Commuted Waveguide Synthesis of the Clavichord

71Valimaki Laurson and Erkut

Vesa Valimaki Mikael Laursondagger

and Cumhur Erkut

Laboratory of Acoustics and AudioSignal ProcessingHelsinki University of TechnologyPO Box 3000FIN-02015 HUTEspoo FinlandVesaValimaki CumhurErkuthutdaggerCentre for Music and TechnologySibelius AcademyPO Box 86Helsinki Finlandlaursonsiba

Commuted WaveguideSynthesis of theClavichord

Computer Music Journal 271 pp 71ndash82 Spring 2003q 2003 Massachusetts Institute of Technology

The clavichord is one of the oldest keyboard instru-ments and it is still often used in performancesand recordings of Renaissance and Baroque musicThe sound of the instrument is pleasant and ex-pressive but quiet Consequently the instrumentcan only be used in intimate performances forsmall audiences This is the main reason why theclavichord was replaced by the harpsichord and -nally by the modern piano both of which producea considerably louder output Attempts have beenmade to amplify the sound of the clavichord usinga piezoelectric pickup (Burhans 1973)

One of our motivations in this research is to givethe clavichord a new life in the digital worldwhere the faint sound level of the instrument canbe ampli ed by simply turning a volume knob Thesuggested synthesis model is based on digital wave-guide modeling of string instruments (Smith 19921998 Valimaki et al 1996 Karjalainen Valimakiand Tolonen 1998) and uses the principle of com-muted waveguide synthesis where the soundboxrsquosresponse is incorporated in the excitation signal(Smith 1993 Karjalainen and Valimaki 1993 Karja-lainen Valimaki and Janosy 1993) Special sam-pling techniques are also employed Musicalexamples produced using the proposed synthesizerwill be included on a forthcoming Computer MusicJournal CD

Acoustics of the Clavichord

A photograph of the clavichord used for the mea-surements in this study is shown in Figure 1 The

instrument is an Anthony Sidey clavichord manu-factured by Heugel in Paris France in 1988 Thisclavichord is an unfretted one so any combinationof notes can be played The range of a clavichord isanywhere from three octaves to over ve octavesOur clavichord has 51 keys ranging from C2 to D6The instrument was tuned about a whole tonelower than the standard modern tuning(A4 4 440 Hz) the nominal frequency of A4 is 395Hz The Werkmeister tuning system was used

For each key of the clavichord a pair of strings istuned in unison as sketched in Figure 2 (see forexample Thwaites and Fletcher 1981 Campbelland Greated 1987) However the two strings are al-ways slightly detuned around the same note be-cause exact tuning is impossible manually Everykey forms one end of a lever that has a tangent at-tached to its other end When a key is depressedthe tangent hits the string pair and initiates vibra-tion One end of the strings has been damped withfelt and the other end goes over a bridge to thetuning mechanism Thus the strings are freely vi-brating between the bridge and the tangent whichworks as both a hammer and a termination Thetangent mechanism is rather noisy as it excitesmodes of the soundboard but also itself causessound from its moving parts owing to friction

When the key is released the tangent falls backwith the aid of gravity and the string vibration isallowed to propagate to the felt-covered end of thestring which ef ciently damps the vibration Atthe end of each note another knock is heard as thetangent returns to its resting position The some-what mistuned strings of each pair are coupled viaa non-rigid bridge and thus both beats and a two-stage decay result (Weinreich 1977)

72 Computer Music Journal

The rst thing that people usually notice aboutthe clavichord is that the sound level is very lowThe maximum sound pressure level at 1 meter isonly about 50 dB or 60 dB depending on the indi-vidual construction of the instrument This makesthis ancient instrument a sound source less ef -cient than a human speaker There are many rea-sons for the weak output (see Campbell andGreated 1987 Fletcher and Rossing 1991) Thestrings are thin and their tension is low and theyradiate sound inef ciently The soundboard issmall and light and it cannot amplify the soundmuch

A particularly interesting feature in the clavi-chord is the mechanical aftertouch known as Be-bung When the player increases pressure on a keythe tangent is raised more which in turn increasesthe tension of the string pair resulting in a raise ofpitch This enables continuous control of vibratowhich is a used performance style Aftertouch hasbeen part of keyboard controllers since the 1980sand it is thus easy to include this control in theclavichord synthesizer However a fully poly-

phonic aftertouch which would really be needed isonly available in high-end keyboard controllers

As the key pressure is changed the tangentcauses a slightly different change in the tension ofthe two strings of a pair This feature affects thetimbre of clavichord tones by introducing a ang-ing effect It is heard during and after the attackparticularly in the bass register

Key

Tangent

Felt

Bridge

Tuningpins

Lever

Stringpair

Figure 1 Clavichord usedin this study

Figure 2 Tangent mecha-nism of one key of theclavichord and a stringpair associated with it


0 02 04 06 08 1 1221

0

1

Lev

el

0 02 04 06 08 1 12197

198

199

200

F0

(Hz)

Time (sec)

0 02 04 06 08

2120

2100

280

Time (sec)

Mag

nitu

de (

dB)

Properties of Single Clavichord Tones

Figure 3 shows the envelope of a recorded clavi-chord tone The irregular non-exponential decay ofthe tone can be observed The key mechanism ofthe clavichord generates a loud knock at the begin-ning and end of a tone which is characteristic tothe sound of the instrument In Figure 3 a burst lo-cated at 11 sec corresponds to the thump causedby release of the key at the same time the signalstarts to decay quickly The envelope curves of the rst ve harmonics are presented in Figure 4 Notethat regular exponential decay (ie linear decay ona dB scale) is rare pronounced beating and other ir-regularities are observed in many harmonics (egthe beating of the fourth partial)

