Computer-Aided Estimation of the Athenian Agora Aulos Scales Based on Physical Modeling

Audio Engineering Society

Convention Paper 8695Presented at the 133rd Convention

2012 October 26–29 San Francisco, USA

Computer Aided Estimation of the AthenianAgora Aulos Scales Based on PhysicalModeling

Areti Andreopoulou1, and Agnieszka Roginska1

1Music and Audio Research Laboratory (MARL), New York University, New York, NY, 10012, USA

Correspondence should be addressed to Areti Andreopoulou ([email protected])

ABSTRACTThis paper presents an approach to scale estimation for the ancient Greek Aulos, with the use of physicalmodeling. The system is based on manipulation of a parameter set that is known to affect the sound ofwoodwind instruments, such as the reed type, the active length of the pipe, its inner and outer diameters, andthe placement and size of the tone-holes. The method is applied on a single Aulos pipe reconstructed fromthe Athenian Agora fragments. A discussion follows on the resulting scales, and the system’s advantages,and limitations.

1. INTRODUCTION

The word music had a much wider meaning to theancient Greeks than it has nowadays. It was usedto describe not only the melody of a tune, but alsoits relationship to lyrics and movement. Melody andrhythm were strongly correlated with the prosody ofthe poem they embellished.

In general, ancient Greek music could be divided intotwo categories: the first consisted of pieces that cre-ated a feeling of calmness, and spiritual rising, andwas associated with the worship of Apollo; the sec-ond, which promoted enthusiasm and passion, wasassociated with Dionysus. Apollo’s characteristic in-

strument was the Lyre, which had 5 to 7 strings, andhis poetic forms were the Ode and the Epic, whileDionysus’ was the Aulos, which was used to accom-pany the singing of Dithyrambs and Dramas [4].

Evidence from at least the 6th century B.C. supportthe theory that both the Lyre and the Aulos wereplayed solo, and that ensembles consisting of bothwere very uncommon. It is also widely accepted thatthe music performed was monophonic, with some el-ements of heterophony, played either in unison or inparallel octaves [4]. However, the concept of melodyand accompaniment in ancient Greek music is slowlyregaining respect [7].

This Convention paper was selected based on a submitted abstract and 750-word precis that have been peer reviewed by at least two qualified anonymous reviewers. The complete manuscript was not peer reviewed. This convention paper has been reproduced from the author's advance manuscript without editing, corrections, or consideration by the Review Board. The AES takes no responsibility for the contents. Additional papers may be obtained by sending request and remittance to Audio Engineering Society, 60 East 42nd Street, New York, New York 10165-2520, USA; also see www.aes.org. All rights reserved. Reproduction of this paper, or any portion thereof, is not permitted without direct permission from the Journal of the Audio Engineering Society.

ANDREOPOULOU AND ROGINSKA Estimation of the Athenian Agora Aulos Scales

Fig. 1: Aulete, a musician playing the aulos. Centermedallion of a red-figured cup from Vulci. Around490 BCE Inv. G 313. Louvre.

The earliest written evidence on ancient Greek scaletheory dates back in the 4th century B.C. Two arethe main schools dominating music theory at thetime: the Pythagoreans, and Aristoxenos. Whilethe first specified music intervals in terms of simpleratios, which corresponded to the harmony of thespheres, the latter described music systems based onthe abilities and limitations of the human hearing,allowing for intervals that were considered inappro-priate by the Pythagoreans.

The primary building block of the scales used in an-cient Greek music was the tetrachord, which con-sisted of four notes spanning a perfect 4th. Threedifferent types (genera) of tetrachords have beenidentified: the diatonic, the chromatic, and the en-harmonic (Figure 2). In the first the lowest intervalwas a semitone, and the other two were whole tones;In the second the two lower intervals were semitonesand the higher a semiditone (a whole and a halftone); whereas in the third the lower two intervalssplit a semitone making the top one a ditone (twowhole tones). Different colorization effects could beapplied to tetrachords by slightly adjusting the tun-ing of the inner notes [21].

