Intonation of solo violin performance with reference to equally tempered pythagorean and just intonations

PESAN Intonation of solo violin performance with reference to equally tempered, Pythagorean, and just intonations” Franz Loosen” Psychonomic Laboratory, University ofLeuven, 102 Tiensestraat, B-3000, Leuven, Belgium (Received 26 February 1992; revised 3 September 1992; accepted 16 September 1992) The purpose of this study was to determine which musical scale best models the solo performances of violinists when they play diatonic scales of C major very slowly and without frequency vibrato as accurately as possible. Eight professional violinists played without | stopping three adjacent scales in ascending order (from C, to C,), followed immediately by an analogous return to the initial note C, in descending order. Results show that when violin performances are analyzed taking into account the context of the scale in which they were played, Pythagorean and equally tempered scales are equally good models for the description of performances. Just intonation fits the data significantly less well. When the data analysis iis limited to the intervals between separate pairs of notes, nof taking into consideration thecontext of the scalein which they were played, observed interval sizes are almost identical to the arithmetic means of the corresponding interval sizes as defined in Pythagorean and equally tempered intonations. Octave intervals calculated by summing the sizes of performed adjacent p\Sart SANS major and minor seconds within octaves, show an average stretching of 3.9 cents. However, the arithmetic mean of computed interval sizes between two C’s, deviates only 0.3 cents from the theoretical tonal extent of 1200 cents. Results suggest that when scales of C major are played, the tonic C is adopted as an absolute cognitive reference point. PACS numbers: 43.75.Bc, 43.75.De, 43.75.St, 43.66.Hg INTRODUCTION The great challenge a beginning violin student has to face is to play in tune. However, what is the standard? In beginners, a sharply defined internal template of relative pitch relations among the notes is not yet present. That knowledge must be progressively acquired from the directives and examples given by the string teacher. However, what model does he offer? The main purpose of this study was to describe how professional violinists played the diatonic scale of C major, when they try to intone as accurately as possible. Playing the scale of C major is a typical exercise for violin students. Even more than 200 years ago the German pedagogue Leopold Mozart (1975), father of Wolfgang Amadeus Mozart, recommended to start with that scale. Traditionally, pitch relationsin musical performances The equally tempered scale (the present standard scale in Western music) is unequivocally defined by 12 equal logarithmic steps, each step representing a frequency that is 2!/12 greater than the one below. In this paper the labels “Pythagorean” and “just intonation” stand for Ptolemy's diatonic ditoniaion and Ptolemy’s diatonic syntonon, respectively. The major mode of the former is defined as one in which the ratios of the frequencies of adjacent notes from the tonic are 9/8, 9/8, 256/243, 9/8, 9/8, 9/8, and 256/243. Analogously, the major mode of the latteris defined by the ratios 9/8, 10/9, 16/15, 9/8, 10/9, 9/8, and 16/15 (for a detailed description of these tunings, see Barbour, 1951, Chap. II). The present study focused specifically on those two historical models because only those two have had any great influence upon modern music theory, and earlier investigators (i.e., Green, 1937a,b, and Nickerson, 1949) studied are discussed with reference to a musical scale, that is, a violin performance explicitly in relation to those two models. physical model for pitch relations in a set of performed musi- Some authors claimed that violin performances are best . described by Pythagorean intonation. They asserted that cal intervals. In the course of time, numerous musical scales have been proposed (Barbour, 1951; Devie, 1990; Ellis, this is a consequence of practicing an instrument tuned by 1954; Lloyd and Boyle, 1963; Partch, 1949). This study is restricted to equally tempered, Pythagorean and just intonations. perfect fifths (i.e., intervals between two frequencies having a ratio of 3:2). Indeed, the tones of Pythagorean intonation ® Results of part of the present research have been presented at the 25th can be easily derived from a sequence of adjacent perfect fifths. It suffices to iterate the tuning upward in perfect fifths from the frequency of a given note to include seven (or more) successive fifths, and to bring back the notes genera- International Congress of Psychology held at Brussels, Belgium, 19-24 ted in this way to the same octave in order to obtain all the July 1992. x; Requests for reprints and correspondence concerning this paper should “ beaddressed to F. Loosen, Department of Psychology, University of Leuven, 102 Tiensestraat, B-3000 Leuven (Belgium). 625 J. Acoust. Soc. Am. 93 (1), January 1993 tones of the Pythagorean scale. Others asserted that the practice of harmonic accompaniment has forced violinists to performin a manner ap- 0001-4966/93/010525-15500.80 © 1993 Acoustical Society of America