The fundamental frequency of clavichord tonesvaries over time as illustrated by an example inthe lower part of Figure 3 This may be partlycaused by tension modulation (Tolonen Valimakiand Karjalainen 2000) which is a purely physicalphenomenon in vibrating strings but also by thepressure of the playerrsquos nger (ie the mechanicalaftertouch) which directly controls string tensionduring playing A time-domain nite-differencesimulation suggests that the tension modulation ef-fect is negligible compared to that caused by themechanical aftertouch (Valimaki et al 2000) InFigure 3 the pitch glide is about 2 Hz which is

lower than the threshold of audibility for this ef-fect According to Jarvelainen and Valimaki (2001)the threshold at 196 Hz (the nearest frequencytested) is 44 Hz In the case of the clavichord usedin our measurements during normal playing thepitch glide can be inaudible for most listeners Au-dible pitch glides can be generated easily howeverFigure 5 gives an example of a tone that containsan exaggerated pitch vibrato which should be obvi-ous for all listeners Note that the vibrato width istypically about 2 Hz or less but can occasionallypeak higher The vibrato rate is about 7 Hz in thiscase which is slightly faster than what would beusual in a musical context The above fundamentalfrequency measurements re ect the effective pitchwhere the contributions of both strings are present

Impulsively driven string instrument sounds arealways at least weakly inharmonic (Fletcher andRossing 1991) This is caused by the stiffness of thestring material It is well known that piano tonesparticularly in the bass range are highly inhar-monic which affects both the timbre and the tun-ing Measurements of inharmonicity have not beenpublished previously for the clavichord so we con-ducted them ourselves This was needed to decidewhether we had to implement the inharmonicityusing an allpass lter as is customary to do inwaveguide synthesis (Jaffe and Smith 1983 Bank2000) We used our parameter estimation softwareto extract partial frequencies of clavichord tones asa function of time Using a least-squares t we es-timated from the data the inharmonicity coef -cient We found that the clavichord used in ourmeasurements produces almost perfectly harmonictones (The inharmonicity coef cient does not ex-ceed 10ndash6) Thus the effect of inharmonicity can beconsidered inaudible (Jarvelainen Valimaki and

Figure 3 (Top) Waveformof a clavichord tone (A3)showing the irregular over-all decay pattern and thethump at the end of the

tone at 11 sec and (bot-tom) the time history of itsfundamental frequency(nominal fundamental fre-quency is 1975 Hz)

Figure 4 Envelopes of thelowest partials of the toneshown in Figure 1 1st(solid line with circles)2nd (solid line) 3rd

(dashed line) 4th (dash-dot line) and 5th (dottedline)


0 100 200 300 400 500

20

40

60

Mag

nitu

de (

dB)

0 100 200 300 400 500

20

40

60

Mag

nitu

de (d

B)

Frequency (Hz)

Karjalainen 2001) and perfectly harmonic synthesisis adequate

Soundbox

An essential part of clavichord tones is the re-sponse of the soundbox The box is strongly excitedevery time a key is pressed This reverberant re-sponse will ring for about ve seconds and itstands out particularly in staccato notes where thestring vibration itself is damped quickly after theattack

We have conducted a series of acoustic measure-ments on the soundbox of the instrument to iden-tify the most prominent modes Figure 6 shows themagnitude spectrum of the soundboard that washit with an impulse hammer when the strings werecarefully damped with soft material Results of twoseparate measurements are plotted on top of eachother The differences between these curves arequite small verifying the repeatability of the mea-surement Many narrow peaks are visible in Figure6 They correspond to long ringing resonancemodes It can also be seen that the excitation ofmodes depends on the location of excitation Forexample the 90 Hz mode is much stronger in thetop part of the gure than in the lower part Asimilar phenomenon occurs in the case of the modeat about 340 Hz

Synthesis Model for the Clavichord

Next we describe the synthesis model for clavi-chord tones The commuted waveguide synthesismethod invented by Smith (1993) and Karjalainenand colleagues (Karjalainen and Valimaki 1993Karjalainen Valimaki and Janosy 1993) is appliedby using inverse- ltered clavichord signals as exci-tation for the synthesis model Inverse lteringhere refers to the processing of a signal with the in-verted transfer function of a waveguide stringmodel In this case the inverse ltering essentiallycancels the partials of a recorded tone The com-muted waveguide synthesis method has been for-merly used successfully for the synthesis of theacoustic guitar (Karjalainen Valimaki and Janosy1993 Valimaki et al 1996 Laurson et al 2001) andthe piano (Smith and Van Duyne 1995)

Figure 7 shows the structure of the synthesismodel developed in this work The coupling of thetwo basic strings models Sl(z) and S2(z) is realizedwith an unconditionally stable technique suggestedby Karjalainen et al (1998) the output of only oneof the string models is fed to the input of the otherand hence there is no feedback In practice thecoupling coef cient gc is selected to have a smallvalue but this is not required because there can beno stability problems The input and output signalsof the string models are scaled by constant multi-

0 05 1 15 2 2521

0

1

Lev

el

0 05 1 15 2 25196

198

200

202

F0 (

Hz)

Time (sec)

Figure 5 (Top) Waveformof clavichord tone A3 withstrong vibrato (mechanicalaftertouch) and (bottom)its fundamental frequencyas a function of time

Figure 6 Magnitude spec-trum of the soundboardexcited with an impulsehammer at the high-frequency (top) and at thelow-frequency end of the

bridge (bottom) Results oftwo measurements (solidand dashed line) are plot-ted one on the other forboth cases


TimbrecontrolExcitation

signals

Sound

Soundboxresponses

Trigger at attack time

Trigger at release time

End knocksgsb

grelease

gc

S2

S1

gi1

gi2

go1

go2

)(zH lLz)(zF

)(zH l)(zF

attackg

(a)

(b)

Lz2

2

plying coef cients gi and go respectively In addi-tion two sample databases are needed for realisticsimulation of the soundbox response and the per-cussive noise caused by key release In the follow-ing we discuss the string algorithm modelparameters and their control and the sample data-bases