1.1. The AulosAccording to Landels, the Aulos, or Tibia in Latin,is the most commonly depicted instrument through-

Fig. 2: The three tetrachord types

out the Greek and Roman history, a fact that mightserve as an indication of its popularity [13]. Bothpictures on the surfaces vases (Figure 1), and thearchaeological remains themselves indicate that theAulos was a woodwind instrument with two cylin-drical bores, held in front of the musician. At earlytimes, the body of the instrument was made in onepiece from reed. However, during the Classical andthe Hellenistic times bones and ivory were also usedas primary material for some instruments, and whenkeywork was developed, manufacturers covered thewooden bore with a layer of bronze or silver [10].

From most pictorial representations it becomes ap-parent that its two pipes, where played simultane-ously. Furthermore, closer examination of the draw-ings indicates that the bores were of unequal lengths.This property was initially attributed to the attemptof ancient painters to represent depth [13]. Nev-ertheless, Psaroudakes, by studying several surviv-ing Aulos fragments, including the Pydna pair, hasclaimed that the left bore was always longer thanthe right one, and the finger-holes were at differentdistances from the mouth-end [16].

Even though the information extracted from the il-lustrated representations is further enhanced by alarge number of archaeological remains (the Brau-ron Aulos [9], the Athenian Agora fragments [10],the Elgin Auloi [19], the Aulos of Argithea [15], andthe Pydna pair [16]), there is still no evidence ofthe type of reed used for that instrument. Accord-ing to West, both the single and double reed tech-nologies were known to the ancient Greeks [21], andeven though the double reed is being treated as themost probable case, the single reed option hasn’tbeen ruled out yet.

1.2. Related WorkThe first well documented attempt to investigate theAulos was by Kathleen Schlesinger [19]. Her exten-sive book The Greek Aulos offers an interpretationof the instrument’s mechanism, based on the exami-nation of the Elgin pipes, and an exploration of theirrelation to the modal system in practice.

Schlesinger is one of the supporter’s of the single reed

AES 133rd Convention, San Francisco, USA, 2012 October 26–29

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Aulos. Her theory was that the single reed replacedthe double in the late 5th century B.C. because itwas fitting better the current modal system [19]. Afew years later Landels commended that this the-ory was based on adequate evidence, and a series ofmisinterpretations [13].

Another very promising theory of Schlesinger wasthat of the “moving mouthpiece”. According to thishypothesis the Aulos tone-holes were equally spaced,and in order for more than one tetrachords to beplayed on each bore, the length of the mouthpiecewas adjusted, such that the two higher holes of thelower tetrachord become the two lower holes of hehigher one. Yet, the assumption of equidistant tone-holes is only supported by the Elgin pipes, but notby any others such as the Brauron [9], the Athe-nian Agora [10], or the Pydna Auloi [16], and shouldtherefore be considered case specific [21].

Landels was the first to propose a very simple math-ematical procedure for estimating the Aulos scales,which is based on simple laws of physics. Any mu-sic interval can be expressed as a ratio between thevibrations of its notes. In the case of wind instru-ments this can be approximated by the ratio of theair column length of the high to the low note. Itrequires an initial assumption that two finger-holeson an instrument bore are a certain interval apart.Most commonly this interval is the perfect fourth,as this is the base of all ancient Greek tetrachords.From this, the correct effective length of the bore canbe estimated, as the one that satisfies this intervaliccondition. This length is then used to estimate thepitch of the remaining finger-holes on the instrumentbore [11] [13].

This approach is still very popular nowadays, andwas applied to a variety of Aulos fragments to de-duce information on their tuning [14]. However, onecan easily see that it is oversimplified, and cannot beregarded as an accurate representation of the scalesproduced by an instrument, since it disregards fac-tors that can considerably affect the pitch of a note,such as the radius of the instrument’s bore, the sizeof the holes, the affect of the reed itself on the re-sulting tone, and the control each player has overthe produced sound.

In general, Landels’ approach to the Aulos music isstrongly driven by the assumption that both pipes

were playing the same melody in unison. This viewis shared by many researchers nowadays. Accordingto them, the reason for that aesthetic choice lies onthe distinct beating sound that is created when twonotes that are very close in frequency are played to-gether. The resulting timbral effect was very suitedto the Aulos dithyrambic character [13].

