,for f ou r hou rs there

was n o perceptible alteration ; b u t afte r that time the difier

en ces began to b e sen sible . Du rin g the se processe s, in con

sequ en ce Of smalle r alteration s in the wood,the Open diapason

sharpen e d u pon the stopped ; the g reate st differen ce du ringthe time o f the experimen t be ing at treble C', where the Opendiapason g rew sharpe r than the stopped by three-e ighths of

a comma. In the en harmon i c organ a rise in the the rm ome te rOf 10

°Cen tig rade ( 18° Fahrenhe it) raised the tu n ing C, te rmed

the “maste r pipe ,” by a comma ; a fal l In the barom e te r of anin ch

,accordin g to P. Thompson , did the same . He states

g en e rally that when the barome te r and the rm omete r m ovethe same way they act in Opposition to each other, when theym ove differen t ways they act tog e the r. Stopped pipe s are

le ss aff e cte d by barom etrical variation s than Open .

The writer has en deavou re d to obtain some more accu ratede te rmin ation s on the su bje ct Of heat , as applied to re eds and

organ -pipe s in the following man n er —A win d che st, capable‘

Of supplying a con tin u ou s stream of air at very equ ab lepre ssu re

,i s con n e cted with two spirals Of metal tubing , on e

of which Is kept con stan tly at the fre ezin g poin t o f water bybeing immersed in m e lting ice, the other at boiling poin t by

134 ON SOUND. (can .

CHAPTER VII.

SCALES, CHORDS, TEMPERAMENT, AND TUNING.

HITHERTO soun ds have been treated as in depen den t of on e

an other,as bearin g n o mu tu al re lation s , an d, except in the

case Of in te rfe ren ce , as exercisin g n o in flu en ce the on e u ponthe othe r. This view repre sen ts on ly a lim ited

,an d what may

b e termed the physical side o f acou stics. Beyon d this lie sthe chief part

,n am e ly the aestheti c or mu sical con ception Of

sou n d ; which differs e ssen tially f rom the f orm e r,in con tem

platin g vibration s as in timate ly lin ked tog ether, e ithe r In closese rie s an d su ccession ,

form ing scales and m e lodie s , or as

simu ltan eou sly e licited , and f u rn ishing the infin ite varie tie s ofchords an d harmony. It i s remarkable that whe reas m e lodyexisted in an cien t times

,an d has be en cu ltivated by all

n ation s , harmon y in an exten ded sen se,an d posse ssing any

pre ten sion to exactn e ss , is comparative ly mode rn in its orig in ,an d l im ited in its diffu sion .

It was long kn own that the rapidity Of vibration Of a stringu n de r con stan t ten sion was in verse ly proportion al to the

length of the string , that is to say, that if we halve thelen g th Of the strin g , we dou ble the n umbe r of its vibration s.To this we owe all power of playing on the violin

,and also all

kn owledg e of the re lative pitch of the n ote s in the Greek and

Arabic scale s , for which the corre spon ding leng ths of the

string were given by Eu clid the mathematician in the fou rthcen tu ry B .C. an d by Abdu l Kadir, the Pe rsian theorist of the

fou rteen th cen tu ry. Helmholtz poin ts ou t that in the mu sicof all n ation s so far as i s kn own

,alteration s of p itch in

melodies take p lace by in tervals, and n ot by con tin u ou s tran

sition s,an d he f u rthe r defin e s all me lodie s as motions withi n

extremes of p itch.

vu .] SCALES, CHORDS, TEMPERAMENT, E'

rc. 135

It ’ wil l b e seen moreover, from the remarks on qu ality, thatt he scale

,as n ow re cogn ised , was poten tial ly con tain ed in all

soun ds from the first,and can b e bu ilt u p from the harm on ics

which accompany mu sical ton e s. For the sen sation of

in terval be twe en ‘ two n ote s is n ot du e to the absolu te differen ce b u t to the ratio of the ir pitche s . In con sidering the

scale therefore w e en ter on a n ew an d in depen den t su bj e ctfrom that of pitch , an d casting aside the e lem en t of time

,

deal en tire ly with proportion .

In the case of the octave , the con n exion with the fu n damen tal ton e i s in timate : a m e lody su ng on the human voiceconveys to the b earer n ot on ly the prim es of the compoun dton es

,b u t al so the ir u pper o ctave s , and with le ss force

,the

othe r u pper partial s . Thu s w hen a voice an octave higher,su ch as that of a woman or a boy

,reprodu ce s the sam e

melody an octav e higher we“hear again a part of what we

heard before,” 1

n othing which we had n ot previou sly heard .

The same is tru e in a le ss deg re e of the twe lfth,though on ly

a smaller part i s repeated , an d in g ivin g the fif th we repeat thesam e ton e s with two n ew on e s

,the third and n in th .

This impe rfe ct repetition of the fifth cau sed the Gre eks todivide the octave in to two te trachords as follows

O B B E G A B O D EF

The se con d te trachord is a reprodu ction of the first with thetran spos ition of a fifth

,the u ppe r ( or third) borrows a n ote

from the se cond . Con se cu tive tetrachords mu st thu s,in an

octave scale , b e su cce ssive ly separate and con n e cted.

In t e rv al s .-The discove ry of the re lation existin g betwee n

the diffe ren t n ote s of the scale date s back to Pythag oras , whodivided a stre tche d strin g in to thre e e qu al parts . On stoppin gthe ju n ction of the se

,the lon g e r se ction was f oun d to g iv e

the lowe r octave of the shorte r. Then dividin g the same

strin g in to two parts bearing the proportion of 2 to 3,he

f ou n d an in te rval of a fifth be tween the two portion s. In a

similar way it may b e shown that the ratio of 3 to 4 g ive sthe f ou rth ; that of 4 to 5 the maj or third

,an d that of 5

to 6 the m in or third. By dou blin g the smal le r n umbe r,

which is equ ivalen t to takin g it an octave highe r, othe r ratios1 Helmholtz , p . 390. Ellis ’

s translation.

136 ON SOUND [CHAR

are obtain ed,nam e ly

,from 4 to 5 al tered to 5 to 8 , we have

the m in or sixth,from 5 to 6

,chan g ed to 6 to 10 3 to 5

,the

major sixth . The se are all the con son an t in terval s which liew ithin an octave . Excepting the m in or sixth

,the most

imperfe ct o f them,all are expre ssed by the first six in tegers.

Hen ce the law first en u n ciated by Pythagoras , that the

simp ler the ratio of the two p arts in to which the string is

divided,the more perf ect the harmony of the two soun ds. It

w as n ot,howev er

,u n til mu ch later

,throu gh the inve stigation s

o f Gal ile o , Newton , Eu ler, an d D. Bern ou ll i that the

n um e rical f ou n dation of the law was d isclosed,and also the

fact that the se ratios g ov ern ed all in strum en ts , as we ll as thestring s on which they had first bee n su b stan tiated. The cau seo f this preferen ce o f the ear for s imple ratios remain edu n explain ed u n til qu ite recen tly. Even Eu ler was satisfiedto believ e that “ th e human m in d had a pe cu liar pleasu rein simple ratios as be in g more easily comprehen ded .

It was re serv ed for He lmholtz to Show that the greatero r le ss smoothn e ss of the compou n d was du e to the

absen ce or pre sen ce of clashing u pper partials,which have

alre ady °b een explain ed to accompany almost every f undam en tal ton e .

Tab le of the prin cipal In tervals , with their Ratios an d Logarithms.

Example . Ratio . Logarithm .

Schisma

Comma

Minor Semiton e

Chromatic Semiton e

Major Semiton e

Min orTon eMajorTon eMin orThird

P erfect Fou rth

Perfect Fifth

ON SOUND.

Major triads with the ir combin ation al ton e s

Min or triads

The prime s are marked as min ims,the ir combin ation al

ton e s as crotchets,those betwe en prime s and first u ppe r partials

as qu avers an d sem i-quave rs . The last in the maj or triadsare practically inau dible in the harm on ium ,

though they mayb e heard in the lou der combination s o f the organ .

In the min or triads,the combin ation ton e s of the first

orde r are easily au dible,an d distu rb the harmon y , as will b e

se en by referen ce to the example .

The same law i s carried ou t in con son an t triads whichexce ed the octave

,an d tran sposition for the pu rpose of

widen ing the in te rvals affe cts the ir harm on iou sn e ss.He lmholtz g ive s as ru le s1 . Those in te rvals in which the smalle r of the two n umbers

expre ssing the ratios of the Vibration al n umbe rs is even ,are

improved b y havin g on e of the ir ton e s tran sposed an octave ,b ecau se the n umbers expressmg the ratio are so d imm lshed

,

thu s

The fif th be come s the twe lfth 2 : 6 1 : 3 .

The maj or third 4 : 5 be come s the maj or ten th 4 10

2. Those in which the smal ler o f the two ratio n umbers isodd are made worse by the same proce ss . For in stan ce the

fou rth 3 4 become s the e leven th 3 8, and so on .

v1r. ] SCALES, CHORDS, TEMPERAMENT, ETC. 139

By a fu rthe r exten sion of this beau tifu l m ethod,which

space doe s n ot here perm it, it may b e shown that the physicaltheory of con sonan ce and disson an ce leads to ru le s whichpreviou s theorie s cou ld n ot con tain

,althou gh they have

often be en aesthetically an d in stin ctive ly followed by g reatcomposers.Th e S cal e .

— It has be en shown that if any n ote which mayb e repre sen ted by C b e playe d on a mu sical in strumen t itin trodu ces

,by the harmon ic law

,two othe r allied n ote s

,

Eand G,the vibration s of which stan d to those o f the first

in the simple ratio of 4 5 6,an d f orm the harmon ic triad .

Similarly,if G

,thu s foun d, b e taken as the basis of a triad

,

it will b e foll owe d by B, D2, bearin g the same re lation . We

shou ld then hav e the f ollowing scale

First Secon d Third Fif th Seven th Eighth6In te rvals g;

9

30 6 5

$5.

the fraction s represen tin g the ratios of the vibration s of eachn ote to that n ext be l ow it. s

B u t it is obviou s that the space be tw e en E an d G ,as w e l l

as the space b etwe en G and B,re qu ire fillin g u p . D has been

sugg e ste d as the basis of a n ew triad , b u t this n ote g ive s verycomplex re su lts. If , howeve r, C2 b e taken as the u ppe re lem en t of a third triad

,the two lower membe rs of which

w ou ld b e F an d A,w e g e t an A g an d an E: with

w hich w e can comple te the scale of e ight n ote s with the

f oll owing in tervals an d vibration n umbe rs, m repre sen tin gthat of the fou n dation n ote

9 5mgm I

m.

C D E

In te rvals 31

37°

He re we have three u n equ al in te rvals on ly employed , whichare te rmed re spe ctive ly,

Maj or ton eMin or ton eMaj or semiton e

140 ON SOUND. [cu m

This form s a su fficien t an d satisfactory scale for a sin glekey. B u t as it is possible to take any othe r n ote be side s C as

the f ou n dation of a scale , term in g it the key-n ote , it become sn e ce ssary to in terpolate in termediate sou n ds betwe en thosethu s fou n d

,so as to pre serve the sam e rotation o f inte rvals .

The se are fiv e in n umbe r,situ ated betwe en the larg e r in te r

v al s,or whole ton es . They are n ot g iv en n ew n ames

,b u t

te rmed the FLATS or SHARPS of the sou n ds betw e en whichthey lie . The amou n t of the flatten in g or sharpen ing is ine ither case represen ted by the fraction

1

3

3, which is te rm ed

a chromatic sem iton e , and which differs1

somewhat from the

maj or sem iton e above g iv en .

It may he re b e n oticed that the difleren ce between the

maj or an d m in or ton e s i s n ot withou t importan ce . Thisd iffe ren ce may b e obtain ed b y in v ertin g o n e f raction an d

mu ltiplyin g it in to the othe r, which i s e qu ivalen t to div ision .

33‘ x ”

126 27

1

7 COMMA,

as this compu tation al in terval i s te rmed . An other fraction ald iffe rence may al so b e adve rted to in this place . The maj orthird f rom C to E is g iv en above as g an d the fif th as g.

If the se b e d ivided in to on e an othe r the diffe ren ce = g, whichis te rmed a m in or third. Now 53 exce eds g b y 2

212

MINOR SEMITONE.

We thu s Obtain thre e semiton es o f differen t size . Thein te rpo lated flats an d sharps may b e conven ien tly tu n ed tothe se con d or chromatic in te rval by alte ring the vibrationn umber in the g iven proportion . We have thu s the u su altwe lve n ote s of the scale te rm ed chromatic

,con stru cted o n

the Simple plan that a n ote i s sharpen ed by in creasin its

v i b ration s in the proportion 1 33, or flatten ed by d imin ishin gthem In the ratio If all the n ote s of the simple scale b ethu s treated w e obtain tw en ty-on e to the octave which are o f

su fficien t importan ce to de se rve tabu lation . The ratios are

g ive n in logarithmic form ,as this re n de rs the actu al steps

more Obviou s than they are in the ir fraction al shape .

142 ON SOUND.

mising , this difficu lty are te rm ed TEMPERAMENTS, and will b econ side re d pre sen tly.

Py th ag or e an S u l a — Pythag oras laid down a somewhatdifferen t se rie s of valu e s

,accordin g to the following scheme

c 1) E F G A B c

i 8 %i% % 8

Here the f ou rth , fifth , an d octave are iden tical with theo rdin ary system ; the maj or third

,sixth

,an d seven th are

g reate r by a comma ; while the small in te rval or sem iton e i sd im in ished by the same qu an tity. In this system the on lyn umbers that appear are 2 an d 3

,whereas in the m odern

,the

n umber 5 appears hen ce the in te rval be tween any two n ote sof the Pythag orean scale can b e expressed as the sum ordifferen ce of a ce rtain n umber Of octave s an d fifths. A

violin tu n ed to tru e fifths really plays the Pythagorean scale ,w ith power, however, of m odifying disson an t n ote s.M in or S cal e .

