# Physical Modeling Simulation of the ancient Greek Elgin Auloi

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Page 00000001 Physical Modeling Simulation of the ancient Greek Elgin Auloi Tsahalinas K. (1) Psaroudakes S. (4) Tzedaki K. (1) Kamarotos D. (1) Cook P. (2) Rikakis T. (3) (1) Program of Psychoacoustics, Aristotle University of Thessaloniki (tsax, tzed)@orfeas.csd.auth.gr, dimik@forum.ars.net.gr (2) Department of Computer Science, Princeton University prc@cs.princeton.edu, http://www.cs.princeton.edu/~prc (3) Music Department, Columbia University than@woof.music.columbia.edu (4) Dept. of Music Studies, Athens University Abstract This paper reflects the first conclusions of a current research project on the simulation of an ancient Greek wind instrument, the Elgin Auloi from the 5th century BC, housed in British Museum under catalog number 1816.6-10. The simulation of this instrument takes place by means of a computer physical modeling application. In this stage of the project we are using the computer for testing different reed volumes in order to come to basic conclusions about the acoustic behavior of auloi in comparison with the known ancient Greek modes as they appear in surviving documents of Ancient Greek Music Theory 1 Introduction This paper reflects the first conclusions of a current research project on the simulation of an ancient Greek wind instrument, the Elgin Auloi from the 5th century BC, housed in British Museum under catalog number 1816.6-10. Elgin Auloi is a pair of cylindrical wooden pipes of unequal length, each of them carrying six tone holes. It is considered that the auloi pair was driven most possibly by a double reed system. The simulation of this instrument takes place by means of a computer physical modeling application based on measurements on the original instrument taken by Pr. St.Psaroudakes. In this stage of the project we are using the computer for testing different reed volumes in order to come to basic conclusions about the acoustic behavior of auloi in comparison with the known ancient Greek modes as they appear in surviving documents of Ancient Greek Music Theory - from now on referring as [AGMT] -. 2 Previous work integrated in this stage of the project First step of this effort was the acoustic input impedance calculation of these -same type- wind instruments. This gives us a considerable understanding of their intonation, tonal quality, and interaction of the reed with the bore. For this purpose, G. Plitnik's - W. Strong's numerical method was applied. Plitnik and Strong originally used it for calculating the input impedance of the oboe [Plitnik & Strong, 1978]. Here this method was adapted on a cylindrical instrument's bore and a computer software which calculates the input impedance was generated. Just below, a synopsis of P-S numerical method's principles and formulas, as applied in our software, is presented: The instrument's cylindrical bore is divided into a series of short circular cylinders, so that between two successive cylindrical sections a tone hole interferes. The impedance at one end of a circular cylinder is written in terms of the impedance at the opposite end and the dimensions of the cylinder. The impedance at the radiating end of the bore (radiation impedance) at a particular frequency becomes the output impedance for the first cylinder along the bore, from which the input impedance (at the same frequency) is calculated. Whenever an open or closed tone hole is encountered, its appropriate impedance is added in parallel to the input impedance of the cylindrical part of the bore. The input impedance for the first system -cylinder and tone holethen becomes the output impedance for the second cylinder and the process is continued until the last cylinder is reached at the reed end of the bore. Then the reed acoustic impedance and the bore acoustic impedance should be added in parallel to arrive at the input impedance function of the actual instrument.