The validity of this assumption was questioned re-cently by S. Hagel, who revisited the evidence ofheterophony in ancient Greek music, and claimedthat the Aulos tuning and scales can only be as-sessed when information on pipe-pairs is combined[7]. His proposed method is based on a computa-tional model that accounts for the physical proper-ties of each studied instrument, such as its lengthbore diameter, position of tone-wholes, and their di-ameter etc.

The process is iterative, and is applied to each holeseparately: an initial assumption on the frequencyon this hole needs to be made; from that the end-corrections of the bore and open hole are calculated,followed by the velocity of the air column inside thebore; these results lead to a corrected estimation ofthe tone-hole pitch; the process is repeated until thedifferences between results become insignificant [7].Unfortunately, the computational model is not pre-sented in the publication, and it cannot be furtherdiscussed here.

This paper is proposing a different approach to scaleidentification by means of physical modeling. It isan extension of Landels’ method, through which theintervalic relationships of the aulos tone-holes arecalculated by wave-guide synthesis techniques. Theprecess is applied to a single pipe, reconstructedfrom fragments of the Athenian Agora excavation.By taking into account not just the length of theair column inside the instrument’s bore, but also itsfull length, inner and outer diameters, the placementand diameters of the tone-holes, and the reed type,the algorithm is expected to offer a more completerepresentation of the pipe’s tuning. A detailed de-scription of the modeled Athenian Agora fragmentsfollows.

2. THE ATHENIAN AGORA FRAGMENTS

2.1. The ExcavationThe Athenian Agora was located in the center ofthe ancient city. It is believed to be one of the most


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Fig. 3: Fragmented bore of an Athenian Aulos [12]

crowded places in Athens, hosting large street mar-kets, performances, athletics, political gatherings,and religious ceremonies. The excavations began in1931, by the American school of Classical Studies,and, as expected, yield an extensive amount of ar-chaeological findings. Among those where nine frag-ments of ancient Greek Auloi, which correspondedto parts of the instruments’ mouthpieces and bores(Figure 3). Landels, who studied those fragments in1964, estimated their origin in the 5th century B.C.[10].

2.2. Selection of PartsFor the creation of an acoustic model of the Aulosin this paper, fragments A (Inv. BI 593), C (Inv.BI 579), I (Inv. BI 630), and H (Inv. BI 594) wereused as models of the bulbs, and the upper, middleand low-end parts of the bore respectively. The se-lection was based on two criteria: the condition ofthe fragments, and the estimated era they belongedto. The physical measurements of the fragmentslengths, radii, number, placement and size of theholes were extracted from Landels’ detailed study,and scaled graphs, available at [10].

2.2.1. The MouthpieceThe mouthpiece consisted of two parts, the cup forthe reed, and the bulb. Unfortunately, no fragmentwas identified by Landels as a cup (upper part of themouthpiece). Also as discussed earlier, there is noinformation available for the type of reeds used. Asa result the only available part of the mouthpiece isthe bulb (Figure 4).

Fig. 4: The Aulos’ bulb [10]

The bulb (Inv. BI 593), which took its name from

its shape, is considered to be the end part of themouthpiece. Its length is around 8.95 cm; it hasa socket at one end, and a spigot at the other, tofit to the surrounding parts. It has been observedthat, at least for instruments made of bone or ivory,the inside diameter of the part did not vary withthe outside curvature, which, probably, served forcosmetic purposes. Similarly for this fragment theinterior diameter is reported to be almost exactly 1cm throughout.

2.2.2. The Upper Part of the BoreSeveral fragments of the excavation belong to thatpart, all of variable length and diameter. As it canbe seen in Figure 5, they contain 4 of the 5 instru-ment holes (3 finger-holes and a thumb-hole).

Fig. 5: The upper part of the Aulos’ bore [10]

The selected fragment (Inv. BI 579) has an over-all length of 11.3 cm., a maximum outside diam-eter of 1.6 cm., and an interior diameter of 1 cmthroughout. The tone-holes are circular, 0.85 in di-ameter, and are all aligned, except for the thump-hole, which is displaced by 5◦ anti-clockwise. Ac-cording to Psaroudakes, this is an indication thatthe part belongs to the right pipe of the Aulos pair[16].