— It was shown that the harm on ic triad,

con sistin g of the ratio 4 5 6,may b e broken u p in to two

in te rvals , den oted re spe ctive ly b y the f raction s g. an d g, whichare termed maj or an d minor thirds The se u n equ ally dividethe con tain ing fifth

,the ratio of which is the product of the se

f raction s, or the maj or scale g ive n above the maj orthird pre cede s, an d i s f oll owed by the m in or. B u t if they b etran sposed, an d the min or third taken first

,w e en tire ly alte r

the m u sical character of the scale, an d produ ce an e ssen tiallydiffe ren t sen sorial or emotion al effe ct : whereas the maj orscale has a cheerfu l an d excitin g ten den cy, the min or , to mostif n ot to all heare rs , i s me lan choly an d pathetic. The alteredposition of the compon en t n ote s

,more ov er, still fu rther com

p licate s the c on stitu en t ratios, an d ren de rs the qu estion o f

temperam en t ev en m ore ardu ou s . The scale thu s f ormed hasthe ratios as follows

A B C D 1 E F G A

1 t i i i g. i 2

T

in which a key-n ote i s assumed a m in or third be low that of

the corre sp on ding maj or scale A,for in stan ce

,in stead of C,

and so on .

Itis to b e remarked that the note D the fou rth of the ab ove scale, is reallya comma too sharp .

V l I. ] SCALES, CHORDS, TEMPERAMENT, ETC. 143

Tem p eram e n t .—It will already have be come apparen t to

the reade r that there is an obv iou s lack of arithmetical ag reemen t between the variou s in te rvals as represen ted fraction ally.

The cau se of this lie s deep in the n atu re of n umbe rs, an d i s

we ll expre ssed b y Mr. El lis . “It is imposs ible,

” he says,in

an appen dix to his tran slation of He lmholtz’s work,

to formOctav e s by ju st Fifths or ju stThirds, or both combin ed , or tof orm ju stTh irds by ju st F ifths, becau se it is impossible bymu ltiplying any on e of the n umb ers 1

21 or g or 2, each by

itse lf , or on e by the othe r, any n umber of time s,to produ ce

the same re su lt as by mu ltiplying any othe r on e of the sen umbers by itse lf any n umber of tim es .

” The physical factmay b e otherw ise stated by sayin g that the octav e an d the

fif th are in commen su rable , j u st as are the diameter,an d

circumfe ren ce of the circle .

The simple st way Of repre sen ting this in commen su rabilityis to take a case . If the octave b e div ided in to tw e lve e qu alsemiton e s

, the fifth ou ght to b e seven Of the se b u t it wasf ou n d ou t very early in the history of mu sic that a fif th isa l ittle more than seven . It i s abou t Con sequ en tly,takin g twe lve

‘

of these fifths,they g ive rathe r more than

se ven octaves. They do n ot re tu rn to the corre spon din goctav e of the starting n ote . The diff e ren ce or departu re i sthe above fig ure mu ltiplied by 12, or 02 3460 of a semiton e .

This old discove ry i§_g en erally attribu ted to Pythag oras, andthe fig u re 023460 is te rmed the “Comma ”

o f Pythag oras.Whethe r Pythag oras deserve s the credit of the discove ry, orwhether b e imported it from Egypt, is matter of doubt b u t

,

at’

any rate,the Gre eks kn ew o f the mon ochord , o f the ratios

to b e derived from it,an d o f the div ision s o f the scale .

Eu clid wrote a work called the Sectio Canonis,or the Division

o f the String , which en te rs in to f u ll de tail s . The third of

the Greek scale was made by f ou r fifths taken u pwards,an d

i s still called a Pythag orean third. In the same way sixmajor ton e s exce ed an octav e by the Pythag orean comma.

It shou ld b e distin ctly n oticed that this discrepan cy is a

law of n atu re , n ot inhe ren t in any particu lar system ormethod,

an d en tire ly beyon d man’s con trol.

Temperamen t may b e defin ed as the division of the octave

in to a n umber of in tervals such that the n o tes which sep arate

them may be suitable in n umb er and arrangemen t f or the pu r

p oses of practical harmony. The possibility o f any othe rd ivision than that re cogn ised in the ordin ary pian o an dharmon ium will b e’

n ew to man y reade rs f or the u su al f ormof keyed in strumen t is so eng raven on ou r min d that most

144 ON SOUND. term .

pe rson s are u naware that an y othe r arran gemen t exists. Thecommon in strumen t has of cou rse its own system o f tem

peramen t, on e that,thou gh n ot the olde st

,is certainly the

simple st, an d which is u su ally te rm e d equ al.Equ al t em p e ram e n t aims at dividin g the octave in to twe lve

equ al parts or sem iton es . If it so happen ed that the octavecou ld b e divide d thu s, and the other in te rvals

,su ch as the

fifth an d third,re tain ed in tun e

,it wou ld b e a g reat b e en .

Un fortun ate ly n atu re has n ot so ordain e d it.The attempts to remedy this in heren t in commen su rability

o f the m u sical scale have been n umerou s an d varied,some

datin g b ack to an cien t t ime s,others of ve ry m odern con stru c

tion . The ir prin cipal varietie s may b e g iven best in a tabu larf orm. Th ey have chiefly b e en applied to keyed in strumen ts

s u ch as the organ , pian o,

an d harmon ium ,where the ir

n ece ssity is mos t fe lt.

Table of Temp eramen ts .

Systems retain ing the ordin ary keyboard .

a Un equ al or mean -ton e tempe ramen t.b Equal temperamen t .

2. The ordin ary keyboard with addition al keys .

a Han de l’s Fou n dling organ .

b The Old Temple organ .

0 The dig ital s of the con certin a.

3 . Addition al keyboards.He lmholtz’s harm on ium with Gu ero ult’s modification s .

4 . Addition al keys an d keyboard .

Perron e tThomp son ’ s e n harmon ic organ .

5. The ordin ary keyboard w ith combin ation stops.Mr. Alexan der Ellis’ s harmon ium .

6 . En tire ly n ew arran g emen t of keyboard.

a Poole ’s system.

6 B osanqu et’s g en e ralised keyboard .

0 Colin Brown ’ s voice harm on ium .

l . (a) The old u n equ al or MEAN-TONE SYSTEM was an

attempt to get the more common scale s faIrly accu rate,

146 ON SOUND. [CHAR

( b ) According to the EQUAL TEMPERAMENT the octave isd ivided in to tw e lve pe rfe ctly s im ilar in te rvals, sev e n of whichare taken f or the fif th

,althou gh its real measu re i s 731, o f

the se . It i s thu s somewhat flatten ed , from to

by the in te rval te rme d a Schisma 49,though le ss so than in

the olde r system,which low e rs it to as will b e se e n

from the tab le . On the othe r han d the third is far too sharp,

or n early two-thirds o f a comma,in stead of as

it stan ds corre ctly in both the othe r column s. The sixth,

more ove r,i s ren de re d extreme ly sharp in equ al tempe ramen t

,

n ame ly or e ight schismas,as again st the tru e

The sev e n th is flatten ed in the old m ore than two schismas ,

and con side rably sharpen ed in the e qu al method,by exactly

six schismas . The f ou rth of the scale is le ss alte red in proportion to its sen sitiven e ss , be ing raise d rathe r m ore than tw o

schismas in the old , an d on ly on e in the e qu al system . These con d of the scale stan ds in a pe cu l iar position

,b e in g a

dou ble n ote . The old tempe ramen t place s it abou t halfwaybe twe en its g rave and acu te forms

,whe reas the e qu al method

remove s it n in e schismas above the g rav e f orm ,thu s con

stitu ting the larg e st departu re from accu racy to b e met

with .

(2, a) Even as early as the time of Han de l the advan tag eto b e de riv e d f rom addition al keys was obv iou sly appre ciated

,

fo r it i s kn own that he presen te d to the Foun dlin g Chape l ano rgan thu s fu rn ished. (b) The Orig in al organ in the TempleChu rch, bu ilt by Fathe r Smith in 1688

,posse ssed f ou rte en

sou n ds to the octave in stead of twe lve , the Ab an d G# as we llas the Eb and Di; be in g distin ct an d d iv ided .

‘ The keysthem se lve s we re spl it across in the m iddle

,the back halve s

risin g above the f ron t portion s,so that the fing e r cou ld b e

placed on e ithe r at the playe r’s discre tion . The ran ge of

g ood keys on the u n equ al system was thu s mate rially ex

ten ded .

2

(4) The dev ice o f addition al keys was,howeve r

,

carried to its f ul le st deve l opmen t by Colon e l Pe rron e tThompson ih his en harmon ic organ , which may still b e se en at the

Sou th Ken sin g ton Mu seum . He u sed the larg e n umbe r of .

sev en ty-two to the octave , which we re fu rthe r d istribu ted o n

thre e differen t keyboards,b u t which also differed am on g

themse lve s in colou r, shape , appearan ce , an d in n ame . Beside s

1 See Cu rwen’

s Tract on Mu sical Statics, pp . 11 , 103 .

2 ( c) The same con trivan ce has b een app lied to the ju st English concertina,which is t n ed to the mean -ton e temperamen t, With duplicate stu ds for I)?ab an d G Ab.

SCALES, CHORDS, TEMPERAMENT, ETC. 147

Fig . 52—Perron et Thompson’

s Keyb oard.

148'

ON SOUND. [CHAP.

the ordin ary dig ital s there w e re others te r med F lu tals, Quarrills

, an d B u ttons. By this me an s,t hough stil l retain ing the

ordinary arran gem en t of the keyb oard , he was e n abled to

produ ce accu rate ly twen ty-on e scale s w ith a m in or to e ach o f

them . He employs a cycle of fifty-thre e sou n ds, of which

he u se s abou t forty,the f u l l cycle be ing discon tin u ed at a

ce rtain poin t .

(3 ) The difficu lty o f adding n ew sou n ds withou t u n du eme chan ical complication has be en attacked in a diffe ren t w ayby He lmholtz . The keyb oards are in this case in creased totwo

,so as to Obtain tw en ty-f ou r in stead of twe lve n ote s to

the octav e . They are of half the u su al depth,placed on e

ab ove the othe r,as in the organ . This has always se em ed to

the write r a practical an d simple system . The in strum e n tmade for He lmholtz w as so tu n ed that all the maj or chordsf rom Fb to Fitcou ld b e played o n it. On the lowe r man u alw e re the scale s f rom Ob maj or to G ,

an d on the u ppe r thosef rom Eb maj or to B maj or. To m odu late b eyon d B maj or ono n e side an d Cb majOr o n the othe r it was n e ce ssary to makethe en harmon ic chan g e be twe en the se two n ote s

,which per

ceptib ly a lte rs the pitch by the in terval of a comma, 35a

The min or m ode s on the lowe r man ual were B or Cb m in or,On

‘t e u pper D# or Eb m in or.(4) The same idea has be en carried ou t w ith sl ight variation

in an in strumen t shown at Sou th Ken sin g ton , n ame ly Gu ero u lt

’s m odification o f He lmholtz ’s harmon ium

,of which the

f ollowin g i s the make r’s own de scription .

This in strumen t has a f ron t and back keyboard, each div ided in to twe lve semiton e s , l ike that of a pian o

,an d each

posse ssing fiv e octave s . They are both tun ed to tru e fifths,b u t the back keyboard is throu ghou t a comma flatter than thefron t

,which is o n the n ormal diapason . The black keys on

e ach keyboard the refore do du ty for a flat an d a sharp,b u t n o t

in the same seri e s . On the fron t keyboard, for in stan ce , Ebrepre sen ts the D# of the back. Con s ide red as flats , the blackkeys of the se con d keyboard repre sen t sharps of a thirdboard which wou ld b e tu n ed a comma low e r than the se cond .

By thu s f u sin g the flat of on e se rie s w ith the sharp of the

othe r, an e rror is c omm itted equ al to the in terval which

is at the extreme limit of au dib le . diffe ren ce s .

On the fron t keyboard , starting from the followin g n ote sare tu n ed to tru e fifths

,so as to g iv e n o heat whatev e r :

A’E ; E) B i D’ A i

l

Gr

'

D i C) G i F1 0 3 Eh} F ; Eb: Eh The

1 In Helmholtz, gag.

150 ON SOUND. [CHAR

the fron t se rie s of the in strum en t are damped, an d the corresponding vibrators of the back se rie s come in to ac tion , u n tilthe n ote s speaking are those of the old-fashion e d man u al .Betwe en the se extreme s any requ ired combin ation of n ote scan b e produ ced , f rom seve n flats to seven sharps

, accordingto the keys employe d . This me thod, which e n tire ly remove sthe difficu ltie s of complex fing erin g , has the d isadvan tag e Of

requ iring a con stan t alte ration of ste ps, which in tran sitorymodu lation s is occasion ally laboriou s .

(6) The last class of con trivan ce for produ cing tru e in ton ation doe s away with the ordinary form of keyboard altog e the r. It is impossible he re to g ive fu ll details of the sein strumen ts

,which practically in trodu ce a n ew prin ciple in to

mu sical exe cu tion . Poole ’s,B osanqu et

’

s,a n d Colin Brown ’ s

f orms may b e take n as typical repre sen tative s of many le ssperfe ct device s . In all

,the se rie s of ton e s are arran g ed

diag onally on e beyon d an othe r, so “ that the form of a chordof g iven key re lation is the same in eve ry key. B u t the n ote sare n ot all symmetrical, and the same chord may b e stru ck ind iffe ren t f orm s accordin g to the view which is taken of itskey re lation ship .