Page 00000002 Cylindrical section Radiation impedance resistive part: R,=0.25co2p/1c (a: pipe radius, p: air density, c: sound speed in air) Radiation impedance reactive part: Xr=0.6133po/7ca Input impedance of a cylindrical section: Zo =Zc[Zi+Zctanh(yl)]/[Z +Ztanh(yl)] (Zi: output impedance, 1: section length) Characteristic impedance of a cylindrical section: Zc [(R+jcoL)/(G +jo C)]'/2 Propagation constant: y=[(R+jcoL)(G+jowC)]1/2 L =p/S representing inertance (S: cross sectional area) C=S/pc2 representing compliance R =2(plUf/S3)12 representing viscous losses (a: viscosity of air) G=[2(n-1)/pc2](QJS/ph)'72 representing heat contaction losses (n: ratio of specific heat, X: coefficient of heat conduction, h: specific heat at constant pressure) For the specific situation of aulos we assumed tone holes of type (c) of Plitnik & Strong's numerical method, where there are no chimneys and the outer surface of the instrument serves as flange. Open tone hole impedance resistive part: Rr=(pc/wa2)Rl(y) (a: hole radius) Ri(y) (1/8)y2 - (1/192)y4 + (1/9216)y6, where y=47a/2 (y<0.85) Open tone hole impedance reactive part: Xr=(2pf/a2)Al Al=b*0.64[1+0.321n(0.30R/b)] (b: tone hole radius, R: outern radius of cylindrical bore at the place of the tone hole) Closed finger hole input impedance: Z=Zc/tanh(yl) Input impedance of the (m+l)st cylindrical section with a tone hole at its output: Zm+ =Zc[Z' +Zctanh(yl)]/[Zc+Z'mtanh(yl) Z'm=ZmZt/(Zm +Zt) (Zm: input impedance of cylindrical section m, Zt: input impedance of tone hole) Acoustic impedance of reed cavity: Z=-jpc2/co V Input impedance of actual instrument (reed coupled to the bore): Zi=ZbZr/(Zb+Zr). 3 Project software For this stage of the project, specific software - programs Nimped and Aulos, written in ANSI C - was generated. The core of this software is an implementation of Plitnik & Strong's numerical method, as previously described. Input data are all possible dimension measurements of each aulos' part required from the method. Since we lack information about the reed volume of the original instrument - as no reed has ever survived from the ancient times due to their construction material - as about the exact length of that part of the bore which lies between the reed and the first finger hole encountered from the reed, we set that values as variable parameters of the program. Through this software we can test a variety of values for the above parameters and select those ones which give us best results in terms of acoustical theory and [AGMT]. For each simple fingering the software generates the input impedance curves of both auloi and finds the frequency locations of the first four significant maxima (resonance peaks) and their lining up (overtone to fundamental ratio). It also calculates all possible intervals in cents between each combination of two simple fingerings out of seven we totally have, as there are six tone holes on each aulos. There is also a software option of inputting a hole status (open/closed) sequence that responds to any given complex fingering and getting the input impedance curve, the maxima and their lining up as well. Below, two shortened example routines of program Aulos are presented. /* Implementation of P-S numerical method. Here, we consider that status of a closed hole is 1, while that of an open one is 0 */ void getimped(void) { int ij; double freq; complex gamma, Zc, Zl, Zo, load, inputimped; /* here, structure complex's 1st member: modulus, 2nd: phase */ for (i=MIN_BIN; i<NUMFREQS; i++) { freq=i*BIN_WIDTH; Zl=bellma(freq,bellradius); for (j=0; j<NUM_SECTIONS;j++) { Zc=Zcma(RLGC(freq,sect rad[j]),freq); gamma=gammama(RLGC(freq,sectrad[j]),freq); Zo=Zo_ma(Zc,Zl,gamma,sectlength[j]); if (j==(NUM_SECTIONS-1)) /* reach last section */ { if (reed_stat) /* reed attached */ load=reed_ma(freq); else { input imped=Zo; break; }

Page 00000003 else if(hole_stat[j]) /* closed finger hole */ { Zc=Zc_ma(RLGC(freq,holerad[j]),freq); gamma=gamma ma(RLGC(freq,hole_rad[j]),freq); load=closed_hole_ma(Zc,gamma,holedepth[j]); } else load=openhole_ma(hole_rad[j],sect rad[j]+holedepth[j ],freq); Zl=add_paralma(Zo,load); input imped=Zl; } mag[i]=input-imped.x; /* modulus of input impedance */ } } /* this function calculates a scale by opening finger holes one by one */ void calc_scale(void) { inti,j,m; for (m=0O, i=O; m<NUM_SECTIONS-1, i<NUM_SECTIONS; m++, i++) { for (j=0;j<m;j++) hole_stat[j]=0; /* open finger hole */ for (j=m; j<NUMSECTIONS-1; j++) hole_stat[j]=l; /* closed finger hole */ get imped(); get_peaks(); } intervals(); } Full code can be found at the address below: http://alexandros. sdauth.gr/<ipsa/aulos:/ 4 