2.2.3. The Middle Part of the BoreThe middle part of the bore (Inv. BI 630) holdsthe fifth finger-hole of the Athenian Aulos. The factthat it has a socket in one end, and a spigot in theother indicates that it was not the end-part of theinstrument. Its overall length is reported to be 9.15cm., with an maximum outside diameter of 1.8 cm.,


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and an inside one of 1.1 cm. throughout. Its hole iscircular, 0.95 cm. in diameter (Figure 6).

Fig. 6: The middle part of the Aulos’ bore [10]

Landels [12] offers two explanations for the need ofa separate bore part for the fifth hole: either bonesof suitable sizes for the Aulos pipes were availableonly in small lengths that could not fit all holes intoone piece; or, ancient Greek organology had devel-oped a method similar to modern wind instrumentpractice, according to which, holes operated by thesmall fingers, are slightly displaced to the side, tofacilitate performance for the musicians.

2.2.4. The Low-end Part of the BoreThere is not much information available on the low-end parts of the Athenian Agora pipes. Archaeol-ogists conclude, based on traces, that they had aventing whole, and their overall radius did not varyconsiderably across their body. Yet, later remainsfound in the same excavation indicate that at somepoint the cylindrical low-end part was replaced by amore conical one in the shape of a bell.

Fig. 7: The low-end part of the Aulos’ bore [10]

The selected fragment (Inv. BI 594) has an overalllength of 8.9 cm., a maximum outside diameter of1.7 cm., and an inside diameter of 1.1 cm through-out (Figure 7). This part is significantly damagedtowards its low-end. However, there are clear tracesof a vent hole approximately 5.7 cm from the end ofthe instrument, whose size cannot be easily approx-imated.

3. MODELING THE AULOS

One possibie way of modeling the acoustic behaviorof wind and string instruments is by dividing them

in two basic interacting parts: the resonator, and theexciter. In the case of the Aulos, these will be theair column inside the bore and the reed respectively.Even though in practice, such separation is impos-sible because there are no clear boundaries betweenthese systems, this simple model has been used inmany studies, with considerably accurate results [1][2] [6] [17].

For the purposes of estimating the scales of the Athe-nian Agora pipe, we employed a simple WaveguideModel technique suggested by Scavone [17], which isstructured as follows: a non-linear excitation func-tion models the behavior of the reed, a bi-directionaldelay line models the resonator, and a digital reflec-tion filter models the discontinuities at the open endof the pipe.

3.1. The Mouthpiece and BoreThe reed and the mouthpiece in woodwind instru-ments act as a pressured control valve of the volumeflow entering the reed, which controls and maintainsthe oscillations of the air column in the bore, at fre-quencies corresponding to the resonances of the pipe.The opening area of the reed is controlled by the dif-ference in pressure P∆ between the inside of the reedarea and the mouth cavity and can be approximatedwith the following model

P∆ = Poc − Pr = k(S0 − S) (1)

where Poc and Pr are the pressure of the oral cavity,and reed respectively, k is the stiffness constant ofthe spring, and S0 is the opening area of the reed atrest. For simplicity in these cases the oral cavity isregarded as large reservoir of constant pressure andzero volume flow.

If we assume that the volume flow remains constantover the whole opening area of the reed, we can applythe Bernoulli theorem between the mouth and thereed to determine its flow qr,

qr = S

√2P∆

ρ(2)

where S is the opening area of the reed and ρ thedensity of air. These two formulas can be applied toboth single, and double reed instruments, since stud-ies have shown that the displacement of the reeds inthe latter case is symmetrical [6].


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As it can be seen from Equation 2, the reed-outputflow qr, which establishes the coupling of the exciterwith the resonator (bore), is created by the pressuredifference between the player’s mouth cavity Poc andthe reed Pr.

For single reed instruments it is safe to assume con-tinuity between the reed and the bore pressures.

Psr = Pb (3)

In the case of double reed, though, increased headlosses introduced to the entrance of the resonator,due to the grown complexity of the exciter’s geome-try, do not permit such assumption. Therefore, thepressure at the end of the reed can be derived by

Pdr = Pb +1

2ρΨ

q2

S2r

(4)

The Aulos bore can be approximated by a Gaussianfunction, with its peak centered at around 2cLb (Lb

is the bore length, and c the speed of air). The widthof the peak, which equals the time it takes for theacoustic wave to travel from the reed to the end ofthe bore and back, is according to Almeida [2], anindication of the visco-thermal losses on the bore.