Modied String Algorithm

The strings of the clavichord can be simulated us-ing digital waveguide string models In principletwo digital waveguide models should be used foreach string since they have two polarizations of vi-bration (ie horizontal and vertical with respect tothe soundboard) but at present we only use onedigital waveguide per string For a pair of clavi-chord strings we thus have two digital waveguidestring models not four This choice results in anef cient algorithm and still generates some beatingin the synthetic tones

Usually a digital waveguide string model has thefollowing transfer function (Jaffe and Smith 1983Valimaki et al 1996) as also illustrated in Figure8(a)

1S (z) 4 (1)

1 L1 1 F (z )H (z)z1

where F(z) 4 h0 ` h1z ndash1 ` h2z ndash2 ` h3z ndash3 is a third-order Lagrange interpolation lter (Laakso et al1996) L is the (integer) length of the delay line and

Hl(z) is a loop lter with the following transfer func-tion (Valimaki et al 1996)

1 ` aloopH (z) 4 g (2)1 loop 1 11 ` a zloop

with 0 gloop 1 and ndash1 aloop 0 The values ofthese parameters can be estimated for each stringusing previously developed methods (Valimaki etal 1996 Erkut et al 2000)

While testing an early version of the clavichordsynthesizer we noticed that the contribution of thetangent knock was not suf ciently prominent Inparticular the attack of high notes sounded toosoft To x this de ciency we modi ed the abovestring algorithm to allow the attack sharpness to becontrolled with one parameter The block diagram

Figure 7 Clavichord syn-thesis algorithm for onekey consists of three data-bases of samples and twobasic string models thatare coupled

Figure 8 Block diagram of(a) the standard and (b)modied string algorithmThe latter allows control-

ling the sharpness ofattack using parametergattack


0 10 20 30 40 500985

099

0995

1

g 1oop

0 10 20 30 40 50

204

202

0

Key index

a 1oop

of the new algorithm is shown in Figure 8b Itstransfer function can be written as

1 LF (z) H (z) zlS (z) 4 g `attack 1 L1 1 F (z) H (z) zl (3)1 Lg ` (1 1 g ) F (z) H (z) zattack attack l

41 L1 1 F (z) H (z) zl

Note that with gattack 4 1 this algorithm is identi-cal to the standard plucked-string algorithm ofEquation 1 A value larger than unity such asgattack 4 4 may be used to imitate the aggressive at-tack of acoustic tones In the current version thevalue of the attack sharpness parameter increaseslinearly with key number so that for C2 it is 20and for D6 it is 40 The linear trend and the actualvalues were adjusted by informal listening There isno physical motivation for this new model struc-ture other than that it enables separate control oftwo different components of the tone the tangentknock and the vibrating string sound

String Model Parameters

Parameter values for the synthesizer are obtainedfrom analysis of recorded signals We use a methodbased on the short-time Fourier transform and si-nusoidal modeling of the clavichord tones Thesemethods have proven successful earlier in the case

of the acoustic guitar Details can be found else-where (Valimaki et al 1996 Erkut et al 2000)

Figure 9 shows the gloop and aloop parameters forall 51 keys of the clavichord These values were es-timated from recorded clavichord tones The sec-ond string model uses the same parameters butslightly different delay lines lengths for mistuningThe fundamental frequencies of the lowest (1) andhighest (51) key are 587 Hz and 1056 Hz respec-tively Note that the loop gain parameters gloop havea nearly linear trend as a function of key index (ielogarithm of frequency) with some variations and afew outliers Jarvelainen and Tolonen (2001) haveinvestigated the tolerance of parameters value vari-ation Their results show that ndash30 to ` 40changes in the time constant (which depends ongloop) are inaudible Also in the aloop parameter vari-ations between ndash17 and ` 16 cannot be per-ceived It appears that the gloop parameterspresented in Figure 9 can be replaced with valuesobtained by linear regression

g rsquo 0989 ` 0000189k (4)loop

where k 4 1 51 is the key number starting fromC2 (k 4 1) and continuing up to D6 (k 4 51) Thisstraight-line t is illustrated in Figure 9 (top) with asolid line

The values of parameter aloop may be approxi-mated by a low-order polynomial as well Howevera linear regression to the data shown in the lowerpart of Figure 9 yields intolerably large errors Toreduce the error we attempt a piecewise linear re-gression with a knee point between key indices 23and 24

1 0514 ` 00176k for k 23a rsquo (5)loop 51 0175 ` 000225k for k $ 24

This two-part straight-line t is shown in Figure 9(bottom) with solid lines one for the low end of thekeyboard and another for the high end This ap-proximation for the aloop values introduces minorchanges in timbre that can be perceived but it doesnot necessarily render the sound quality worse Infact the overall tone color behaves more uniformlywith the linear approximations than with the un-processed parameter data

The damping of the string vibrations as a resultof releasing a key is simulated by changing the loop

Figure 9 Estimated valuesof string model parametersgloop (top) and aloop (bottom)(dots connected with

dashed line) together withtheir straight-line approxi-mations (solid lines)


lter characteristics (eg by setting loop gain gloop

equal to 01 momentarily) A similar method hasbeen used successfully to simulate damping of aguitar string by nger (Erkut et al 2000 Laurson etal 2001)

Sample Databases

Excitation signals for the synthesis have been ob-tained by processing anechoic recordings of clavi-chord tones using the methods described by Erkutet al (2000) The procedure consists of sinusoidalanalysis subtracting partials equalizing the resid-ual and truncating the resulting signals with theright half of a Hanning window One such excita-tion signal is used for each of the 51 keys Cur-rently all excitation signals are 20000 sampleslong (about 045 sec at a 441 kHz sampling rate)It would be possible to reduce their length furtherbut we have found it unnecessary in our currentimplementation since there is no lack of memory