” They the refore posse ss the g reat advan tag eo f similarity of man ipu lation

,although this is qu ite diffe ren t

f rom that ordin arily tau ght. It w ou ld appear,howeve r

,that

the new systems are far f rom diflicu lt to learn by any personwho has ob tain e d some e xpe rien ce on the Olde r f orm of

in strumen t .(a) The first attempt in this dire ction was made by H . W.

Poole, o f Sou th Dan ve rs, Massachu se tts , U.S. The in stru

men t appears to have been con stru cted, an d i s de scribedin Silliman

’s J ou rnal for 1850 . His organ was in tended to

con tain 100 pipe s to the octave , and the scale to con sistof ju st fifths and thirds in the maj or chords , an d also then atu ral or harmon ic sev en ths . The arran gemen t o f keysis be st g iven by a diag ram extracted f rom Mr. B osan quet

’

s

work.

B osanqu et, op . cit. p. 45.

VIL ] SCALES, CHORDS, TEMPERAMENT,ETC. 15 1

152 ON SOUND. [0m m vu z

According to Mr. B osan qu et’

s n otation here u sed,n otes

are arran g ed in se rie s in orde r of su cce ssiv e fi fths . Eachserie s con tain s twe lv e fifths from F# u p to B. On e se rie s isu nmarked . It con tain s the standard C. Each n ote o f the

n ext se rie s o f twe lv e fifths u p is affe cted with the markwhich is called a mark o f e levation

,an d is drawn u pwards in

the dire ction of writing . The n ext serie s ‘

has the markand so on . The serie s be low the u nmarked serie s is affe ctedw ith the mark which is cal led a mark of depre ssion , and isdrawn downwards

,in the dire ction of writing ; the su cceed

ing serie s is marked and so on . Whe re , as in Poole’s key

board,pe rfe ct thirds are tu n ed indepen den tly o f the fifths

,

they are he re repre sen ted by the n ote e ight fifths distan t inthe serie s ; this is a close approximation to the perfe ct third ,according to a re lation which has been called He lmholtz’

s

The orem . Thu s C— \Em ean s a pe rfe ct Third ; \E— \ G is

also a pe rfe ct third ( chord o f domin an t of \A m in or). he

place s of harmon ic sev en ths are marked by circle s

( b) B osanqu et’

s Gen eralised Keyboard.

In the’

enharmon i c harmon ium exhibited at the Loan Colle ction o f Scien tific In strumen ts

,Sou th Ke n sing ton , 1876 ,

there was a keyboard which can b e employed with all systemsof tu n ing redu cible to su cce ssion s of u n iform fifths ; fromthis prope rty it has be en called the g en e ralised keyboard . It

w ill b e conv en ien t to con side r it first w ith refe ren ce to perfectfifths. The se are actu ally applied in the in strumen t in qu estionto the division of the octav e in to fifty

-thre e equ al in te rvals,

the fifths of which system diff er from perfe ct fifths by le ssthan the thou sandth part o f an equ al tempe ramen t semiton e .

Itwill b e remembe red that the e qu al temperamen t semiton ei s the twe lfth part o f an octave . The lette rs E. T. are u sedas an abbrev iation for the w ords “

equ al temperamen t.”The arran g emen t o f the keyboard i s based u pon E. T.

position s taken f rom left to right, and deviation s or departu re s from those position s taken up and down . Thu s then ote s n early on any leve l are n ear in pitch to the n ote s of an

E. T. serie s ; n ote s highe r up are highe r in pitch ; n oteslow e r down l owe r in pitch .

The octave i s divided left to right in to the twe lve E. T.

division s,in the same way, and with the same colou rs

,as if

the broad fron ts o f the keys of an ordin ary keyboard we reremoved

,and the backs left .

Proceedings of the Musical Association , 1874-75, p . 14.

154 ON SOUND. [CHAR

The deviation s f rom the same leve l follow the serie s o f

fif ths in the ir steps of in crease . Thu s G is placed on e -fou rtho f an in ch fu rthe r back , an d on e-twe lfth o f an in ch highe r,than C ; D twice as mu ch

,A thre e time s

,an d so on

,till we

come to /C, the n ote to which w e re tu rn afte r twe lve fifthsu p ; this n ote is placed thre e in che s f u rthe r back

,and on e

in ch highe r, than the C from Which we started.

With the system of pe rfe ct fifths the in te rval C C is aPythag orean comma. With the same system

,the third

determ in ed by two n ote s e ight steps apart in the se rie s o ffifths (C E) is an approximate ly pe rfe ct third . With thesystem of fifty

-thre e the state of thing s is ve ry n early thesame as with the system of perf ect fif ths .The prin cipal practical simplification which exists in this

keyboard arise s from its arran g emen t be ing strictly accordingto in te rvals. From this it follows that the position -re lationof any two n ote s fo rm ing a g iven in te rval i s always exactlythe same it doe s n ot matte r what the key re lation ship is, orwhat the n ame s of the n ote s are . Con sequ en tly a chord of

g iven arran g emen t has always the same fo rm u n de r thefinge r an d

,as particu lar case s , scale passag e s as we ll as

chords have the same form to the han d in whatev e r key theyare played

,a simplification which g ive s the beg in n e r on e thin g

to learn , where as the re are twe lve on the ord in ary keyboard .

The keyboard has be en explain ed above w ith referen ce tothe system of pe rf e ct fifths and all ied systems ; b u t there i san othe r class o f systems to which it has spe cial applicabil ity ,the mean -ton e an d its kin dred systems. In the se the third ,made by tu n ing fou r fifths up, is perfe ct or approximate lype rfect. The m ean -ton e system is the old u n equ al tempe ramen t. The defe cts of that arrang emen t are g ot rid of bythe n ew keyboard

,an d the fin g e rin g is remarkably easy.

The u nmarked n atu rals in the diagram pre sen t the scale of 0

when the mean -ton e system i s placed on the keys. ‘

( 0 ) Co lin B rown’

s Natural F ingerboard with P erfect Intona tion .

The dig itals con sist o f three separate sets, of which thosebe long ing to fou r re lated keys , repre se n ting the n ote s 2

,5,1,

4,are white ; those be long ing to three re lated keys

,and

I For further details see An Elemen tary Treatise on Mu sical In tervals andTemperamen t, with r

an accou n t of an En harmon ic Harmon ium exhibited in the

Loan Collection of Scien tific In strumen ts , Sou th Kensin gton ,1876 ; also of an

En harmon ic Organ exhib ited to the Musica l Association of Lon don , May , 1875 , byR. H. M. B osanquet, Fello wof St. J ohn

’

s College , Oxford. (London : Macmillan ,

vn . ] SCALES, CHORDS, TEMPERAMENT, Ere . 155

repre sen ting 7 ,‘

3,6,are colou red ; the small rou n d dig ital s

repre sen t 7 minor, or the maj or seven th of the m in or scale .

The se are the sam e in all keys .This fin gerboard can b e made to con sist of any n umb e r o f

keys. The scale s ru n in the u su al orde r in dire ct lin e horizontally f rom le ft to right along the fin g erboard .

The keys are at right an g les to the scale s. and ru n vertical lyacross the keyboard

,from \Cb in the f ron t to /C# at the back,

C be in g the cen tral key .

The scale to b e playe d is always f ou n d in dire ct lin e horizon tally betwe en the key

-n ote s marked on the finge rboard ,b u t the dig ital s may b e tou che d at any poin t .

The orde r of su cce ssion i s always the same,an d oon ss

qu en tly the prog re ssion of finge ring the scale is iden tical ine ve ry key .

The first,se con d

,fou rth

,an d fif th ton e s of the scale are

playe d by the white dig itals , the third, s ixth , an d seven th bythe colou re d .

0

The sharpen ed sixth an d seven th of the mode rn m in orscale are played by the rou n d dig itals . The rou n d dig ital ,two rem ov e s to the le ft as in the key of B flat

,is re late d to

that in the key of C as 8 : 9, and su pplie s the sharpen edsixth in the relative m in or o f C ; so in all keys Sim ilarlyre lated .

Playing the scale in each key the f ollowing re lation s appear(see diagram on p . 157)From white dig ital to white , say from the first to se con d

an d fou rth to fifth of the scale , an d from colou red to colou red,

or f rom the s ixth to the seven th of the scale,the re lation i s

always 8 9.

From white to colou red , be ing from the se con d to the third,and from the fifth to the sixth of the scale

,9 10 .

From colou red to white , be ing f rom the third to the fou rth,and f rom the sev en th to the e ighth of the scale

,15 16 .

From white to white,or co lou red to co loured , is always the

maj or ton e, 8 9.

From white to co lo u red is always the min or ton e, 9 10 .

From co lou red to’

white, the diaton i c sem iton e , 15 16.

The rou n d dig ital i s re late d to the colou red which su cceedsit as 15 16 , and to the white which pre cede s it as 25 24

,

b e ing the impe rf e ct chromatic sem iton e .

Lo okin g across the fing e rboard at the dig ital s endwise, fromthe e nd of e ach white dig ital to the e n d of each colou re d immediate ly above it , in dire ct lin e , the re lation is always 128 135 ,o r the chromatic sem iton e ; and from the e nd of each colou red

156 ON SOUND. [CHAIn VII.

“

dig ital to the white immediate ly above it,in dire ct lin e

,the .

comma is f ou n d,80 81.

Betwe en all en harmon ic chang e s , su ch as be tw e en A flat404§gx to G sharp 405,the in terval of the schisma always occu rs

,

the diffe ren ce b e ing 37 .

The se . simple in te rvals an d differen ce s , 8 9,9 10 , 15 16

,

24 25, 80 81, 128 135,an d comprise all

the mathematical an d mu sical re lation s of the scale . The large rin te rvals of the scale are composed of so many of 8 9

,9 10

an d 15 : 16 , added tog e the r. The “ comma of Pythag oras ,being a comma an d schisma added toge the r, is fou n d be twe enevery e nharmon ic chan g e of key , as f rom Kit, to B

,or twe lv e

remove s of key.

The dig itals rise to higher levels at each en d,diffe rin g b y

chroma an d comma,or comma an d chroma

,altern ate ly. This

cau se s separate leve ls on the fing e rboard at each chan ge o f

colou r. Thou gh the se are n ot e ssen tial,they will b e fou n d

ve ry u sefu l in man ipu lation,an d se rve readily to distingu ish

the diffe ren t keys.The two l on g dig itals in each key are tou ched with great

con ven ien ce by the thumb. The lowe r en d of each colou reddig ital always repre sen ts the seven th in its own key , and theb orrowed

,or . chromatic sharp ton e

,in every o the r ; thu s the

seventh in the key of G is the sharpen ed fou rth, or F sharp ,in the key of C an d so in re lation to eve ry other chromaticsharp ton e .

The white dig ital i s to ev e ry colou red dig ital as its chromaticflat ton e thu s the fou rth in the key o f F is B flat

,or the flat

Seven th in the key of C so in re lation to eve ry other chro

matic flat ton e . In this way all chromatic Sharp an d flat ton esare perfe ctly and conven ien tly supplied withou t en cumberin gthe fing erboard with any extra dig itals , su ch as the blackdig itals on the ordin ary keyboard, the scale in each key borrow ing from those re lated to it every poss ible chromatic ton ein its own place , in pe rfe ct in ton ation . The tu n in g is remarkably easy

,and as Simple as it i s perfect.

It wi l l be ob served that each‘lin e of digital across the fingerb oard

b ears on e gen eric n ame , as Oh, 0 , C ,C#, Cit, So with every

other~ lin e .

ON SOUND. [CHAR

While all the maj or keys u pon the fin g e rboard, according to )

its ran ge , hav e re lative m in ors , the followin g , [\ B b, \F , \C, \Gr,\ D,

A,E,B,Flt, C#, G#, an d D#, can all b e played both as

major and as pe rfe ct ton ic min ors.The se se con dary keys are m ore than appear at the first

in spe ction of the fing e rboard . A serie s of rou n d dig ital splaced u pon the white

,an d a comma highe r, addition al to those

p laced u pon the colou re d dig itals , w ou ld supply the scale ineve ry f orm the m ost exacting mu sician cou ld de sire

,b u t it is

a qu e stion if su ch extreme exten sion s are e ithe r n e ce ssary orin tru e key re lation ship , an d whethe r simplicity in the fin g e rboard is n o t more to b e de sire d than anymu ltiplication of keyswhich involve complexity an d con fu sion .

‘

A d ap tat io n o f Tru e In t o n ation t o th e Or ch e s tra .— The re

can b e n o doub t that the place in which the superiority of

tru e ove r tempe red in ton ation is m ost fe lt,an d in which it

cou ld b e m ost easily attain ed , is the orche stra. Un fortu n ate lythis is exactly where it has been m ost thorou ghly n eg le cted .

To a ce rtain deg re e it is in stin ctive ly an d u n con sciou slyprodu ced the strin g ed in strumen ts hav ing the powe r o f

m odifyin g the ir n ote s accu rate ly by ear accordin g to the

requ iremen ts of a particu lar chord or m odu lation . Evenhe re

,howeve r

,the re su lt is marred by the e rron e ou s practice

o f man y violin i sts. B u t in the wind-in strumen t departmen tthis powe r is limited to the sl ide trombon e s

,an d the corre

spon ding trumpet. It i smu ch to b e regretted that the n atu ralan d pe rfe ct qu arte tt form ed by the slide tr umpet, alto , ten or,an d G bass trombon e s shou ld b e so mu ch disfig u red at the

pre sen t day, the alto be ing almost always replaced by a se con dten or

,the bass of ten omitted or tran sfe rred to the e uphon ium

,

an d the in im itable trumpet spoiled by the com et. B u t beyondthe se two type s n o e n deavou r whateve r se ems to hav e be e nm ade towards tru e in ton ation . Flu te s

,oboe s

,clarin ets

,an d

bassoon s , as made an d supplied to the ir re spe ctiv e playe rs , arerare ly in tun e at all

,even to themse lve s, an d , at most, pre sen t

a fe eble approach to e qu al tempe ramen t . There i s n o in

supe rable obstacle in the w ay o f the ir all be ing so man u

factu red an d tu n ed as to g ive at least the prin cipal en harmon icdifie ren ce s , by u til ising du plicate fin g erin g s fo r the same n ote ,w hich already exist on all of them .