The cylindrical bore can be represented by a one-dimensional waveguide model of its reflection coef-ficient, in cases of linear acoustic propagation. Theimpedance mismatch that the traveling waves en-counter at the end of the finite length bore causespart of the wave to be transmitted to the newmedium and part of it to be reflected back. There-fore the bore of the Aulos is represented as two su-perimposed pressure waves, an incoming p− and anoutgoing one p+.

pb = p+ + p− (5)

For the creation of the single-reed computationalmodel of the Athenian Agora Aulos, the output ofthe instrument is calculated as the physical pressureinside the mouthpiece, and is based on the design ofG. Scavone [17] [18]. For the double reed one, theoutput of the instrument is estimated at the exitof the resonator (bore), according to A. Almeida’smodels [1] [2] [3].

3.2. The Tone-holesThe waveguide synthesis of the holes is based ona model described by Keefe [8] that accurately

presents a tone-hole without taking into account in-teractions from adjacent holes. Each one is repre-sented by a two-port junction function, which mod-els both series and shunt impedances. This methodrequires four second-order filters per hole.

This mathematical description can only describe thetwo extreme states “open”, and “closed”. For all in-termediate cases, two approaches can be followed: tovary in time the resulting filter coefficients in orderto compensate for the graduate opening and closingof the holes; or to model that action with a graduatecross-fade technique between the opening and clos-ing filters [17] [18]. For this specific model we are nottaking into account partially-closed hole techniques.

4. CALCULATION OF SCALES

As mentioned earlier, this proposed approach is anextension of Landels’ method on Aulos scale estima-tion. The key difference is that instead of basingthe intervalic relationships of the instrument’s tone-holes exclusively on the length of the air column in-side the bore, a physical model of a full pipe is imple-mented. The model requires information about theactive length of the bore, its inner and outer diam-eters, the placement and size of the tone-holes, andthe reed-type, in order to estimate the fundamentalfrequencies of the produced notes. Once the fre-quencies are calculated, possible matches to knowntetrachord tunings are investigated.

4.1. Mouthpiece Correction and Reed TypeUnfortunately, there are no intact mouthpiece partsamongst the Athenian Agora excavation. Only afracture of a bulb has been identified. Therefore, inorder to calculate the instrument’s effective lengthwe need to produce a mouthpiece correction dis-tance. The process begins identically to Landels’.A set of holes is assumed to be a certain intervalapart (for this specific case the interval of a perfectfourth was chosen). Their frequencies are inverselyproportional to the distances from the centers of theholes to the top of the instrument. These distancesare the sum of the measured lengths Llow and Lhigh,and the mouthpiece correction x (Equation 6) [12].

x+ Lhigh

x+ Llow=

3

4(6)


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Once the bore length and distances of all tone-holesare adjusted by the mouthpiece correction estima-tion, the Aulos physical properties are entered tothe model, and the frequencies of all notes are cal-culated. The difference in our method is that thecorrection estimation is then adjusted to the result-ing frequencies of the two notes, and the whole pro-cess is re-evaluated iteratively, until it results in aperfect fourth tuning.

Even though most research nowadays leans towardsthe hypothesis that the Aulos was a double-reed in-strument, the single-reed option hasn’t been ruledout completely. For completeness, two differentphysical models of the Aulos were implemented, al-lowing for observation of the effect of the reed typeon the notes, and tuning of the instrument.

4.2. The Athenian Agora Aulos ScalesSix different tuning assumptions of a perfect fourthwere made, which resulted in different mouthpiececorrection lengths. The process was repeated forboth reed options (Tables 1, and 2).

Single-Reed ModelSelected Tone-holes Mouthpiece correction (m)

All open → II 0.141All open → III ��0.522All open → IV 0.2679

I → III ��0.426I → IV 0.1719T → IV - ��0.0719

Table 1: Resulting mouthpiece correction distances(m), from 6 tuning assumptions for a single-reed Au-los model.