The reverberation caused by the soundbox is in-corporated by triggering a soundbox response sam-ple at a low level each time any note is played (seeFigure 7) This sample must be long enough (about ve seconds) so that it provides the reverberantcharacter of the instrument This is particularlyimportant for short notes such as staccato playingfor which the output signal would otherwise stopsuddenly in an unnatural manner Note that inprinciple we could instead use very long excitationsignals for each note in which case the full sound-box response would automatically sound with eachnote However we prefer using short excitation sig-nals because we can avoid certain problems relatedto the inverse ltering (eg ringing and beating dueto imperfectly cancelled partials) This also savesmemory because we use fewer soundbox samplesthan excitation signals Using a separate soundboxsample also allows adjusting the relative levels ofsoundbox reverberation and string sound which isan advantage

The soundbox samples can be obtained by hittingthe bridge of the clavichord with an impulse ham-mer at various points The response must be editedto remove the sharp attack caused by contact of

metals that is the tip of the hammer and a tuningpin This is easily done with a fast fade-in Notethat this editing is not extremely critical becausethe impact noise from the tangent mechanismwhich is included in the excitation signal domi-nates during the attack and masks the beginning ofthe soundbox response For each key a soundboxsample should be chosen which is obtained closeto the point where the strings are attached How-ever it is practically impossible to excite the bridgeof the clavichord between the tuning pins and theonly clean samples that we could obtain are fromboth ends of the bridge (see Figure 6) This leavesus with possibilities to use one of the samples forthe left half of the keyboard and the other for theright one or to interpolate between the high- andlow-end samples along the keyboard Although thesampling-based reverberation modeling is clearlysimpli ed it yields a natural-sounding ambience inthe synthetic output and is very cheap to imple-ment because it requires no ltering

The knock database for note endings consists of afew short samples (duration less than 05 sec)These samples may be obtained from recorded longtones where the sound of string vibration is al-lowed to decay below the threshold of audibilitybefore the key is released Now the isolated thumpcan be simply edited from the recording One prac-tical problem was caused by the poor signal-to-noise ratio of the release samples It was possible tohear when a key release occurred from the in-creased hiss during synthesis This led us to apply ade-noising algorithm to suppress the backgroundnoise from the key release recordings which solvedthe problem

It is unnecessary to record a release sample foreach key Because the tangent mechanism for everykey is identical and the resulting sound is similarit is plausible to use a single sample for all syn-thetic notes To avoid a boring similarity as a com-promise a small collection of release samples maybe used We have observed that the largest varia-tion among different tangent releases is caused bythe velocity of release which slightly changes therhythm of different noises within the knock Thusone choice is to gather a collection of tangent re-leases for different release velocities and select one


2050

05 (a)

2050

05 (b)

2010

01 (c)

2010

01 (d)

0 02 04 06 08 1 12 14 16205

005 (e)

Time (sec)

randomly or based on some rule such as the play-ing tempo We are currently using two release sam-ples and we slightly randomize the release times ofdifferent keys More sophisticated but still fairlysimple synthesis models employing physically in-formed randomization have been proposed by Cook(1997)

Delay Line Variations and the Flanging Effect

Although the variation of the fundamental fre-quency in clavichord sounds can be inaudible asdiscussed earlier it is also easy to cause pitchchanges with mechanical aftertouch Because anychange in the string tension will be different for thetwo strings of a pair a anging-like effect can ap-pear also in tones where the pitch change is negli-gible Variation of the fundamental frequency canbe simulated in the following way a decaying con-trol function for example a scaled impulse re-sponse of a leaky integrator (a one-pole lter) issubtracted from the delay-line length of both stringmodels This imitates the change of pressure on thekey by the playerrsquos nger right after depressing thekey (or the decrease of tension modulation depthwhile the string vibration begins to decay) andbrings about the progression of the fundamentalfrequency similar to that shown in Figure 3 Note

that this is a simpler way of producing a time-varying pitch than the tension-modulation tech-nique Tolonen et al (2000) used previously for thesynthesis of the kantele (Valimaki et al 1999 Er-kut et al 2002) and the tanbur (Erkut and Valimaki2000) The tension-modulation algorithm would re-quire computing the elongation of the string in realtime and controlling the delay-line length of thestring model according to it

For the mechanical aftertouch a more sophisti-cated control signal is required The pitch contourof a note is generated with a help of a pitch scalerin the same fashion as in Laurson et al (2001) Ifthere is neither an initial pitch glide nor vibratothe scaler value is 1 The initial pitch glide is im-plemented as a ramp starting from a value largerthan 1 and ending at 1 after around 01 sec (Theinitial value depends on the dynamic level of thecurrent note so that forte playing yields a highervalue than piano playing) The vibrato control inturn is realized with separate parameters for rateand maximum depth and a temporal envelope fordepth (for more details see Laurson et al 2001)

A speci c feature of the clavichord synthesizer isthat the pitch scaler affects a detuning parameter ofthe two string models dynamically Thus a moredrastic pitch drift results in a tone that is more outof tune This mimics the imperfect mechanism ofthe tangent that pulls the two strings unequally Adatabase of detuning parameters was generated bytrial and error An alternative way would be tomeasure detuning parameters from the original re-cordings which turns out to be dif cult High-pitched tones tend to be more detuned than lowones The sound resulting from the time-varyingdetuning is suggestive of the anging effect whichbrings warmth and variation to the synthesis

Synthesis Example

Figure 10 presents the components for the synthe-sis of a single clavichord tone The excitation sig-nal shown in Figure 10a is inserted into the twostring algorithms The name lsquolsquocommuted synthe-sisrsquorsquo comes from the principle that this excitationsignal includes both the contribution of the excita-