I A fu ll description of the voice harmon ium may b e fou nd in the sp ecificatio nfor paten t . The prin ciples u pon which it ' is con stru cted and tu n ed will b e

fou n d fu lly stated in Mu sic in Common Things parts i. and ii. , pu b lish ed b yMe ssrs . William Collin s , Son s , an d 0 0 Glasgow,

an d Bridewell Place , NewB ridge Stre et, Lon don , an d the Ton ic So lfa Agen cy, 8, Warwick Lan e , E.C .

VIL ] SCALES, CHORDS, TEMPERAMENT, ETC. 159

The write r has himse lf dire cted the con stru ction of a

clarin et an d bassoon which, w ithou t any appre ciab le in creaseof me chan ism

,an d w ithou t the slighte st alteration of the

system of fin ge rin g u su ally adopted,can b e made to g ive

n in e te en n ote s to the octave , a.

n umbe rm ore than su fficien t toprovide for all the commatic diffe ren ce s in v olv ed. The clarin et,thre e f orm s of which are comm on ly u se d in the orche stras‘len ds

itse lf part icu larly w e l l to su ch an arran g em en t, from the factthat it is rare ly if ev e r requ ired to play in high flat or sharpkeys. Those actu ally in u se can there fore b e tu n ed accu rate ly,an d a duplicate n ote provided in case of su dden an d temporarym odu lation s .It is

,how ever

,e ssen tial that the pre sen t mu sic shou ld b e

carefu lly g on e ove r in score by a competen t harm on ist, an dthe m od ified n ote s marked in to the band parts acc ording toMr. B osan qu et

’

s or any equ ally good n otation,w ith a mark of

e le vation or depre ssion,according to the ir spe cific key re lation

ship . An orche stra in which pe rfe ct in ton ation w e re thu sse cu red wou ld in stan tly obtain what is ve ry obviou s inlisten in g to keyboard in strum en ts of corre ct scale

,su ch as

that of Mr. Colin Brown,n ame ly

,a larg e ly in creased volume

of ton e in proportion to the in strumen tal re sou rce s empl oyed .

An exce llen t adaptation of the en harmon ic prin ciple to brassv alved in strumen ts has be en dev i sed byMr. H . Bassett,in what he te rms the COMMA an d TELEOPHONIC trumpets . In

all in strum en ts fu rn ished w ith the ordin ary valve s the re are

g reat fau lts in in ton ation .

“It is n ot difficu lt,” says Mr. B as

sett, “to Show , by calcu lation from the varying len gths of

tu b e brought in to action by the valv e s,that man y of the

in te rvals re su ltin g from the ir combin ation are n ot in accordan ce with the ju st or tempered scale s . The u n fortu natepractice of tran sposing parts written in w ide ly diffe ren t keys,so as to u se on ly on e or two crooks

, g reatly in crease s the see rrors , be sides sacrificing the ben efit o f the n atu ral in terv als

,

an d distin ctiv e qu ality of ton e of the diffe ren t crooks .”He first con stru cte d a valv e trumpe t in which the se fau lts

o f in ton ation shou ld b e avoided . In it the first an d se con dv alve s remain as u su al ; that i s to say they lower the pitch bythe in te rval s of a maj or ton e an d a d iaton ic sem iton e re spe ctiv ely. The third valve raises the pitch o f an y n ote produ cedo n the first valve by the in te rval o f a comma

,or

, in othe rwords, the first an d third valve s together low e r the pitch a

m in or ton e . This system of valve s,which is also applicable

3 Proceed ings of the Mu sical Association ,

'

1876-77 . p . 140.

160 ON SOUND. [CHAP J

to the Fren ch horn , e nable s the playe r to produ ce a practicallype rfe ct diaton ic scale in the ton ic , dominan t, an d subdom in an tkeys

,w ith the advan tag e o f having on ly two valve slide s to

tun e whe n chan g ing the crook,the alte ration the oretically

requ ired in the third or comma valve ” be ing so small as tob e in appre ciable .

In his se cond , which he te rm s the “teleophon ic in stru

men t, he retain s the orig in al slide , thu s keepin g the pow e r ofadju sting each n ote to accu rate in ton ation ; b u t he adds a

sing le valve tu n ed in u n ison w ith the open D,or harmon ic

n in th— in othe r w ords,lowe ring the pitch a m in or ton e . This

valve is worked by the forefin ge r of the left han d , the in strumen t be in g he ld exactly in the Usu al man n e r. B y the u se o f

this sing le valve and the slide , separate ly or tog ethe r, it ispossible to produ ce a complete scale , maj or or m in or

,with a

pe rf e ction of in ton ation lim ited on ly by the skill o f the playe r.The v alve n ot on ly su pplie s those n ote s which are false ore n tire ly wan ting on the ordin ary slide trumpet , in clu ding the

low A flat and E flat on the higher crooks, b ut it g reatlyfacil itates tran sposition and rapid passage s.

162 ON SOUND [CHAR

con tin u ou s m e lodic re citation,bearin g spe cific sign ifican ce . in

particu lar form s an d phrase s . Fam iliar example s are the falling in fle ction of the voice at the en d of a phrase

,an d the

risin g caden ce of in te rrogation .

Polyphon i c mu sic appeared m the form of discan t,in which

d ifferen t v oice s,each proceeding in depen den tly an d Sin g ing

its own m e lody,had to b e u n ited in su ch a way as to produ ce

e ithe r n o disson an ce,or on ly tran sien t disson an ce s

,which we re

readily re solved . In this way can on ic im itation arose , as earlyas the twe lfth cen tu ry. The old e ccle siastical m ode s werere tain ed

,an d in 1547 Glarean u s distingu ished tw e lv e of them

,

Six au then tic,an d six plagal , assign ing to them ,

som ewhat in correctly , the Gre ek n am e s

,Ion ic

,Doric

,Phryg ian , Lydian ,

Mixolydian,an d E olic

,by which they have been kn own

S1n ce .

Harm on ic or modern mu sic i s marked by the in depen den ceof its con stru ction

,and the artistic con n e ction of its parts ;

w ith this system it has be come possible to compose works of

g re ater exten t,

and more en e rge tic expre ssion than itsprede ce ssors . Its e ssen tial law i s that “the whole mass of

ton e s, an d the con n e ction of harmon ics mu st stan d in a close

an d perceptible re lation ship to some arbitrarily se le cted ton icand that the mass of ton e which f orm s the whole compositionmu st b e dev e loped from this ton ic

,an d mu st fin ally retu rn

to it.N o tatio n o f M u s ical To n e s .

—Sev eral system s of n otationhave be en proposed with a view to distingu ish the difieren t

octave s from on e an othe r. That adopted by He lmholtz,an d

common ly u sed in Ge rmany,begin s with the g reat or 8-foot

octave,the C of which wou ld b e g iven by a pipe of this

size,or by the l owe st string of the Violon ce llo . This is marked

by capital le tters as follows

The n ext octave above this is te rmed the small or u naccen tedoctave of 4 fe et pitch distin gu ished by small le tters .

v111 . ] SPECIAL APPLICATIONS TO MUSIC. 163

The 2-foot octav e has small letters marked with a sing leaccen t thu s , 0

’d

’e

’

f g’

a’ b

’

an d is te rmed the on ce -accen tedoctave . The l -foot octav e has two of the se accen ts c”

and i ste rm ed the twice -accen te d octav e .

Be low the g reat octave i s the 16 foot or con tra octave ,distingu ished b y capital lette rs with an in v e rted accen t be l owthem

,thu s 0

,D

,E, E,

G,A

, B ,an d be low this again the still

d e eper,or 32-foot octave , which is marked by capitals With

two inverted accen ts D,E

,F,,G,A

Mr. Elli s su gg e sts an other an d very conv en ien t n otationf ou n ded on the pow ers of 2.

“All mu sician s ,” he says , “are

fam il iar with the octave , an d accu stomed to div ide the wholeran g e of mu sical sou n ds thu s . This amou n ts to se lecting a

se rie s of ton e s on the prin ciple of con tin u ally mu ltiplying the

corre spon din g n umber of vibration s by 2. Arithmetically w e

are therefore bou n d to beg in with le ss than 2 vibration s in a

se con d,which

,n ot be in g mu ltiplied b y two at all

,may be said

to comm en ce the zero octave, and the simple st su ch n umbe rthat can b e selected is 1 itse lf . Then from 2 u p to 4 w e

mu ltiply the form e r n umber of vibration s on ce by 2 an d hav ethe first octave . At 4 or twice 2 w e mu ltiply twice by 2

,and

have the se con d octave . The octave s will corre spon d withthe n umber of vibration s with which they beg i n, thu s

8,

11, 12.

Vih. 1 , 2, 4 , 8, 16, 32, 64 , 128, 256, 512, 1024 , 2048, 4096 .

The n umber of the octave is the in dex of the power of 2,

g iv ing the n umber of vib ration s with which the octave beg in s.Thu s 256 28 beg in s the 8th octave .

Su pposing the n ote s 0 , D ,E to occu r in any octave

,the

n umbe r of the octav e is prefixed . Thu s 8Em ean s Ein the8th octav e . The difficu ltie s arising from u sing differen tstan dards of pitch an d temperamen ts are m e t b y u sing some

fraction as 1 in place of 1 for the in itial n umbe r of vib ration s. B u t wat is common ly called the the oretical pitch ,80 256 vib i s the on ly on e sugg e sted by pu re arithm e tic .

This simple con trivan ce for marking the octave obviate sa vast n umber of difficu ltie s .4G 16 vib . mean s, 0 in the su b con tra octave , or

double u n der-accen ted or u n de rl in ed g reat octave or C,or twice in dexe d g reat octave 0 2, or twice n egative lyindexed g reat octave 0 - 2,

or C ”,or 32-foot octave

,or

COG octave .

104 ON SOUND. [CHAR

5 0 32 v ib . m ean s 0 in the con tra octave,or an octav e

ab ov e all those n am ed,or 0 0 .

6 0 64 v ib . me an s 0 in . the 8-foot octave , the l owe stn ote of the Violon ce llo .

7 0 128 vib . mean s 4-foot C or small c,the lowe st n ote

of the viola or ten or violin .

8 0 256 vib . m ean s 0 in the on ce accen ted or 2-footoctav e

,or c

’

, or c ov erlin ed, or cc w ith . two small o

’

s com

m on ly called “m iddle be ing on the ledg e r l in e betwe enthe b ass an d treble stave s .

V arie t ie s o f Or g an p ip e s .-Pipe s are employed in the organ

which aptly illu strate many of‘

the prin ciple s previou sly ”

men tion ed . As regards material , they are e ithe r, of me tal orof w ood ; the f orme r be ing composed of tin

,with a g reate r

or le ss adm ixtu re o f lead , some times rwhen .ve ry larg e theyare made of zin c ; ' the latte r of pin e w ood , cedar o r:mahogany.

In both case s they are divided in to open and' stopped , as

before de scribed , w ith an in term ediate form . te rmed halfstopped ; the stopped pipe Speaking an octave low e r than an

open pipe of’

the same len g th. B u t the m ost importan tclassification i s in to flu e or‘ fiu te pipe s , reeds , and m ixtu re s .

The two first have al ready be e n fu lly adv erted to ; the m ixtu restops

,in stead of a sing le pipe s ou ndin g to each n ote

,possess

sev eral,from tw o or thre e

,up to a mu ch larg er n umbe r, su ch

as twe lve or ev e n -fou rteen . Thes e are arran g ed in tie rs u ponthe same supply of w ind , an d thu s all sou n d tog ethe r. Theyare tu n ed to the in div idu al ton e s of the harmon ic se rie s g iv enabove

,or to some of the ir octav e s , su ch as the third

,fifth ,

an d e ighth of the fou n dation -ton e . The se squ ialte ra, on e of

the m ost u su al m ixtu re s , was orig in ally a stop of two rankson ly

,composed of the twe lfth and seven teen th in te rvals abov e

the g rou n d -ton e,thu s giv in g prom in en ce to the third an d

fou rth u ppe r partials. The se squ ialte ra is n ow often made withthree u p to six ran ks o f open m etal pipe s . The m ixtu reprope r is more shrill an d acu te , comprising the seven te en th

,

n in ete en th , twen ty-se con d , tw en ty-Sixth,and tw en ty -n in th .

As in the treble , the pipes bec ome very small,keen

,an d

prom in e n t,the smal ler ranks are discon tin u ed ab ou t m iddle

0 , an d larg er pipes , sou n ding an octav e l ow e r,are Substitu ted .

This alte ration is called a break,an d take s place also in the

se squ ialte ra The obje ct of su ch stops is obv iou sly to in creasethe brillian cy o f e ffe ct b y re in forcing the u ppe r partial ton e swhich He lmholtz has of late years shown to b e alwayspre sen t in sou n d of a m e l odiou s qu an tity. It is v ery remarkable that the fact itse lf had been long ago discovered as a

166 ON SOUND [CHAR

board of its own ) , in which the prin ciple of re con stru ctin g a

bold qu al ity of ton e from its affiliated harmon ics was earlycarried to a high deg re e of deve lopmen t . In deed this cu riou soccu rren ce ju stifie s the remark in the in trodu ctory chapte rthat in strumen tal applian ce s in ten ded orig in ally for eff e ct, an dd iscovered fortu itou sly ,

'

have in the en d proved importan t corrob oration s of scien tific research . In the organ in the chu rch o f

St. Sophia at Dre sden bu ilt by Silbe rman n abou t 1750 , there arebe sides a com e t of five ranks

,a cymbe l of sim ilar charac

ter of three ranks, and a m ixtu re of fiv e ranks. In the pedalorgan of the same fin e in strumen t is a com et of n o le ss thane ight ranks , the bu ilde r ev iden tly kn owing by in stin ctiv eexp erien ce what has n ow b een proved by theory

,that the

u pper partial s of the deepe st n ote s wou ld carry with them a

larg e r train of atten dan t vibration s still within the rang e of

mu sical sou n d,than those m ore acu te .