Double-Reed ModelSelected Tone-holes Mouthpiece correction (m)

All open → II 0.051All open → III ��0.32All open → IV 0.1779

I → III ��0.36I → IV 0.0819T → IV - ��0.0181

Table 2: Resulting mouthpiece correction distances(m), from 6 tuning assumptions for a double-reedAulos model.

As it can be seen in the tables above, the secondfourth and sixth correction lengths had to be ex-cluded as highly improbable, or unfeasible. For theremaining tunings, all possible intervals were calcu-lated and converted in cents. The resulting tetra-chord options were then compared to known tetra-chord genera.

More specifically, the “All holes open to SecondFinger-hole (II)” tuning assumption resulted in thefollowing intervals (cents), for the single and doublereed respectively (Table 3). By allowing for a ±20cents deviation between intervals, the tetrachordsof both reed versions can be said to approximatethe Philolaus’ Chromatic tetrachord, as describedby West [21].

All open → IIReed Type Intervals (cents)

Single Reed 93.603 121.77 305.0759Double Reed 93.2152 118.77 299.0759

Table 3: Resulting tetrachords in cents for bothreed options, based on the “All holes open to SecondFinger-hole (II)” tuning assumption.

Similarly, the “All holes open to Fourth Finger-hole(IV )” tuning assumption resulted in the followingintervals, which, by allowing for the same deviation,can be said to approximate the Ptolemy’s Tonic -Diatonic and Soft - Diatonic respectively (Table 4).

All open → IVReed Type Intervals (cents)


Table 4: Resulting tetrachords in cents for bothreed options, based on the “All holes open to SecondFinger-hole (IV )” tuning assumption.

Finally, the ‘First (I) to Fourth Finger-hole (IV )”case resulted, as it can be seen in Table 5, in anapproximation of the Aristoxenos’ Chromatic tetra-chord for both reed options.

This latter tuning assumption was also used by Let-ters when he applied Landels’ tuning theory on frag-ment C of the Athenian Agora archaeological re-mains [14] (Table 6).


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I → IVReed Type Intervals (cents)


Table 5: Resulting tetrachords in cents for bothreed options, based on the “First (I) to FourthFinger-hole (IV )” tuning assumption.

I → IV (Landels)Reed Type Intervals (cents)

Unspecified 153.00 76.00 269.00

Table 6: Resulting tetrachords in cents of Landels’tuning theory, based on the “First (I) to FourthFinger-hole (IV )” assumption.

One can easily see the variations between the sug-gested scales in Tables 5 and 6, which result fromthe differences in the instrument representation inthe two scale estimation systems. A discussion fol-lows on the importance of a complete representationwhen studying the characteristics of an instrument,the limitations posed by the lack of all necessary in-formation, and their affect on the suggested system.

5. DISCUSION

This paper discusses a different approach to music-scale estimation of the ancient Greek Aulos, basedon waveguide synthesis. A model of the instrumentis derived from a set of physical properties that areknown to affect its sound, such as the reed type, theactive length of the pipe, its inner and outer diam-eters, and the placement and size of its tone-holes.We consider this design to be more comprehensivethan the most known approach introduced by Lan-dels [12], as it creates a more complete representa-tion of the instrument, and can also form the basisfor a more informative analysis of the Aulos proper-ties. This can lead to estimations of the Aulos inter-vals, tuning, and timbre, and provide assessments ofthe effects of the reed, the use of vent holes, and theneed for double bores.

The validity of the concept was evaluated throughinvestigation of the Athenian Agora Aulos scales. Asdescribed earlier, the goal was approached throughsimple physical models, which do not take into ac-

count the effect of the mouth cavity on the excita-tion mechanism, the ability of performers to adjusttuning by partially closing tone-holes, or differentperformance techniques used during that era. Thisdecision was necessary in order to create a simplemodel that would focus directly on scale estimation.As a result our model is not capable to fully repro-duce the possible timbre of the extinct instrument,as this is beyond the scope of this research.

This decision results from the insufficient infor-mation, available to us nowadays, on performanceand fingering techniques applied to the Aulos, themouthpiece and reed, and the absence of surviving,fully-functional instruments. This partial informa-tion on the music performance has led to the viewthat models of the ancient Aulos will always be in-conclusive [12] [13]. Even though, this perspectivehas some merit for tasks like timbre reconstruction,in the case of interval identification, it might be con-sidered over-pessimistic, for reasons that follow.