Figure 10 Signals involvedin the synthesis of a singleclavichord tone (a) excita-tion signal (b) string algo-rithm output (c) soundbox

sample (d) end noise sam-ple (triggered at about 1sec) and (e) synthetictone which is sum of sig-nals (b) to (d)


tion mechanism (the tangent) and the soundbox re-sponse However as can be seen in ourimplementation the input signal contains only thevery beginning of the soundbox response The tailof the soundbox response is produced separately us-ing the sampling-based method

The summed output signals of the string modelsconstitute the basic vibrating string component ofthe synthetic tone which is displayed in Figure10b The key release occurs at 097 sec note howthe string tone in Figure 10b decays quickly afterthat Figure 10c shows the beginning of a ve-second-long soundbox sample which has beenedited from the impulse response of the low-frequency end of the bridge A tangent release sam-ple which is started at the release time is given inFigure 10d Finally Figure 10e presents the sum ofthe signals shown in Figures 10b 10c and 10d andit is the output of the clavichord synthesizer

Although remaining almost invisible in Figure10e the soundbox sample rings several seconds af-ter the string tone and the key release sample havedied out The soundbox sample will also be clearlyaudible only after the string tone has been ringingand decaying for some time and particularly afterthe key has been released We may note that iden-tity resynthesis is impossible using the proposedsynthesis model and parameter estimation meth-ods Nevertheless the obtained similarity is con-sidered to be suf cient for high-quality synthesisThis is con rmed by musical examples producedwith the synthesizer

Musically Interesting Modications

The proposed synthesizer structure allows the userto modify the timbre in numerous ways Many ofthe possible variations have a clear intuitive inter-pretation For example varying the gain gloop of theloop lter affects the decay time but otherwisethe timbre remains unchanged However varyingthe loop lter coef cient aloop changes the decaytime at different frequencies resulting in a curiousevolution of the spectrum over time The attacksharpness parameter and the gain of the note-offnoise also enable meaningful and useful controls

Exaggerated detuning of the two string models foreach voice leads to a low-qualitymdashor possibly avery oldmdashclavichord As a consequence of theabove parametric modi cations the synthetictones will still be reminiscent of the clavichord

Extending the range of an instrument is fascinat-ing and useful in the case of the acoustic guitarDesign of a synthetic lsquolsquosuper guitarrsquorsquo that has awide range almost like that of a grand piano hasbeen documented recently (Laurson Valimaki andErkut 2002) While an arbitrary change of pitch iseasy in a waveguide synthesizer (simply vary thelength of a delay line) the loop lter parametersand excitation signals should be extracted from re-cordings but for non-existing fundamental frequen-cies they are unavailable The extension of thepitch range thus involves extrapolation problemsthat must be solved somehow We are planningtests on extending the range of the proposed clavi-chord synthesizer

Naturally it is also possible to turn the clavi-chord synthesizer into a previously unheard virtualinstrument For instance modifying or replacingthe signals in the sample databases allows dramaticvariations Exploring all such transformations wasnot included in the goals of this study and the po-tential of these modi cations thus remains mostlyunknown

Software Implementation of the Synthesizer

The real-time implementation of the proposed clav-ichord synthesizer has been realized using aPatchWork (Laurson 1996) user-library calledPWSynth (Laurson and Kuuskankare 2001) Thehigh-level part of PWSynth is based on Lisp andCLOS but the low-level and time-critical DSP-routines have been implemented in C The systemcan be played either from musical notation andcontrol software called Expressive Notation Pack-age (ENP described in Kuuskankare and Laurson2001 Laurson et al 2001) or from a MIDI key-board

The starting point for the clavichord implemen-tation was to use a guitar synthesizer implementedearlier (Laurson et al 2001) These systems have


several features in common Both use basically thesame dual-polarization string model although inthe case of the clavichord it models a pair ofstrings and not the orthogonal polarizations of asingle string The database of samplesmdashneeded forthe excitation signals and the noise effects in theclavichord simulationmdashcould be used in both casesin a similar way

The main difference between the implementa-tions is the fact that the clavichord consists of 51strings while the classical guitar has only six Tocreate a synthesizer that could be used in real timewe designed a voice-allocation algorithm that al-lows us to effectively play typical pieces of theclavichord repertoire We found that a synthesizerfor six simultaneous strings could be played in realtime on a Macintosh G3 400 MHz portable com-puter This is suf cient for a large percentage of oldkeyboard music New faster computers will allowplaying more voices simultaneously

The control information for the clavichord syn-thesizer is calculated from an ENP input score Theuser enters rst the basic musical information (iepitches and rhythms) of the piece using the graphi-cal front-end of ENP After this the user typicallyadds expressions and tempo functions that allow ne tuning of the timing information in the scoreAlso one can include expressions speci c to theclavichord such as the amount of vibrato to be ap-plied to a given note Furthermore a special rulemodi es the end time of notes (ie the time whenthe player lifts the ngers from the keyboard) Thiswas done to make the performance more livelyThe characteristic noise burst of the instrumentwhen the ngers are released was found to be es-sential for the realism of the synthesis

Figure 11 shows the three last measures of apiece by Guillaume Dufay (1400ndash1474) Besidespitch and rhythm information the piece includesseveral expression markings that control both vi-brato (see the labels starting with lsquolsquovbrsquorsquo) and articu-lation (see labels starting with lsquolsquostrsquorsquo which standsfor staccato) The numbers after the expressionmarkings denote how much vibrato or staccatoshould be applied to a given note This example hasalso a special expression containing a tempo func-tion (see the function above the staff marked withlsquolsquoritardandorsquorsquo in Figure 11) which controls thetempo of the piece

Our current MIDI keyboard implementation isfairly rudimentary and serves only as a basic toolfor testing The control parameters of the MIDIkeyboard include key number key velocity attacktime and end time of notes Furthermore we usethe MIDI aftertouch parameter to simulate the n-ger vibrato effect of the clavichord A more usableMIDI implementation would require a keyboardwith polyphonic aftertouch Also we found thatthe MIDI keyboard at our disposal was not respon-sive enough when simulating the nger vibrato Aplayable keyboard simulation of the clavichordmodel requires more sensitive hardware than whatis commonly available today