In the chu rch of St. B ernhardin,at Bre slau

,is an organ

bu ilt by Gasparin i in 1705,con tain in g two cymb els , each of

two ran ks , an d two m ixtu re s of fou r an d five ran ks re spec

tively. In the chu rch o f St. Mary Magdalen , in the sam e

t own,is an othe r by a difieren t bu ilder

,dated 1725, con tain ing

n o le ss than e ight su ch compou n d stops. The same prin cip lei s howeve r more distin ctly to b e trace d in what are te rmedmu tation stops

,which thou gh composed of Sing le pipes do n ot

g ive a sou n d corre spon din g to the n ote pre ssed,b u t which

Sou n d e ither the fifth or the third above it,the first be in g

called qu in t stops,the se con d tierces . Othe r form s su ch as

the ten th , twe lfth , an d larig ot , the last sou n ding a n in e teen thab ove the diapason s , are con stan tly m et with .

The Qu in ta ton or Qu in tadena , in addition to the fu n dame n talton e , sou n ds al so

,softly b u t distin ctly

,the twe lfth above

,as

its n am e implie s. It IS a sing le ste pped me tal pipe of smal l”

b ore , in which the first possible u ppe r part ial is un u su allypredom inan t.The Qu in t or Diap en te, in Spite of the similarity of its

n am e to that last g iven , is of a difieren t characte r. Thatb e long ing to a diapason of 32 fe e t lon g i s itself 103: in length ;for an open diapason of 16 fe et it is 5g in e ach case it w illb e seen that it g ive s the twe lfth of the fou n dation ton e

,the

se con d u pper partial,or the fifth above the octave

,when ce its

n ame . It is n ev e r drawn withou t its correspon ding low e rstop , an d eviden tly acts by en richin g the qu ality of thislatter by au xiliary harmon ics. The 103 stop can howev e r b eu sed in a som ewhat differen t man n e r

,by be in g drawn with a

16 foot open diapason . The resu lt is v ery remarkable for a

SPECIAL APPLICATIONS TO MUSIC. 167

g rave harmon ic is thu s su gge sted to the e ar, which produ ces

b y diff eren ce of v ibration s an artificial 32 feet ton e withou tan y pipe of this calibre be ing pre sen t.C orr e ct io n o f B e rn o u ll i’

s Law .- It has l on g be en kn own

that if an open pipe b e stoppe d at on e end,its n ote i s n ot

exactly an octave be low that g iv en by it when open , b u ts omewhat le ss ; the diffe ren ce be in g abou t a maj or seven thin stead of an octave .

In a cylin drical tube open at both en ds o f len g th t

an d diam .

Eflectiv e len gth l 2a .

a b ein g the correction for on e open en d. If aflat stopper b eapplied at on e en d it i s equ al to a pipe of length 2 (l a).The ratio of the n ote s the refore is

Mr. Bosan qu et in inve stigating this subje ct expe rim en tally,took an organ pipe 95 in . lon g , an d °

95 in . diameter. It

be in g difficul t to obtain octav e s from the same pipe open an d

stopped, the octave an d twelfth , or se con d harm on ic we re

compare d. This Was fou n d to b e 2 commas sharper than itshou ld b e , or an in te rval of abou t 40 41 .

The corre ction for the mou th was dete rm in ed by sawingacross a sim ilar pipe .

The valu e of a was fou n d to b e °635 R for the open en d

of the pipe , an d ‘59 r n early for the mou th .

He remarks that in Bern ou ll i’ s the ory of organ -pipe s, thehypothe sis is made that the chan g e f rom the c on strain t of

the pipe to a con dition in which n o remain s of con strain t areto b e perce ived take s place su dden ly at the poin t whe re thewave-system leave s the pipe . It is howeve r ev iden t that thediv erg en ce which take s place may b e con ce ived of as sen din gb ack to the pipe a serie s of refle cted impu lse s in stead of the

sin g le refle cted impu lse which retu rn ed from the open en d

of the pipe accordin g to Bern ou lli’s theory, an d that the se

e lemen tary impu l se s,com in g from difieren t distan ce s , may

b e altog ethe r equ ivalen t to a sin g le refle cted impu lse from a

poin t at a l ittle distan ce from the end of the pipe .

In s tru m e n ts o f M u s ic p r o p e r .— In speakin g of tu e innu

merab le methods of e licitin g sou n d,it was stated that some

,

168 ON SOUND. [CHAR

fo r mu sical pu rposes , had su perseded the o thers. The harp ,pian oforte , org an , and harmon ium n eed n ot b e m ore m in u te lyde scribed in a work like the pre sen t. It se em s

,h owever,

advisable to g ive a few details as to the various in strumen tsu sed in the orche stra.

The combination of s ou n d-produ cers , form ing a fu ll .band ,

i s a comparative ly m ode rn invention ; more so :in deed thanthe organ. The in dividu al in strumen ts we re many of themkn own and u sed separate ly 3 b ut the ir u n ion in to .an organizedbody is far more re cen t. N 0 su ch con ju n ction existed -in

classical tim e s. It has been shown that an cien t mu sic waspu re ly m e lodic

,an d that even ~vocal harm ony arose from

the olyphon ic con stru ction of the m iddle ag e s. Even downto an de l’s days , strin g ed in strumen ts , comb in ed with oboe s ,we re exclu sive ly employed

,occasion ally re in forced by bassoon s

and trumpets. Haydn began to u se a larg er ban d . Mozartmay b e said to have orig inated the presen t arrang emen t .

Beethoven expan ded and strengthen ed it, and has left 0 11

re cord his views as to its prope r size and con stitu tion . Han de ldied on April 21st, 1759. The first Han de l commemorationw as in

“1784, in We stm in ster Abbey. The ban d con sisted

of

49 first violin s . 13 first oboe s.46 se con d violin s . 13 se con d oboe s .

26 violas. 6 flute s .21 violon ce llos . 26 bassoon s.15 double basse s . 1 con tra fag otto.

The sacbu ts we re the instrumen ts n ow te r med trombon e s,

b e in g bass trumpe ts 3 the n alm ost obsolete b u t n ow abu n dan tly“u sed .

At the Han de l Fe stival in 1871 the ban d consisted of

93 first viol in s. 8 flu te s.72 se cond violins . 8 oboe s .

56 Violas. 8 Clarinets .58 violon ce llo s. 8 b as soons .

57 dou b le-basses. 1 contra-fagotto.

336 stringed instru 33 wood in stru

men ts merits . 38 brass in strumen ts.

Drums 8 making 415 in all.

6 come ts.6 trumpets .12 horn s.9 trombon e s.3 ophicle ide s.2se rpen ts.

170 -ON SOUND. [CHAR

Th e V io l in has f ou r string s tu n ed in fifths,and n amed

downwards as follows

First string tu n ed in E.

Se con d A .

Third D.

Fou rth G.

The in tervals between the se n ote s are obtain ed by shortening the string b y pre ssin g it w ith the tips of the fin ge rs o f

the left hand on the prol ong ed “fing er-b oard.

” The lowe stn ote i s obviou sly the G of the 4 foot or small octave

,the seven th

or 27,in Mr. Ellis’ s n otation . The u pper limit 1s le ss d istin ctly

defin ed , sin ce , by mean s of harmon ics , ve ry acu te n ote s can

b e produ ced .

Th e V io la , or ten or violin , more properly term ed the altoabroad , is a violin somewhat larg er in size , with the highe ststrin g removed , an d. replaced by an othe r b e low

,at the sam e

in te rval of a fifth,therefore giving 0 of the 4-foot octave .

Its string s are as f ollows

First tu n ed in A .

Se con d D .

Third G.

Fourth C.

The qu ality of the viola is more hollow and m ou rnfu l thanthat of the violin .

Th e V io l o n ce l l o is an octave be low the viola. Its fou r string sare tu n ed to the octave of those on the f ormer. It con

sequ en tly reache s down to the C of the g reat, or 8-foot octave ,the sixth or 26 of Mr. Ellis’s n otation .

Th e D o u b l e B as s is said to have be en inven ted by MicheleTodin i at Rome in 1676 , b u t it probably existed in some formlon g be fore this date . Be ing e ssen tially a viol it has a diff e ren t shape from its n eare st kin dred the Violon ce ll o. It u su allyhas thre e string s , an d i s tu n ed in fou rths in stead of in fif ths.The n ote s are as follows

First string tu n ed in G.

Se con d D .

Third A .

It thu s reache s on ly two real n ote s be low the Violon ce llo , thatis to the A of the con tra or 16-foot octave the 25 or fifth

VIII. ] SPECIAL APPLICATIONS TO MUSIC. 171

of Mr. Ellis’s n otation . Its larg e r body o f ton e make s itse em of a de eper pitch than that it actu ally posse sse s .

In Ge rman y a fou rth strin g , tu n ed to E,has lon g be e n

u sed,a fou rth low e r than that of the ordin ary in strum en t .

B u t it is perf e ctly easy an d m ost de sirab le to carry the pitchstill l ow er to the C, or fou n dation -n ote of the 16-foot octav e .

In this part the b an d is de cidedly weak, an d far in fe rior tothe organ or even the pian oforte . Be ethoven mu st have fe ltthis defe ct

,for he write s dou ble -bass parts

,su ch as those in

the Pastoral an d C m in or symphon ie s, down to this n ote . A

dou b le -bass stru ng down to con tra 0 was exhibited b y thew rite r at a recen t In tern ation al

,an d at the Loan Exh ibition s .

The prin ciple has be en already de scrib ed .

Orch e s tral W in d In s tru m e n t s hav e already be en de scribedas regards the ir prin ciple . Those In actu al mu sical u se are of

thre e kin ds.

1 . Flu te s .2. Reeds.3 . In strum en ts w itn cu pped m ou thpie ce s.

They all requ ire two e ssen tial parts : I. the win dche st ;II. the embou chu re .

The w in dche st in thi s case is invariably the human thorax.

The write r made a se rie s of experim en ts som e years ag o f or

the pu rpose of d eterm in in g what the pre ssu re s w ithin the

thorax actu ally were . A water g au ge was con n e cted with a

smal l cu rve d pie ce of tube by m ean s of a lon g flexible in diaru b ber pipe . The cu rv ed tu be b e in g in serted in the ang le of

the mou th,did n ot

, afte r a little practice,in terf e re with the

ordin ary playin g of the in strum en t. The variou s n ote s we rethen soun ded su cce ssiv e ly

,an d the he ight at which the

column stood was n oted . The f o llowin g table of pre ssu re swas obtain ed as an averag e of man y expe rimen ts :

Tab le of P ressu res.

ON SOUND.

Longitu dinal and tran sversal section of the mouthp iece.

174 ON SOUND. I'

CIIAP. VIII.

i s fashion ed in the oboe an d bassoon in to a broad spatu la-likef orm with two thin n ed plate s of the can e in close approxima

tion to on e an othe r.It is therefore te rm ed a dou ble re ed in Opposition to that

of the clarin e t an d some other in strumen ts where the vibratin gplate of can e is sin g le . It has be en materally redu ced in sizeof late years with a corre spon din g improv emen t in the ton eof the in strum en t. Even as late as the visit of the compose rRossin i to this cou n try a reed re sembling that of the bassoonwas in u se for the oboe .

The bore of the oboe is con ical, enlarg in g at the low er '

extrem ity in to an expan ded be ll . Its scale is foun ded on the

in terval o f the octave,beg in n ing at the Eh or Eh of the

fou r fe e t or small octave,an d exten din g to F in alto in the .

twice accen ted , or on e f oot, or 9th octave .

Th e C larin e t i s an in strum en t of fou r-foot ton e , with a

sin g le reed an d sm ooth qu ality,common ly said to have

been in v en ted in 1690 at Nu rembe rg . It is probable , howeve r, that in on e f orm or an other it existed long before . Its

n ame is eviden tly a dim in u tiv e of Clarin o, the Italian n am e

of the trumpet, to which its ton e has séme sim ilarity.

The '

clarin et con sists of a pe cu liar mou thpie ce f u rn ishedwith a sing le b eating reed

,a cylin drical tu be term in atin g in

a he ll , with e ighteen open ing s in the side,half of which are

closed by the fin gers, and half by keys. The lowe r scalecomprise s n in ete en sem iton e s

,from E in the base stav e to

El; in the octave above . The lowe st n ote is em itte d throu ghthe he ll

,the G of the two-foot octave throu gh a hole at the

b ack of the tube,pecu liar to this in strum en t. This reg iste r

is te rmed Chalumeau . By Open in g a lever above the backhole n amed above

,the pitch is raised a twe lfth

,so that the E

of the small or fou r-foot octave be come s the EHof that above .

By the su cce ssive rem oval of fing ers , fifteen m ore sem iton e sare obtain ed , reachin g to high and above this n ote i san othe r octave obtain ed by cross-fingering .

The mou thpiece i s a con ical stopper flatten ed at on e side

to form the table for the reed , an d thin n ed to a chise l edg eo n the other for the conven ien ce of the lips. From the borea late ral orifice is cu t in to the table which is closed in playing by the thin end of the reed . The table on which the

re ed lie s , in stead of be ing flat,i s cu rved backwards towards

the poin t, so as to leave a gap or slit abou t the thickn ess of

a sixpen ce be tween the en d of the mou thpie ce and the poin tof the reed.

He lmholtz has analyzed the ton e an d mu sical characte r

Fig . 59.—Hau tb ois .

Fron t and side view of reed .

Pi 60.-Clarin et .