First stage computer work In this stage of work we test the intervals that Elgin auloi can produce with different reed volumes, with no additional length on the original instrument's measurements. Our intention is to make assumptions about the reed volume that can best produce some of the intervals mentioned in [AGMT] references. The fitness of each reed volume is decided by its ability to: a. Align correctly the resonance peaks for all the single fingerings ofAulos. (i) b. Produce some of the known intervals mentioned in [AGMT] references (ii) c. Produce some of known tetrachords as mentioned in [AGMT]. (ii) (i) In this work, an alignment of the first upper three resonance peak frequency positions is considered acceptable when they are within + - 1 semitone from those of the odd harmonics series (ii) Estimations based on lists of known intervals and tetrachords in [AGMT] by Tzedaki.K [11]. Figure 1 Diagram of aulos reed I _T II III IV _V_ a b c d e f g ( I-V are the tone holes of aulos / a-g are the sections of aulos between toneholes) As both auloi have the same number of holes, table 1 refers either to short or to long aulos. Short aulos: Acceptable alignment of the resonance peaks exists for reed volume values ranging from 0.5 - 2.3 cm3. Reed volumes 1 cm3 and 1.1 cm3 result in a fourth (-500 cents) between tone holes II - IV and the intervals are ranging from 241 to 370 cents between toneholes I-T, T-II, II-III, III-IV, IV-V. Reed volumes 1.3 cm3, 1.4 cm3, 1.5 cm3 give a fourth (-498 cents) between the T and III tone holes while intervals are ranging from 235 to 367 cents between toneholes I-T, T-II, II-III, III-IV, IV-V. Except for the fourth no other interval of [AGMT] exists. For reed volume values higher than 2.3 cm3 the fundamentals form known [AGMT] intervals although the correct alignment of the overtones is not acceptable. Long aulos: Acceptable alignment of the resonance peaks exists for reed volume values ranging from 0.5 - 3.1 cm3. Reed volumes 1.4,1.5,1.7 cm3 result in a fourth between II - IV tone holes. For reed volume 1.6 cm3 an octave between I-V, a fourth between I-II, and a fourth between II - IV are produced. Some more [AGMT] intervals are produced within reed volume range of 0.9- 2.5 cm3 but they do not form any known [AGMT] tetrachord. Reed volumes higher than 3.1 cm3 produce a lot of 9/8 intervals between I-T, T-II, II-III toneholes but we have no fourths and no good alignment of the resonance peaks. 5 Future Work As acoustical theory is still incomplete about the behavior and effect of the double reed when coupled to the woodwind bore, a better approach than considering the reed cavity as pure compliance is needed for integration in software. Moreover there must be a

Page 00000004 software development so that it can cover the case of half hole -fingerings. We are currently testing results of complex -fingerings which are not reported in the paper. We are concurrently testing assumptions about Elgin Auloi, such as addition of a potential extra bulb or possible displacement of the already existed bulb in other positions. 6 Acknowledgments This work was realized in the Program of Psychoacoustics of the Aristotle University of Thessaloniki and was supported by the Stanley J. Seeger Viting Research Fellowship awarded to K.Tzedaki from the Program of Hellenic Studies of Princeton University. 7 References [1] Benade, A. 1976 "Fundamentals of Musical Acoustics", New York, Oxford, [2] Cook, P. 1995 "Greek Aulos Project Status Report: Acoustics of Double Reed Cylindrical Bore Instruments", * [3] Hall, D. "M~usical Acoustics", Second Edition, Brooks/Cole Publishing Company [4] Keefe, D. 1982 "Theory and Experiments of the Single Woodwind Tonehole", Journal of the Acoustical Society of America, 72(3) p3p.677-699 [5] Parker, 5., 1988 Editor "Acoustics Source Book", McGraw-Hill [6] Plitnik, G. - Strong, W. 1978 "Numerical Method for Calculating Input Impedances of the Oboe", Journal of the Acoustical Society of America, 65(3), pp. 8 16-825 [7] Psaroudakes, 5. 1995 "Project report on: Acoustic Study of aulos by the method of physical modeling", * [8] Rossing, T., Fletcher, N. 1995 "Principles of Vibration and Sound", Springer-Verlag, [9] Schlesinger, K. 1939 "The Greek Aulos", London [10] Schumacher, R. 1981 "Ab Initio Calculations of the Oscillations of a Clarinet", Acustica 48(2), pp. 7 1-85 [11] Tzedaki, K. " Project report: Physical modelling of Ancient instruments, Elgin Auloi", * [12] West ML, 1982 "Ancient Greek Music", Oxford University Press *Proceedings, " Physical Mlodeling in Music", Workshop II, Program of Psychoacoustics of A.U.Th, Thessaloniki 1995