Conclusions and Future Plans

A simpli ed physical model was proposed for thesynthesis of the clavichord It follows the principleof commuted waveguide synthesis but also usesmany samples obtained by editing and processingrecordings The excitation signal can be obtained

Figure 11 ENP screenshotfrom the nal cadence ofPar le regard de vos beauxyeux by Guillaume Dufay


by inverse ltering and truncating a recorded clavi-chord tone Two coupled digital waveguide stringmodels are used for synthesizing each voice A mi-nor modi cation to the plucked string model wasintroduced to enable control over attack sharpnesswhich was found necessary during testing of thesynthesis algorithm The reverberation caused bythe soundbox that has many resonant modes wasimplemented in a computationally economic wayby triggering a soundbox response sample witheach note The ending of a note requires anothersample to be played because there must be alsquolsquoknockingrsquorsquo sound which is characteristic to theclavichord The mechanical aftertouch of the clavi-chord can be implemented by varying the delayline lengths of the waveguide string models overtime When these changes are different for the twostring algorithms modeling a pair of strings a ang-ing effect is generated which is also heard in thereal clavichord

In the future we plan to develop another versionof the clavichord synthesizer using the directphysical modeling approach where the excitationstrings and the soundbox have a counterpart in themodel To accomplish this the sampling-based im-plementation of the reverberant response of thesoundboard must be replaced with a soundboxmodel that lters the output of the string modelsThe soundbox can be considered a small reverber-ant room and be simulated with an arti cial rever-beration algorithm (eg Gardner 1998 Bank 2000)

In addition further measurements of the real in-strument should be taken Particularly the acceler-ation of the tangent needs to be registered duringthe attack of the tone because this information isneeded for an algorithm simulating the tangent ac-tion similar to piano hammer simulations (Bank2000) This might also reveal the importance of thetension modulation effect in the sound productionof the clavichord The tangent modeling algo-rithmmdashwhich will be different from physical mod-els available for the piano hammer as Hall (1993)has pointed outmdashshould be developed to be able tocontrol the excitation using physical parameters

AcknowledgmentsThe work of M Laurson is part of the projectlsquolsquoSounding ScoremdashModeling of Musical Instru-

ments Virtual Musical Instruments and their Con-trolrsquorsquo and the work of C Erkut is part of the projectlsquolsquoTechnology for Speech and Audio ProcessingrsquorsquoBoth projects are funded by the Academy of Fin-land The authors are grateful to Dr Tero Tolonenfor his contributions in the initial phases of thisstudy to Mr Paulo A A Esquef for his help in thede-noising of the sample databases and to MrJonte Knif for his help in the production of musicalexamples using the clavichord synthesizer Specialthanks to Professor Matti Karjalainen who provideddigital photographs of the clavichord and our mea-surement session

References

Bank B 2000 Physics-Based Sound Synthesis of the Pi-ano Report no 54 Helsinki University of TechnologyLaboratory of Acoustics and Audio Signal ProcessingEspoo Finland Available on-line atwwwacousticshutpublications

Burhans R W 1973 lsquolsquoClavichord Ampli cationrsquorsquo Jour-nal of the Audio Engineering Society 21(6)460ndash463

Campbell M and C Greated 1987 The MusicianrsquosGuide to Acoustics New York Schirmer Bookspp 234ndash236

Cook P R 1997 lsquolsquoPhysically Informed Sonic Modeling(PhISM) Synthesis of Percussive Soundsrsquorsquo ComputerMusic Journal 21(3)38ndash49

Erkut C et al 2000 lsquolsquoExtraction of Physical and Expres-sive Parameters for Model-Based Sound Synthesis ofthe Classical Guitarrsquorsquo Proceedings of the AES 108thConvention New York Audio Engineering Society

Erkut C and V Valimaki 2000 lsquolsquoModel-Based SoundSynthesis of Tanbur a Turkish Long-Necked LutersquorsquoProceedings of the IEEE International Conference onAcoustics Speech and Signal Processing Vol 2 Pis-cataway New Jersey Institute of Electrical and Elec-tronics Engineers pp 769ndash772

Erkut C et al 2002 lsquolsquoAcoustical Analysis and Model-Based Sound Synthesis of the Kantelersquorsquo Journal of theAcoustical Society of America 112(4)1681ndash1691

Fletcher N H and T D Rossing 1991 The Physics ofMusical Instruments New York Springer-Verlag

Gardner W G 1998 lsquolsquoReverberation Algorithmsrsquorsquo In MKahrs and K Brandenburg eds Applications of DigitalSignal Processing to Audio and Acoustics DordrechtThe Netherlands Kluwer pp 85ndash131


Hall D E 1993 lsquolsquoString Excitation Piano Harpsichordand Clavichordrsquorsquo Proceedings of the 1993 StockholmMusic Acoustics Conference Stockholm SwedenRoyal Swedish Academy of Music pp 309ndash314

Jaffe D A and J O Smith 1983 lsquolsquoExtensions of theKarplusndashStrong Plucked-String Algorithmrsquorsquo ComputerMusic Journal 7(2)76ndash87

Jarvelainen H and T Tolonen 2001 lsquolsquoPerceptual Toler-ances for Decay Parameters in Plucked String Synthe-sisrsquorsquo Journal of the Audio Engineering Society 49(11)1049ndash1059

Jarvelainen H V Valimaki and M Karjalainen 2001lsquolsquoAudibility of the Timbral Effects of Inharmonicity inStringed Instrument Tonesrsquorsquo Acoustics Research Let-ters Online 2(3)79ndash84

Jarvelainen H and V Valimaki 2001 lsquolsquoAudibility of Ini-tial Pitch Glides in String Instrument Soundsrsquorsquo Pro-ceedings of the 2001 International Computer MusicConference San Francisco International ComputerMusic Association pp 282ndash285