ON SOUND.

o f the clarin e t, as has been stated above . It stan ds apartfrom all

‘

other in strum en ts,both in its qu ality

,in its scale

,

fou n ded on the twe lfth , an d accordin g tothe w riter’ s expe rimen ts

,in the wind-pressu re

requ ired for its variou s reg isters. The clarin et is made in many keys

,to m e e t diffi

cu lties of exe cu tion . This fact en ables a

very close approach to tru e in ton ation to b eobtain ed on it , as de scri b ed in the bhapteron Temperamen t.Th e B as s o o n is a double -reed in strumen t

of e ight-foot ton e , as implied in its n ame ;it be in g the n atu ral bas s of the oboe . Inon e form or an othe r it is probably of g reatan tiqu ity ; thou gh it is said to have be eninven ted in 1539 by Afran io

, a can on ofFe rrara. . A class of in strumen ts n amedb ombards

’

, p ommera,or b rummers se em to

have be en the immediate prede ce ssors of thebassoon . It is a con trivan ce which has evi

de n t]y orig in ated in a f ortu itou s man n e r,

deve loped by su ccessive improvem en ts of anempirical character. Variou s attempts hav e been made tog ive greater accu racy and completen ess to its singu larlycapriciou s scale

,b u t w ith on ly partial su cce ss . Its compass

is from El; in the con tra or sixte en f oot,to A l; in the on ce

accen ted or tw o-foot octave , b u t addition al m e chan ism has

g reatly raised the u ppe r lim it,so that the C or even the

E above that n ote can b e obtain ed .

Like the oboe,it g ive s the con se cu tive harmon ics of an

open pipe .

Fig . 62.— Bassoon .

In s trum en ts w ith cu p p e d m o u th p ie ce s may b e cited as

the simple st mu sic‘

al in stan ce s of con son an t tu b es . They all

con sist e ssen tially of an open con ical tube , often of great

1-78 ON SOUND.

It will b e seen that in the low e r part o f the se rie s the ihtervals be twe en the sou n ds are larg e , b u t that the u ppe rharm on ics approach close r an d closer together, so that fiom

the m iddle B}, a n early pe rfe ct octave scale of con tinu ou sn ote s can b e obtain ed . It has long be en the cu stom to in te rpe late the m issing sem iton e s on the Fren ch horn by thru stingthe han d in to the be ll , and so l owering the pitch by a variablequ an tity. The in strum en t is hen ce n amed the “han d ” horn ,and the n ote s so m odified han d-n ote s. Of late years , howev e r

,valve s have been applied

,as w ill b e de scribed in a

su bsequ e n t parag raph .

Fig . 64.

— Trump et an d clarion .


Th e Tr um p e t .— Speaking in a higher octave posse sse s the

first e leve n open n ote s of the Fre n ch b orn . In this in stru

men t,and in the trombon e , its n atu ral bass , a totally differen t,

and far more perfe ct system has be en adopted for comple tingthe scale . An U shaped port ion of the tu b e is made to slidewith g e n tle friction u pon the b ody of the in strumen t

,so that

the leng th o f the he re can b e in creased and dim in ished by anyg iven qu an tity w ithin ce rtain l im its

,at the will o f the

playe r. The n ote em itted can thu s b e l owered in sen sibly,

and w ithou t abru pt chan g e s throu gh a variab le in te rval. Theabsen ce of fixed n otes e n able s the in ton ation to b e g u ided at

the w ill of the playe r,b y acc urate ear

,exactly as is the case

on the violin fami ly. We therefore have in the trumpet an dtrombon e qu arte tt a perfe ct comb in ation as regards temperam e n t. and on e equ ally we ll d iv ersified by con trast o f ton e .

It is to b e reg re tted that owing to the diflicu lty of findingcompeten t playe rs , this mo st pe rfe ct departmen t of the

wind-ban d is falling in to n eg le ct an d disu se .

Th e B ar i s divided in to thre e parts , the extern al,m iddle

,

an d in te rn al The latte r con tain s the n e rvou s apparatu scon ce rn ed in hearin g ; the two f or me r act as con veyan ce s o faerial vibration s they are n ot absolu te ly n e ce ssary to the act

of hearing , for person s su ffe rin g from deafn e ss du e to diseaseof the m iddle ear can often hear a mu sical n ote by applyin gthe sou nding body , su ch as a tu n ing -fork

,to the adjacen t bon e s

of the head,through which the v ib ration is tran smitted .

The exte rn al ear con sists of the au ricle or p in na, an d theau ditory can al

,or meatu s.

Con side rable dou b t exists as to the part played by the

au ricle in the act o f hearing . That it has some in flu en ce isclearly shown by the fact that in the lowe r an imals it ism ovable , an d is in stin ctive ly dire cted towards the sou rce of

sou nd . He n ce is de rived the comm on phrase Pricking upthe e ars . In the human su b je ct the mu scles which actu ate itare all b u t atrophied

,and the powe r of motion is sl ight an d

u n u su al . The ears howeve r of su ch an imals as the horse, cat,

and ass are of a trumpe t shape , an d em in en tly fitted to discrim inate the direction of son orou s u n du lation s . Even in man

it re tain s some su ch pow e r,which is often assisted u n

con sciou sly b y su pplem en tin g it w ith the han d , an d ben ding itf orward in to a more e fficien t position . It retain s the fun n e lshape of the lowe r an imal s to a con siderable deg ree .The m eatu s is abou t an in ch an d a qu arte r long , dire cted

s lightly f orwards as we l l as inwards , closed somewhat obliqu e lyat its in n e r extrem ity b y the tympan um or drum o f the ear.

182 ON SOUND. [CHAR

The in tern al ear i s en tirely embedded In the hard or petrou sportion o f the temporal bon e . Its complicated an d an frac

tu on s con tou r is lin ed with a membran ou s sac. It is n o t

in aptly te rmed the labyrin th,an d i s divided in to the ve stibu le

,

the cochlea, an d the three sem i-circu lar can als. The ou te r wallof the ve stibul e is perforated by an othe r open ing beside s thefen e stra ovalis

,n am e ly the fen estr ' rotu n da. The inn e r wall

has a n umber of small hole s tran sm itting bran che s of the

au ditory n e rv e,the P ortia mo llis of the seven th cran ial n erve .

The two f en e strae thu s commu n icate with the m iddle ear, the

smalle r orifice s with the in side of the cran ium .

The sem icircu lar canal s commu n icate by five open ing s withthe u pper an d poste rior part of the ve stibu le ; two extrem itie s,those of the superior, an d of the poste rior can al s en tering bya comm on term in ation . Each can al is expan ded at on e en d

into a g lobu lar en larg emen t termed an ampu lla.

Be side s the superior an d posterior, the third canal is n am edfrom its position

,the exte rn al. They stan d in three re ct

an gu lar plan e s to on e an other ; thu s repre sen ting the thre edim en sion s of space .

In f ron t of the v e stibu le,111 form of a prolongation , is the

cochlea, sim ilar In shape to a sn ail-she ll, a flat con e with apexou twards , con sistin g of a spiral taper tu be of two and a halftu rn s arou n d the axis. The cen tral column of bon e sen ds a

partition,n amed the lam ina spiralis

,ou twards from its cen tre ,

sim ilar to the thread of a taper screw,b u t de fe ctive at the

apex . This lam ina of bon e is met an d completed by twom embran ou s prolongation s termed the basilar membran e an d

the m embran e of Re issn er.In this way three he lical passag e s are cu t off

,termed

seve rally the scala tympan i, the scala vestib u li an d the du ctu s

cochlearis, the last b e ing the smalle st and the least distin ct.The two f ormer are con n e cted at the apex, and at the bottome nd

,on e

,the scala vestibu li , with the ve stib u le , an d the other,

the scala tympan i,w ith the in terposition o f a membran e ,

op en ing by the f en e stra rotu nda in to the tympan ic cavity.

The m embran ou s labyrin th lie s free ly in the bon y cavity ,except as regards the du ctu s cochlearis , which is in closeapproximation to the bony wal l o f the labyrin th. Flu id iscon tain ed In its sac

,an d also b e tween it and the wall . The

m emb ran ou s lin in g of the ve stibu le i s divided in to two n on

commu n icating parts,the larg er term ed the u trieu lus , the

I These modes of describ ing the can alis memb ranacea and organ of Cortiare ju diciou sly adopted b y Mr. Champn eys in Stainer an d Barrett's Dictionaryof Mu sicalTerms.

SPECIAL APPLICATIONS TO MUSIC. 183

an te rior an d smaller the saccu lu s, which latter epen s in to thedu ctu s cochlearis .Within the m embran ou s can al or cochlear du ct, separated

from it by a laye r of membran e,l ie s what is n amed after its

discoverer the organ of Corti.This remarkable apparatu s is placed betw e en a l ower or

basilar membran e , stretched ov e r the scala tympan i from the

bon y lam in a spiralis to the ou te r wall , an d the membran en am ed abov e

,separating it f rom the cochlear du ct. It con

sists of a dou ble se rie s of dim in ishing rods,followin g the

spiral of the cochlea, abou t in n umber. The ir base s areseparate , b u t the ir u pper en d s m e e t at an ang le , “ f orm ing a

sort of spiral gable roof .

” It has latterly be en su pposed thatthe se are the in strumen ts by which w e distingu ish the pitcho f sou nds . Each on e is con sidered to vibrate in sympathyw ith on e n ote

,an d to tran sm it its vibration s to a special tw ig

of the n e rve o f hearing . The sem i-circu lar can als are su pposedto enable u s to determ in e the dire ction o f sou n ds

,a the ory

which the ir re lation to the thre e dimen sion s of space se ems to

ju stify. B u t the poin t can n ot b e con side red as fin ally se ttled.

Ed u cat io n o f th e Ear .-It i s somewhat sin g u lar that in the

maj ority o f acou stical re searche s the eye s have be en made todo m ore service than the ears ; and when the eye s have failed ,that they have been replaced by calcu lation or the ory. On e

re sult o f this cu riou s fact is that sin ce the discovery of in

g en iou s m ethods for ren de rin g the vibratory m otion s o f solidsv isib le to the eye , man y facts have been substan tiated as to

the vibration s of string s , plates , and reds,b u t far fewer as to

those of liqu ids an d g ase s . In deed , the mov emen t of the air

in an organ -pipe is still matter of dispu te,although it may b e

stu died optically by mean s of a ve ry simple con trivan ce .

It is n ot, however, suflieien tly appreciated that the edu cationof the car may ren de r that organ competen t to u n dertakehighe r fu n ction s. In this re spe ct the ear re semble s all theothe r sen se s.In many case s it is simply au ral memory which n e eds

cu ltivation . Even an ordin ary cou n tryman can re cog n ize thesou n d of his villag e be ll after twen ty years’

absen ce ju st as

we all recogn ize a frien d whom w e have n ot seen for l ong ,and whom tim e

’s ravag e s have made a strang er to the eye , by

the sou nd o f a sing le spoken word . To this categ ory be long sthe ton al memory which distingu ishe s absolu te pitch . The recan b e n o doubt that the facu lty exists in some person s to a

ve ry high deg re e . It is an en tire ly differen t qu e stion whe the rit is what is common ly termed or an acqu iremen t.

184 ON SOUND. [CHAR

In a g reat deg ree , doubtle ss , it is m ixed with what has be enlate ly re cogn iz e d as the sixth sen se , n ame ly , the “mu scu larsen se by which we appre ciate the amou n t of n ervou sin flu en ce or mu scu lar con traction n e ce ssary to produ ce a

ce rtain resu lt. Ju st as , by mean s of the “mu scu lar sen se,

”

we can approximate to an e stimate of we ight or spe ed , so bythe in stin ctive con traction of the laryng eal mu scle s we men

tally produ ce a u n ison to any heard soun d,an d

,as it were ,

w e igh it in the balan ce again st ou r own vocal pow e rs. Thatthis is the real process in man y case s is ev iden ced by the factthat the fortu n ate own ers of clear, accu rate voice s , are speciallye ndowed with the pow e r in qu estion . B u t it i s n ot confin edto the se ; an d , in some in stan ce s

,appears to b e a pu re act o f

memory. B u t it will u su ally b e fou n d to exist in this f orm in

old practised mu sician s,whose pian o

,organ , or o ther mu sical

in strumen t,has be com e

,by long fam iliarity, almost a part o f

them se lve s . An othe r frequ en t indication of its acqu iredn atu re

,as again st its in n ate posse ssion , is the fact that dif

fe ren t form s o f acu ten e ss can by practice he deve loped in cars

o f otherwi se equ al sen sibility. Whe re the g lee -sing er, orchoru s-master

,or organ ist, acqu ire s the powe r of g iv ing the

key-n ote

,the pian o or organ -tu n e r cu ltivate s a se n sibility to

beats,or diff eren ce s of ton e , which the form e r hardly hears at

all. Upon this poin t the obse rvation s o f Prof . Preyer, qu otedby Mr. Ell is , are of very g reat valu e .

Th e S e n s ib ility o r th e B ar for the appre ciation of weaksou n ds is very g reat. Ev ery fly which bu zzes arou n d g ive s toa Iriu sical ear the pow er of in stan tly te sting the n umbe r ofvibration s pe r se con dwhich itswin g s g o through . Even b etweenthe lim its of high to de ep vib ration s the car has vast latitu de ,from -32 in a se con d to ten thou sand , or in exception al case sfar beyon d . The eye possesse s at me et an octave o f sen sation s , b u t the ave rag e ear easily ru n s throu gh seven or e ight.

'

Even in the matte r of accu racy, a comparison between theeye an d e ar en ds in favou r

'

of the latter. An archite ct ordrau ghtsman

'

who be twe en tw o l in e s,n e ither paralle l n or: In

o n e plan e , made an e rror of e stimation by eye n ot exce edin g13—6 wou ld gain credit for u n u su al pre cision . B u t In the ear ah:amou n ts to a qu arter of a ton e , and b y car 3

15 of a ton e 1s

easily dete rm in ed . Still more remarkable is the car In the

rapidity of its appre ciation s . They are all b u t In stan tan e ou s .