The ability of woodwind musicians to adjust the in-tonation of their instruments through a variety oftechniques, is well-known, and very common nowa-days. It would be naive to assume that ancientGreek Aulos players did not employ similar meth-ods during their performances. However, the act offine-tuning nowadays comes as a slight short-termadjustment against the designed tuning of the in-strument. In other words, there exist clear manu-facturer standards, widely accepted and employed,which account for the current tuning and scale needs.

Similarly, there is evidence that Aulos makers in an-cient Greece followed certain patterns when buildingtheir instruments. For example, in the majority ofdiscovered pipes, the distribution of tone-holes onthe surviving fragments is similar, and the lengthsof the pipes fall within certain ranges. From thiswe can assume that those construction choices werein accordance to the aesthetic standards, and mu-sic reproduction needs of that era. Therefore, thepossibility that the Aulos performers constantly dis-regarded the inherent tuning of their instruments,can be considered improbable, and we have to inves-tigate the possibility that the scales of the Auloi canbe estimated from the instruments themselves.

The designed physical model of the Athenian AgoraAulos managed to approximate, given a ±20 cent


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deviation range, three different tetrachord types,namely the Philolaus’ Chromatic, the the Ptolemy’sTonic - Diatonic and Soft - Diatonic, and the Aris-toxenos’ Chromatic, based on different mouthpiececorrection length estimations. One interesting obser-vation from these results is that the choice of reed(single vs. double) did not have a major effect onthe resulting scales. Nevertheless, close examinationof the resulting intervals reveals an advantage of thedouble-reed model in scale approximation accuracy.

One should take into consideration, though, that thefragments available from this specific excavation site,suffice to reconstruct a single pipe of an instrumentthat was designed to operate on two bores. Therole of the two bores has been under debate, butif we move away from the “two bores in unison”theory [13], towards the more progressive assump-tion of “melody and accompaniment ” [7], we can-not make clear assessments on the Athenian AgoraAulos scales. Therefore, future work will involve theapplication of this computer model to more com-plete surviving pairs, such as the Pydna, and theElgin pipes.

6. REFERENCES

[1] A. Almeida, C. Vergez, and R. Causse. Qua-sistatic Nonlinear Characteristics of Double-Reed Instruments. Journal of the AcousticalSociety of America, 121(1):536 – 546, January2007.

[2] A. Almeida, C. Vergez, R. Causse, andX. Rodet. Physical Study of Double-Reed In-struments for Application to Sound-Synthesis.In Proceedings of the International Symposiumin Musi- cal Acoustics,, pages 221–226, MexicoCity, Mexico, 2002.

[3] A. Almeida, C. Vergez, R. Causse, andX. Rodet. Experimental Research on DoubleReed Physical Properties. In Proceedings of theStockholm Music Acoustics Conference,, 2003.

[4] J. P. Burkholder, D. J. Grout, and C. V.Palisca. A History of Western Music. W. W.Norton, New York, 6 edition, 2010.

[5] J. S. Clay. Pindar’s Twelfth Pythian: Reed andBronze. The American Journal of Philology,113(4):519–525, 1992.

[6] P. Guillemain. A digital synthesis model ofdouble-reed wind instruments. EURASIP Jour-nal on Applied Signal, ii:990–1000, 2004.

[7] S. Hagel. Calculating Auloi. The Louvre AulosScale. Studien zur Musikarchaologie IV, Orient-Archaologie, 15:373 – 390, 2004.

[8] D. H. Keefe. Woodwind air column models.Journal of the Acoustical Society of America,88(1):35 – 51, 1990.

[9] J. G. Landels. The brauron aulos. Annual ofthe British School at Athens, 58:116–119, 1963.

[10] J. G. Landels. Fragments of auloi found inthe Athenian Agora. Hesperia, 33(4):392–400,1964.

[11] J. G. Landels. A newly discovered aulos.The Annual of the British School at Athens,63(1968):231–238, 1968.

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[13] J. G. Landels. Music in Ancient Greece andRome. Routledge, 11 New Fetter Lane, London,1999.

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Computer-Aided Estimation of the Athenian Agora Aulos Scales Based on Physical Modeling

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