Karjalainen M and V Valimaki 1993 lsquolsquoModel-BasedAnalysisSynthesis of the Acoustic Guitarrsquorsquo Proceed-ings of the 1993 Stockholm Music Acoustics Confer-ence Stockholm Sweden Royal Swedish Academy ofMusic pp 443ndash447

Karjalainen M V Valimaki and Z Janosy 1993 lsquolsquoTo-wards High-Quality Sound Synthesis of the Guitar andString Instrumentsrsquorsquo Proceedings of the 1993 Interna-tional Computer Music Conference San Francisco In-ternational Computer Music Association pp 56ndash63

Karjalainen M V Valimaki and T Tolonen 1998lsquolsquoPlucked-String Models From the KarplusndashStrong Al-gorithm to Digital Waveguides and Beyondrsquorsquo Com-puter Music Journal 22(3)17ndash32

Kuuskankare M and M Laurson 2001 lsquolsquoENP MusicalNotation Library based on Common Lip and CLOSrsquorsquoProceedings of the 2001 International Computer Mu-sic Conference San Francisco International ComputerMusic Association pp 131ndash134

Laakso T I et al 1996 lsquolsquoSplitting the Unit DelaymdashTools for Fractional Delay Filter Designrsquorsquo IEEE SignalProcessing Magazine 13(1)30ndash60

Laurson M 1996 PATCHWORK A Visual Program-ming Language and Some Musical Applications Doc-toral dissertation Sibelius Academy HelsinkiFinland

Laurson M et al 2001 lsquolsquoMethods for Modeling Realis-tic Playing in Acoustic Guitar Synthesisrsquorsquo ComputerMusic Journal 25(3)38ndash49

Laurson M and M Kuuskankare 2001 lsquolsquoPWSynth A

Lisp-based Bridge between Computer Assisted Compo-sition and Model-based Synthesisrsquorsquo Proceedings of the2001 International Computer Music Conference SanFrancisco International Computer Music Associationpp 127ndash130

Laurson M V Valimaki and C Erkut 2002 lsquolsquoProduc-tion of Virtual Acoustic Guitar Musicrsquorsquo Proceedings ofthe AES 22nd International Conference on VirtualSynthetic and Entertainment Audio New York Au-dio Engineering Society pp 249ndash255

Smith J O 1992 lsquolsquoPhysical Modeling Using DigitalWaveguidesrsquorsquo Computer Music Journal 16(4) 74ndash91Available on-line at www-ccrmastanfordedujospmudw

Smith J O 1993 lsquolsquoEf cient Synthesis of Stringed Musi-cal Instrumentsrsquorsquo Proceedings of the 1993 Interna-tional Computer Music Conference San FranciscoInternational Computer Music Association pp 64ndash71Available on-line at www-ccrmastanfordedujoscs

Smith J O 1998 lsquolsquoPrinciples of Digital Waveguide Mod-els of Musical Instrumentsrsquorsquo In M Kahrs and K Bran-denburg eds Applications of Digital Signal Processingto Audio and Acoustics Dordrecht The NetherlandsKluwer pp 417ndash466

Smith J O and S A Van Duyne 1995 lsquolsquoCommuted Pi-ano Synthesisrsquorsquo Proceedings of the 1995 InternationalComputer Music Conference San Francisco Interna-tional Computer Music Association pp 319ndash326

Thwaites S and N H Fletcher 1981 lsquolsquoSome Notes onthe Clavichordrsquorsquo Journal of the Acoustical Society ofAmerica 69(5)1476ndash1483

Tolonen T V Valimaki and M Karjalainen 2000lsquolsquoModeling of Tension Modulation Nonlinearity inPlucked Stringsrsquorsquo IEEE Transactions on Speech andAudio Processing 8(3)300ndash310

Valimaki V et al 1996 lsquolsquoPhysical Modeling of PluckedString Instruments with Application to Real-TimeSound Synthesisrsquorsquo Journal of the Audio EngineeringSociety 44(5)331ndash353

Valimaki V et al 1999 lsquolsquoNonlinear Modeling and Syn-thesis of the Kantelemdasha Traditional Finnish String In-strumentrsquorsquo Proceedings of the 1999 InternationalComputer Music Conference San Francisco Interna-tional Computer Music Association pp 220ndash223

Valimaki V et al 2000 lsquolsquoModel-Based Synthesis of theClavichordrsquorsquo Proceedings of the 2000 InternationalComputer Music Conference San Francisco Interna-tional Computer Music Association pp 50ndash53

Weinreich G 1977 lsquolsquoCoupled Piano Stringsrsquorsquo Journal ofthe Acoustical Society of America 62(6)1474ndash1484


The rst thing that people usually notice aboutthe clavichord is that the sound level is very lowThe maximum sound pressure level at 1 meter isonly about 50 dB or 60 dB depending on the indi-vidual construction of the instrument This makesthis ancient instrument a sound source less ef -cient than a human speaker There are many rea-sons for the weak output (see Campbell andGreated 1987 Fletcher and Rossing 1991) Thestrings are thin and their tension is low and theyradiate sound inef ciently The soundboard issmall and light and it cannot amplify the soundmuch

A particularly interesting feature in the clavi-chord is the mechanical aftertouch known as Be-bung When the player increases pressure on a keythe tangent is raised more which in turn increasesthe tension of the string pair resulting in a raise ofpitch This enables continuous control of vibratowhich is a used performance style Aftertouch hasbeen part of keyboard controllers since the 1980sand it is thus easy to include this control in theclavichord synthesizer However a fully poly-

phonic aftertouch which would really be needed isonly available in high-end keyboard controllers

As the key pressure is changed the tangentcauses a slightly different change in the tension ofthe two strings of a pair This feature affects thetimbre of clavichord tones by introducing a ang-ing effect It is heard during and after the attackparticularly in the bass register