A sou n d lasting on ly a qu arte r of a se con d 1s as we ll or be ttergau g ed than if it lasted a m in u te , an d in th1s short period

.

If

sev e ral sou n ds b e simu ltan e ou sly produ ced ,

“

the car can 1n

stin ctiv ely separate them, withou t any risk of con fu smn . It

186 ON SOUND. [CHAR

b e forced from the lu n g s with a view to se t them in to vibration .

The solid portion s of the frame in which the se membran e s ,e rron eou sly termed vocal chords , are set, con sist of cartilag e .

Fig . 67 . The human voice in terior view o f the larynx. Glottis ;vocal chords.

A Oral Cavity B Soft Palate ; C Tongu e ; DTon sil : EEpiglottis ; F Thyro idCartilage ; G Arytaen oid Cartilage ; H I Vocal Chords ( Sup erior an d In fe rior)K Ven tr1cle L RimaGlottidis MCricoid Cartilage NTrachea ; OEsophagus .

They are n in e in n umber,thre e symme trical , and occu pying

the m iddle lin e of the body, n ame ly

,the epig lottis , the

thyroid , and the cricoid cartilag e s ; six in pairs,namely

,the

two aryten oid cartilag e s, and two smaller acce ssory carti lage s,named after San torin i and Wrisberg .

VIIl . ] SPECIAL APPLICATIONS TO MUSIC. 137

The thyroid cartilage f orm s the u pper part of the hard massto b e f elt in the n e ck , an d comm on ly called “Adam ’s apple .

”

It rests u pon the cricoid cartilag e , with which it is articu late dat e ither extrem ity. It i s in shape somewhat l ike a partlyopen ed book with the back dire cted forwards. At the posterioredg es are f ou r horn s , the lowe r pair con n ected , as n amedabove , with the cricoid

,the u ppe r pair su spen ded to the

hyoid bon e at the root of the ton gu e .

The cricoid cartilag e is n ot u n like a sign e t ring , with a

broad facet dire cted backwards , u pon which are two pairs of

smooth su rface s , the lower pair f or the attachmen t of the

horn s of the thyroid , the u pper for the loose articulation o f

the two aryten oid cartilag e s .

The aryten oid cartilag e s are of an irregu lar trian gu lar orpyram idal shape , with the ir apice s u pwards, the ir base s re stin gon the broad part of the cricoid ring . Two other smallcartilag es , those of San torin i

,are set on the apice s of the se .

The epig lottis is an oval leaf-shaped flexible plate of cartilag e ,situ ated at the back of the tongu e . The tail or stalk is attachedju st within the n otch of the thyroid body

,an d the fron t is con

n e cted by an e lastic ban d with the back of the hyoid bon e .

The vocal chords them se lve s are attached in fron t ju stbe low the tail of the epiglottis , to the ang u lar open ing betweenthe sides of the thyroid cartilag e . This attachmen t is therefore fixed . The ir posterior attachmen t is to the apice s of thearyten oid cartilag e s , and is therefore fre e to m ove in twodire ction s. When the aryten oid cartilag e s are drawn backward by mu scu lar action

,a ten sion i s pu t u pon the vocal

chords when they are drawn ou twards or inwards,the back

extremitie s of the vocal chords are e ithe r wide ly separatedor close ly approximMed . From this approximation an d

ten sion the vibration and con sequ en tly the mu sical n ote of

the voice is produ ced.

It has become possible to g ive a m u ch more satisfactoryde scription of the position an d action of the vocal chords sin cethe inven tion of the laryng oscope , a simple b u t effe ctu al reflectin g apparatu s , by mean s of which the in te rior of themou th an d laryn x can b e examin ed. It mu st b e rememberedthat the epig lottis is the poin t at which the re spiratory an d

alimen tary passage s cross or decu ssate .

I The proper orificeo f the form er is the n ostril

,when ce the dire ction

,at first

horizon tally backwards, tu rn s downwards behin d the ve il of1 The two organ s an d their appen dages are physio logically d istingu ished b y

the differen ce of the epithe lium, that of the resp iratory tract b eing ciliated ,that of the alimen tary, tesselated.

188 ON SOUND. [CHAR vm .

the palate . The open ing of the food passag e is the m ou th ,when ce it passe s ove r the epig lottis to the posterior fau ces ;the oesophag u s lying behin d the trachea. Du rin g swal low ingthe laryn g eal open ing is tightly closed , an d it is aston ishinghow accu rate ly the mechan ism for preven ting the e n tran ce o f

food in to the trachea works .Du ring ordin ary b reathing the epig l ottis l ie s back an d the

tips of the aryten oid cartilag es can b e se en w ith the laryn g o

scope . The Open ing of the g lottis is diamond-shaped . In

pron oun cing the vowe l s it rise s an d shows the larynx, the

v ocal chords be in g brou ght tog ether. For the produ ction of

low che st n ote s,the aryten oid cartilag e s ben d u n de r the o ve r

han g ing epiglo ttis ; b u t du ring the u tteran ce o f high che stn ote s

,it is easy to se e them ,

with the vocal chords close lyapproximated .

“In shrill cries the cu shion o f the epig lottisappears to b e pre ssed down on the fron t part o f the vocalchords so as to shorten the ir vibrating portion , ju st as a strin gis shorten ed by the fingers on the fing er-board of a string edin strumen t.The n ame of chords obviou sly sugg ests an e rron eou s

hypothe sis as to the ir fu n ction . No string so short cou ldprodu ce so low a n ote as those of the male voice

,an d the laws

o f string -ten sion are n ot observed . Ne ithe r can they b e

su pposed to act o n the prin ciple of a d iapason pipe . Theyfu lfil far m ore closely the condition s of a fre e re ed of a

membran ou s character with a dou ble vibrator. They posse ssa tu be be low them in the trachea an d bron chi

,with a complex

re son an ce tu be ab ove them in the cavity of the mou th,

pharynx,n ose

,an d fron tal sin u se s . This is farthe r de scribed

in speaking of the vowe l sou n ds . He lmholtz con sidersthe head voice to b e produ ced by a thin n ing of the edgeof the chords by draw ing aside the mu cou s coat be low them .

The produ ction o f the falsetto is still Open to con siderabledoubt, som e hold ing that the chords on ly vibrate in part of

the ir leng th ; others that they vib rate in segmen ts, g iving

harmon ics ; othe rs again ascrib e them to the sam e cau se as

that of the head voice , and they have been attribu ted to thefolds of mu cou s membran e above them , cal led false vocalchords. Pe rhaps the most probable the ory is that of harm on ics ,althou gh the exact mechan ism is u nkn own .

Czermak , as qu oted in Stain er’

s Dictionary of Mu sical Terms . ArticleLarynx.

”

um

Gase s , velo city o f sou n d in ,49

reflectio n from , 59

G raphic m ethods , 85

Harmon icon , 20

Harm on ics,118

Harmon y , 162

Hau tb ois , 173

H earin g . me ch an ism o f, 180

Heat— e ffe ct on so u n d , 127

evolve d in v 1b 1 ation 53

efi‘

ect on strin gs , 129

tu n in g forks, 130re ed s , 131

organ p ip e s, 131

H elmh o ltz’s vowe l th eory, 116

H o rn, 177

Hydrogen , sou n d velocity in , 49

In ten sity , 62

In te rie ren ce s o f sou n d , 70

In tervals , 135

In strum en ts o f mu sic , 167Irregu lar vib ration , 4

Keyb oards , 144B osan qu e t ’

s , 152

Co lin B rown ’

s , 154

G u erou lt’s 148

He lmh o ltz s , 148P oole

’s , 151

Th omp son , P e rron e t , 147

Kaen ig’s man om e tric flam e s , 94

Ku n dt,his experim en ts , 55

Laplace , h is corre ction o f Newton ’

s

fo rmu la. 53

Laryn x, 185

Lateral vib ration o f rods , 16Laws o f B ern o u lli,40

Limits o f au dib le sou n d, 75Liqu ids

,ve lo city in ,

50‘

temperatu re m o dification,129

Lissajo u s , h is figu re s , 90Lon gitu din al vib ration s,15

o f strin g s , 13o f rod s , 16

INDEX.

Nail-fiddle , 17Natu re o f mu sical sou n d , 110Newton ’

s ve locity of sou n d , 53

Node s , 9No tation , 162

Major scale , 139Man ome tric flame s

, 94Marimb a, 21

Mayer’s e le ctrical ton ometer, 102

Mean ton e temp erame n t . 144

Me chan ical m e thods o f ton ometry,77

Me lde , his exp erimen ts , 12

Mem b ran e s , vib ration o f , 33

Me tals . velo city o f sou n d in , 14

Microph on e , 48Min or scale , 142Mixtu re s

,164

Moistu re . in flu en ce o f, 129Mon och o rd

,84

Mo tion o f air in p ipe s , 39Mon th

,reson an ce o f

,116

Mu sic,sp ecial application to , 161

Mu sical b ox, 19

glasse s , 31

Mu sical sou n d an d n oise , 1

Ob o e , 173O ctave . th e , 135

Open pip e s , 39

Optical ton ometry , 90cu rve s , 91 , 93

Orch e stra, th e , 158Orch estra. tru e in to n ation in

,158

Orch e stral win d in strum e n ts,171

Organ p ip e s , varietie s o f, 164qu ality o f, 122e ffects o f h eat on , 131

Over ton e s , 111

P .

Partial ton e s , 111

Ph on au tograph , 87Ph o tography o f vib ration s , 99Pip e s , see Organ pip es , 165Pitch , 75

alteration o f,b y motion

,109

Plates , vib ration o f , 25

P re ssu re,effe cts o f , 128

Preyer’

s re search e s , 185

Propagatio n o f sou n d , 46b y e arth , 46

Pyroph o n e , 44

Pythagorean tu n in g, 136

Q u ality. 110

Q u in taton ,166

Rayle igh’s exp erimen t, 103

Re e ds , 34

Refle ction o f sou n d , 56

Re fraction o f sou n d , 56

Reg ular vib ration , 5

INDEX.

Torsion al vib ration s , 14Tran sverse vib ration s, 6

Treve lyan’s ro ck er , 41

Tromb on e,179

Trump et , 18 0Tu n in g-forks , 21

effects o f h eat on ,130

Upper partial ton e s , 113re ality of , 113

Re lative harmon iou sn e ss of ch ords ,

137Re son an ce ,

th e ory o f, 68

Re son ators , 66 , 112

Re su ltan t ton e s , 123

Rise in p itch , cau se s o f, 108

Rod , vib ration s o f . 15

Rods fre e at b oth e n d s , 19

Savart’s whe el , 78

Scale , 139

Pythagorean ,142

che ib ler’s ton om e ter, 100

Sen sitive flam e s,44

Shadows , sou n d ,59

Sh o ck as a sou rce o f sou n d , 4

Sin gin g flame s , 42

Siren , 79dou b le , 82

Se e b e ck’s . 80

So lid s,ve locity in ,

51

Son dhau s’

s exp erim en t, 42

Son ome te r , 11

fo r lo n gitu din al vib ration ,

Sou n d , d efin ition o f, 1

So u n d b oards, 69So u rce s o f so u n d ,

2

Stan dard s o f pitch , 105Stopp ed pipe s, 40

Strin gs , 6e ff e cts of h eat on , 129

Summation ton e s , 124

Syn th e sis o f ton e s, 114

Tartin i’s ton e s , 3Te leph on e ,Wh e atston e ’

s , 47

G rah am B e ll ’s , 48

Temp e ram en t , 143

tab le o f , 144

Th omp son . P erron et,h is organ , 147

Tim b re . 110

Ton om e ters , 100

To oth ed wh eels o f Savart, 79

Water. wave s in , 63

ve locity in , 50Wave m o tion . 52

Wave s , le n gth o f , 54We rth e im ’

s exp e rim e n ts , 54

Wire s , s ee Strin gs , 6Wood , ve locity in ,

47

Zan z e . 20

Z amb om b a, 34

191

Variation o f pitch . 105

Ve lo city . o f sou n d , 48

in gase s,129

in flu en ced b y temp eratu re , 127d en sity , 49

e lasticity, 49

Newt on ’

s calcu lation o f , £3

gase s , 49 , 129

liqu id s . 50

in solids , 51

Ve n tral se gme n ts, 9

from h eat , 41

Vib ration lon gitu din al , 5tran sverse , 6torsion al , 14

Vib ration o f strin gs , 6ro d

‘

15

p lat e s . 25

b e lls , 27m em b ran e s , 33re ed s , 34

co lumn s o f air , 36Vib ratio n m icroscop e , 122

0 Vib ro scope , 86Vio la , 170

Vio lin , 170

Violo n ce llo , 170Vo ice , 185

Vowe l sou n ds , an alysis o f,syn th e sis o f, 117

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I . Large Post 4to . Price 4d. each .

2 . Post Ob long . Price 3d . each .

*i . INITIATORYEXERCISES 65° SHORTLETTERS .

*2 . WORDS CONSISTING OF SH ORTLETTERS.

MISCELLANEOUS . 4 7

M ACM ILLAN‘

S C OPY—B OOKS Con tin u ed

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3 . LONG LETTERS . With words contain ing LongLetters— F igu re s .

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4 . WORDS CONTAINING L ONG LETTERS .

4a. PRACTISIN G AND REVISING COP Y-B OOII’

. Fo r

No s. I to 4 .

*

5. CAPITALS AND SH ORT HALF -TEXT: Wordsb egin n ing with a C apital .*6. HALF TEXT WORDS

,b egin n ing with a Capital

F igures .

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7 . SMALL-IIAND AND HALF TEXT. With Capitalsand F igures.

*8 . SMALL-HAND AND HALF TEXT. With Capitalsand F igu res .

8a. PRACTISING AND REVISING COP Y-B OOK F o r

Nos. 5 to 8.

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9. SMALL-HAND SINGLEHEADLINES— F igures.

Io . SMALL-HAND SINGLE HEADLINES— Figures.

*I I. COMMERCIAL AND ARITHMETICAL EX

Iza. PRACTISING AND REVISING COP Y-B OOK Fo r

NOS. 8 to 1 2 .

TAese nmn é e mmay Oe lzaa’

wit/z Good/n an’

5 Paten t Sliding

Copie5. Large Post 4to . Price 6d. each .

B ya simple device the copie s, which are printed upon separateslips, are arranged with a m ovab le attachmen t, b y which theyare adju sted so as to b e directly b efore the eye o f the pupil at

48 MACMILLAN’


M A CM ILLAN ’ S C OPY -B OOKS Con z‘

in u ed

all poin ts Of his progre ss . It en ab le s him,also

,to keep his

own fau lts con cealed, with perfect models con stan tly in viewfor imitation . Every experien ced teacher kn ows the advan tage

o f the slip copy, b u t its practical application has n ever b eforeb een su ccessfully accomplished . This featu re is secured ex

e lu sive ly to Macmillan ’

s Copy-b ooks u n der Goodman ’

s paten t.An in spection o f b ooks written on the o ld plan

,with copies

at the head Of the page , will show that the lin es last written at

the b ottom are almost in variab ly the poorest. The copy hasb e en to o far from the pupil’s eye to b e of any practical use ,an d a repetition and e xaggeration of his errors have b e en the

resu lt.

MA C M ILLAN ’S P R OG RES S IV E F REN C H C OUR SE— B y

G . EUGENE-FASNACHT, Sen ior Maste r o f Mode rn Languages,Harpur F ou n dation Mode rn School, B edford .

I.— F IRST YEAR , con tain ing Easy Lesson s on the Regu lar Ac

ciden ce . Extra fcap . 8y o . 1 5 .

II.

— SECOND YEAR , con tain in g C on versational Le sson s o n

Systematic Accide n ce an d Elemen tary Syn tax. With Philological.Illu strations and Etymological Vocab u lary.

'

1 5 . 6d.

M A CM ILLAN'

S P R O G RES S IV E GERM AN C OUR SE— B y

G . EUGENEFASNACHT.

Part I.—F IRSTYEAR . Easy Le sson s and Ru les o n the Regu lar

Acciden ce . Extra fcap . 8vo . 1 5 . 6d.

Part II.—SECOND YEAR . Conversational Lesson s in Sys

tematic Acciden ce and Elemen tary Syn tax. With PhilologicalIllu stration s and Etymological V ocab u lary. Extra fcap .

8vo . 25 .

MARTIN THE P OET’

S H OUR Poetry sele cted and

arran ged for Children . B y FRANCES MARTIN . ThirdEdition . I8mo . 25 . 6d .

SPRING-TIME WITH THE P OETS : Poetry selected b yFRANCES MARTIN . Secon d Edition . 18mo . 35 . 6a

'.

(50 MACMILLAN’


PY LODET— NEW GUIDE TO GERMAN CON VERSATION con tain ing an Alphab etical List o f n early 80 0 Fam iliarWords followed b yExercises, V ocab u lary ofWords in frequen tu se Fam iliar Phrases and Dialogu es a Sketch of Ge rman

Literatu re , Idiomatic Expressions, &c. B y .L. PYLODET.

18mo . cloth limp. 23. 6a’.

A S YNOPSIS OF GERMAN GRAMMAR. From the

ab ove . 18mo . 6a'.

REA D IN G B OOKS— Adapted to the English and Scotch Codes.B ound in C loth.

18mo . (48 pp. )

for Standard I.

II.

III.

IV .

V .

VI.

B ook VI. is fitted for higher C lasses, and as an In trodu ction toEnglish Literatu re .

“They are far ab ove any o thers that have appeared b o th in form andsu b stan ce . Th e editor of the prese n t se rie s has rightly se e n thatreadin g b ooks mu st ‘

aim chiefly at giving to the pupils the power o faccu rate , and , if po ssib le , ap t and skilfu l expre ssion ; at cu ltivatin g inthem

’

a good literary taste , and at arou sing a de sire o f fu rthe r reading .

This is don e b y taking care to’

se le ct the extracts from tru e English classics ,goin g u p in Stan dard VI. course to Chau cer, H o oke r, and B acon , as we llas Wordsworth , Macau lay , and Fro u de . This is quite on the ri

ght

t

Grack . an d indicates justly the ideal which we ou ght to se t b efore us.

UARDIAN.

SHAKES PEARE— A SHAKESPEARE MANUAL. B y F . G .

FLEAY, M.A. ,

late Head Master o f Skipton Grammar Schoo l.Secon d Edition . Extra fcap. 8vo . 6a

'.

AN,ATTEMPT TO DETERMINE THE CHRONO

LOGICAL ORDER OF SHAKESPEARE’

S PLA YS .

B y the Rev. H . PAINE STOKES, B .A. Extra fcap . 8vo .

THE TEMPES 7 With G lossarial and Explanatory Note s .

B y the Rev. J . M. JEPHSON . Second Edition . U .

2d.

18mo . (96 pp . ) 4d.

18mo . ( 144 pp . ) 561.

18mo . ( 160 pp .) 6a’.

18nio . ( 1 76 pp . ) 8d.

’

18mo . (380 pp .) 15 .

Crown 8vo . (430 pp . )

MISCELLANEOUS. 51

SON NEN SC H EIN an d M B IK LEJ OH N — IHE ENGLISHMETH OD OF TEACHIN G TO READ . B y

'

A. SON

NENSCHEIN an d J . M. D. MEIKLEJ OHN, M .A. Fcap .

'

8vo .

COMPRISING

THE NURSER Y B OOR'

, con tain ing all the Two -Le tterWords in the Language . 1d . (Also in Large Type on

Sheets for School Walls.

IHE FIRSTCOURSE, cons isting of Short Vowels withS ingle C ons onan ts . 6a

'.

IHE SECOND COURSE,with Comb in ations and B ridges .

con sisting of Short Vowe ls with Doub le C onsonan ts. 6d .

THE THIRD AND F OURTH COURSES , con sisting o f

Long V owels, and all the Doub le Vowe ls in the Language .

641.

These are admirab le b ooks , b ecau se they are co nstru cted on a prin

ciple , and that the simp les t prin ciple on which it is po ssib le to learn to read

English.— SPECTATOR.

TANN B R— FIRSTPRIN CIPLES OF A GRICULTURE. B yH . TANNER ,

Professor o f Agricu ltu ral Scien ce ,Un iversity C ollege , Ab e rystwith, &c. 18mo .

TAY LOR WORDS AND PLACES ; or, Etymological Illu stration s o f H istory, Ethn ology, and Geography. B y the Rev .

ISAAC TAYLOR , M .A. Third an d cheaper Edition , revisedand compressed . With Maps. G lob e 8vo . 6S.

A HISTOR Y OF THE ALPHA B ET. B y the same

Author.

TAY LOR— A PRIMER OF PIANOF ORTE PLA YING. B yFRANKLIN TAYLOR. Ed ited by GEORGEGROVE. . 18mo . Ir .

TEG B TM B IB R — H OUSEH OLD MANAGEMENT ANDCOOKER Y. With an Appendix o f Recipe s u sed b y the

Teachers o f the National School o f C ooke ry. B y W. B .

TEGETMEIER. Compiled at the requ est_of the Schoo l B oardfo r London . 18mo . Ir.


’

TH R IN G— Works b y EDWARD THRING ,M .A. , Head Master o f

Uppingham .

THE ELEMENTS OF GRAMMAR TA OGET IN

ENGLISH With Question s. Fou rth Edition . 18mo .

THE CHILD’

S GRAMMAR . B e ing the Sub stance o f‘fThe Elemen ts o f Grammar taught in English , adapted forthe Use o f Jun ior C lasse s. A New Ed ition . 18mo . Is.

SCH OOL SONGS . A Collection o f So ngs for Schools.With the Mu sic arranged fo r fou r Voice s. Ed ited b y theRev . E. THRING and H . RIC CIUS. F olio . 7s. 6a’

.

TREN C H (AR CH B ISH OP )— Works b y R . C . TRENCH , D.D.

,

Archb ishop o f Du b lin .

HOUSEH OLD B OOK OF ENGLISH POETR Y. Sele ctedand Arranged , with No te s. Third Edition . Extra Fcap . 8vo .

55 . 6a'.

ON THE STUD Y OF WORDS . Lectu res addre ssed(originally) to the Pupils at the Diocesan Train ing Schoo l,Win che ster. Se ve n tee n th Editio n , revised . Fcap . 8vo . Sr.

ENGLISII, PAST AND PRESENT. Ten th Ed ition,

revise d an d improved . Fcap . 8vo . 55 .

A SELECTGLOSSAR Y OF ENGLISH WORDS,u sed

formerly in Sen ses Difi’

e ren t from the ir Presen t. Fcap .

8vo . 651. [New Edition in tfieprerr.

V AUG H AN (c . M . ) WORDS FROM THE P OETS . B y

C . M . VAUGHAN . Eighth Ed ition . 18mo . cloth. 1s .

WEIR— HARRISON WEIR ’

S DRAWING COP Y-B OOKS .

Ob long 4to . 1. An imals .

W H ITNEY — Works b y WILLIAM D. WHITNEY,Profe ssor o f

San skrit and In stru ctor in Mode rn Lan gu ages in Yale Collegefirst Pre side n t o f the Ame rican Philological Association

,and

hon . memb e r o f the Royal Asiatic Socie ty of Great B ritain an d

Ire land and Corre spon den t o f the B erlin Academy o f Scien ces.

A COMPENDIOUS GERMAN GRAMMAR . Crown8m . 4 3. 6 1.

MACMILLAN’

S

G L O B E L I B RA R Y .

B eau tifu lly prin ted on ton ed paper, price 35. 6a'. each. Also kept

in variou s morocco and calf b ind ings, at moderate prices.

The Satu rday Review says The G lob e Ed ition s are admirab le,

fo r ‘the ir: scholarly editing , their typographical exce llen ce ,”

the ircompendiou s form ,

and the ir cheapn e ss.

The Ddi/yfilefiapk calls it a series ye t unrival ledf or its,com ~

b ination o f exce llen ce and cheapn ess .

”

SHAKESPEARE’S COMPLETE WORKS. Edited b yW. G .

A . With G lossary.

0 k o f fKing

du ction , Note s, and G lossary, b y Sir EDWARD STRACHEY.

B URNS S COMPLETE WORKS : the Poems, Songs

‘

,and

Le tters. Edited,with G lossarial Index and B iographical

Memoir, b y ALEXANDER SM ITH .

ROB INSON CRUSOE. Edited afterthe Origin al Edition s, withB iographical In trod uction , b y HENRY KINGSLEY .

SCOTT’S P OETICAL WORKS . With B iographical and Critical

Essay, b y FRANC IS TURNER PALGRAVE.

GOLDSMITH ’

S MISCELLANEOUS WORKS . With'

B io

graph ical In trodu ction b y Professor MASSON .

SPENCER’

S COMPLETE WORKS . Ed ited , with G lossary,b y R . MORRIS

,and Memoir b y J . W. HALES .

P OPE’S P OET[ CAL WORKS . Edited , with Notes an d Intro

du ctory Memoir, b y Professor WARD.

DR YDEN ’S P OETICAL WORKS . Edited , with a Revised

Text and Notes, b y W. D . CHRISTIE, M .A . , Trin ity C ollege ,Camb ridge .

COWPER ’S P OET[ CAL WORKS . Ed ited , with Note s and

B iographical In trodu ction , b y W. B ENIIAM .

VIRGIL’S WORKS . Rendered in toEnglish Prose . With In tro

du ction s, Note s, Analysis, and Index, b y J . LONSDALE, M.A . ,

and 8 . LEE, M.A .

HORACE. Rende red in toEnglish Prose . With run n ing An alysis,In trodu ction , and Notes, b y J . LONSDALE, M.A . ,

and S . LEE,M .A.

MILTON ’S P OETICAL WORKS . Edited , with Introdu ction s,

&c . , b y Professor MASSON .

P ub/irked even ) fl ursa’ay, price 6a

’. Mom‘fily parts,

2s. and as. Hagf I/ecmy Volumes, 1 55.

AN ILLUSTRATED yOURNAL OF SCIENCE.

NATURE expounds in a popu lar and yet au then ticman n er

,the GRAND RESULTS OF SCIENTIFIC RESEARCH ,

discu ssing the most recen t scien tific discoveries, and

poin ting ou t the b earing of Scien ce upon civilisationand progress, and its claims to a more gen eral recog

n ition,as well

’

as to a higher place in the educational

system of the coun try.

It con tain s original articles on all su b jects within the

domain o f Scien ce ; R e views se tting forth the nature and

valu e of recen t Scien tific Works ; Corresponden ce

C o lumn s, forming a medium o f Scien tific discu ssion and

o f in tercommu n ication amo ng the mo st distingu ishedmen o f Scien ce ; Serial Co lumn s, giving the gist of the

mo st importan t papers appearing in Scien tific J ournals,b o th Home and Foreign ; Tran saction s of the prin

cipal Scien tific Socie ties and Academies o f the World ,

No tes, & c.

In Schoo ls where Scien ce is in cluded in the regu lar

course of studie s,this paper will b e mo st acceptab le , as

it te lls what is doing in Science all over the world , is

popu lar withou t lowering the standard of Science,and b y

it a vast amou n t of in formation is b rought within a smallcompass, an d stu den ts are directed to the b est sources

for what they n eed . The variou s que stion s conn ected

with Science teaching in scho o ls are also fu l ly discu ssed ,and the b est me thods of teaching are indicated .

Elementary Lessons Sound - Forgotten Books

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