Key

Tangent

Felt

Bridge

Tuningpins

Lever

Stringpair

Figure 1 Clavichord usedin this study

Figure 2 Tangent mecha-nism of one key of theclavichord and a stringpair associated with it


0 02 04 06 08 1 1221

0

1

Lev

el

0 02 04 06 08 1 12197

198

199

200

F0

(Hz)

Time (sec)

0 02 04 06 08

2120

2100

280

Time (sec)

Mag

nitu

de (

dB)











0 100 200 300 400 500

20

40

60

Mag

nitu

de (

dB)

0 100 200 300 400 500

20

40

60

Mag

nitu

de (d

B)

Frequency (Hz)


Soundbox






0 05 1 15 2 2521

0

1

Lev

el

0 05 1 15 2 25196

198

200

202

F0 (

Hz)

Time (sec)






signals

Sound

Soundboxresponses



End knocksgsb

grelease

gc

S2

S1

gi1

gi2

go1

go2

)(zH lLz)(zF

)(zH l)(zF

attackg

(a)

(b)

Lz2

2





1S (z) 4 (1)

1 L1 1 F (z )H (z)z1










0 10 20 30 40 500985

099

0995

1

g 1oop

0 10 20 30 40 50

204

202

0

Key index

a 1oop



41 L1 1 F (z) H (z) zl






g rsquo 0989 ` 0000189k (4)loop











Sample Databases








2050

05 (a)

2050

05 (b)

2010

01 (c)

2010

01 (d)

0 02 04 06 08 1 12 14 16205

005 (e)

Time (sec)







Synthesis Example































References





































0 02 04 06 08 1 1221

0

1

Lev

el

0 02 04 06 08 1 12197

198

199

200

F0

(Hz)

Time (sec)

0 02 04 06 08

2120

2100

280

Time (sec)

Mag

nitu

de (

dB)











0 100 200 300 400 500

20

40

60

Mag

nitu

de (

dB)

0 100 200 300 400 500

20

40

60

Mag

nitu

de (d

B)

Frequency (Hz)


Soundbox






0 05 1 15 2 2521

0

1

Lev

el

0 05 1 15 2 25196

198

200

202

F0 (

Hz)

Time (sec)






signals

Sound

Soundboxresponses



End knocksgsb

grelease

gc

S2

S1

gi1

gi2

go1

go2

)(zH lLz)(zF

)(zH l)(zF

attackg

(a)

(b)

Lz2

2





1S (z) 4 (1)

1 L1 1 F (z )H (z)z1










0 10 20 30 40 500985

099

0995

1

g 1oop

0 10 20 30 40 50

204

202

0

Key index

a 1oop



41 L1 1 F (z) H (z) zl






g rsquo 0989 ` 0000189k (4)loop











Sample Databases








2050

05 (a)

2050

05 (b)

2010

01 (c)

2010

01 (d)

0 02 04 06 08 1 12 14 16205

005 (e)

Time (sec)







Synthesis Example































References





































0 100 200 300 400 500

20

40

60

Mag

nitu

de (

dB)

0 100 200 300 400 500

20

40

60

Mag

nitu

de (d

B)

Frequency (Hz)


Soundbox






0 05 1 15 2 2521

0

1

Lev

el

0 05 1 15 2 25196

198

200

202

F0 (

Hz)

Time (sec)






signals

Sound

Soundboxresponses



End knocksgsb

grelease

gc

S2

S1

gi1

gi2

go1

go2

)(zH lLz)(zF

)(zH l)(zF

attackg

(a)

(b)

Lz2

2





1S (z) 4 (1)

1 L1 1 F (z )H (z)z1










0 10 20 30 40 500985

099

0995

1

g 1oop

0 10 20 30 40 50

204

202

0

Key index

a 1oop



41 L1 1 F (z) H (z) zl






g rsquo 0989 ` 0000189k (4)loop











Sample Databases








2050

05 (a)

2050

05 (b)

2010

01 (c)

2010

01 (d)

0 02 04 06 08 1 12 14 16205

005 (e)

Time (sec)







Synthesis Example































References






































signals

Sound

Soundboxresponses



End knocksgsb

grelease

gc

S2

S1

gi1

gi2

go1

go2

)(zH lLz)(zF

)(zH l)(zF

attackg

(a)

(b)

Lz2

2





1S (z) 4 (1)

1 L1 1 F (z )H (z)z1










0 10 20 30 40 500985

099

0995

1

g 1oop

0 10 20 30 40 50

204

202

0

Key index

a 1oop



41 L1 1 F (z) H (z) zl






g rsquo 0989 ` 0000189k (4)loop











Sample Databases








2050

05 (a)

2050

05 (b)

2010

01 (c)

2010

01 (d)

0 02 04 06 08 1 12 14 16205

005 (e)

Time (sec)







Synthesis Example































References





































0 10 20 30 40 500985

099

0995

1

g 1oop

0 10 20 30 40 50

204

202

0

Key index

a 1oop



41 L1 1 F (z) H (z) zl






g rsquo 0989 ` 0000189k (4)loop











Sample Databases








2050

05 (a)

2050

05 (b)

2010

01 (c)

2010

01 (d)

0 02 04 06 08 1 12 14 16205

005 (e)

Time (sec)







Synthesis Example































References







































Sample Databases








2050

05 (a)

2050

05 (b)

2010

01 (c)

2010

01 (d)

0 02 04 06 08 1 12 14 16205

005 (e)

Time (sec)







Synthesis Example































References





































2050

05 (a)

2050

05 (b)

2010

01 (c)

2010

01 (d)

0 02 04 06 08 1 12 14 16205

005 (e)

Time (sec)







Synthesis Example































References































































References




































Commuted Waveguide Synthesis of the Clavichord

Documents

Transcript of Commuted Waveguide Synthesis of the Clavichord