proximating just intonation. This time, the conviction relied frequency ratio between two tones becomes more complex. professional violinists when playing the diatonic scale of.C major, very slowly, without vibrato, and as accurately as possible (i.e, the usual way of playing when a string teacher shows his pupil how a scale sounds). _ (2) The present study focused on the relative size of the musical intervals between notes in performed diatonic scales of C major, considered as a whole. This approach was based Others still advocated that the conditioning influence of equally tempered tuning as employed upon fixed pitchininfluence on intonation (Dowling, 1991; Francés, 1958; upon the association of two arguments: first, the knowledge that the tones of this intonation are obtained on a string instrument by dividing the string into 2, 3, 4, etc., equal sections (and bringing thetn back in the proper octave), and second, the belief that consonance gradually decreases as the on the finding that musical context may have a significant struments (especially keyboard instruments) had brought Krumhansl, 1990; Shackford, 1961, 1962; Small, 1937), and violinists to performin a manner approaching the equally tempered scale. skilled musicians developed expertise for playing and perceiving typical musical sequences as a whole (Abraham, There exist numerous other opinions about intoning by 1923; House, 1977; Krumhansl, 1979, 1990; and Watkins violinists in the musicological literature. However, most of and Dyson, 1985). . (3) The intonation of violinists was studied as a function of two parameters: (i) scale location, that is, the location of the musical scale in the frequency range and, (ii) scale direction, thatis, the ascending or descending direction of the musical scale. (4) This study aimed to evaluate the size of inter- and intra-individual variability in performances. them have never been verified empirically. Willemze (1973) for example, is convinced that violinists play ascending scales in Didymus’ diatonic scale (1.e., a scale in which the ratios of the frequencies of adjacent notes from the tonic are 9/8, 10/9, 16/15, 978, 9/8, 10/9, and 16/15 in the major mode!) and descending scales in just intonation. Lloyd (1940) went even further and rejected, explicitly, every rigidity of intonation. In his opinion, violinists are likely to be guided by the music itself as to what intonation to use. In the past, a considerable amount of experimental research had been directed toward the investigation of the perception of isolated and synthesized tonal intervals in different intonations (e.g., Mathews and Sims, 1981; Roberts and Mathews, 1984; Vos, 1982, 1984; Vos and van Vianen, 1985; Ward, 1970; Ward and Martin, 1961). However, as far as I am aware, until now, only six attempts have been made to determine the intonation used by solo playing violinists during actual performance: Cornu and Mercadier (1869, 1871, 1872, 1873), Corso (1954a), Geringer (1978), Greene (1937a,b), Nickerson (1949), and Small (1937). However, the sizes of the reading unit in the studies of (5) Instead of a stroboscopic analysis for measuring fundamental frequencies of violin tones an electronic technique was used. | I. METHOD The subjects were eight highly trained professional violinists between the ages of 25 and 33 years. All of them had wide experience in solo performance. Moreover, all of them were members of an orchestra and were teachers in an academy of music. The musicians were asked to play without in-: terruption three adjacent scales of C major starting in ascending order from C, up three octaves to C, (i.e., a range of to 20 cents” ). Furthermore, the studies of Corso and Gerfundamental frequencies from about 262 Hz up to 2096 Hz, immediately followed by an analogous return to the initial inger were set up to investigate a psychophysical question and the impact of verbal inducement on intonation, respectively. Nickerson pooled the data of violins, altviolins, and note C, in descending order. Subjects were given a printed form (without fingering) of the musical sequence to be performed (Fig. 1). Every violinist was given three trials. Becellos. Hence, only Green’s investigation is concerned with the question of this study. Therefore, it appeared sensible to start by collecting new data of solo playing violinists. Some tween the trials there was a short pause of about 30 s. Musicians were prepared for the experiment by playing warm-up exercises and practicing diatonic scales of C major individurespects of the present study are different from the previous ally. Cornu and Mercadier, and Small were excessively large (10 ones. The violinists were asked to concentrate their attention to sustain each note about 3 s and to play with an even tone 111% TE Fie os which musical scale best models the solo performances of 119» EES on the accurateness of intonation. They were also instructed Tin— Te TR Tije TIR> (1) The main objective of this study was to determine 3 2327 == FIG. 1. Consecutive scales of C major played three times by eight violinists ( a:= 3s). Each note was played on a separate bow strike. The numbers marked above the notes represent the finger that should be pressed on the string to produce the required note; 1, 2, 3, and 4 refer to the forefinger, the middle finger, the’ ring finger, and the pinkie finger, respectively. 526 J. Acoust. Soc. Am., Vol. 93, No. 1, January 1993

without frequency vibrato. Each note was played on a separate bow strike. Open strings were not allowed. On loudness nothing explicit was mentioned because it was expected that the musicians would play mezzo-piano spontaneously, and because experimental data available indicate that intensity has no consistent effect on the pitch of violin tones or that the effect is so slight as to be counteracted by other factors in performance (Fletcher, 1934; Lewis and Cowan, 1936; Seashore, 1938). Musicians were allowed (1) to tune their instrument at their own discretion and (2) to use the fingering best suited to them. The first option was based on the objective of this study to investigate the intonation of violinistsin everyday conditions, and on the observation that no professional violinist uses a tuning fork or any other device to tune his/her instrument in daily practice. Furthermore, the differences in initial tuning were no disturbing factor in the present study because (1) open strings were not allowed, (2) the research was not interested in the frequency with which a given note was played, but in the ratios between frequencies of notes, (3) advanced players have acquired the skill to play in tune ing. The speed variation between the beginning and the end of a 7-in. reel was + 0.1%. The analysis of the recorded violin tones was based on the assumption that the pitch of violin tones can be related to the frequency of their fundamentals (Plomp, 1976; Rasch and Plomp, 1982). This assumption is plausible because it was found that the frequencies of the partials in the central part of violin tones are harmonic (thatis, integral multiples ), and changesin the harmonic of the fundamental frequency structure are smallin that part of the tone (Fletcher ef al, 1965). Hence, the fundamental frequency of each of the 129 notes played by each violinist was measured in the following way. First, two bandpass filters (Krohn-Hite, model 330N) were used to filter out higher partials. The resulting sinusoidal signal was controlled by means of an oscilloscope. Simultaneously, by means of an electronic digital timer-counter (Hewlett-Packard, model 5326B), as many samples of five consecutive periods as possible were drawn. For every sample, the fundamental frequency was calculated to the nearest 0.01 Hz and stored into a computer. However, samples at the on a violin that is out of tune (Galamian, 1962), and (4) the subjects—due to the slow tempo—had the opportunity to beginning of each note were excluded because the change of Hence, generated tones were always the result of the violinist’s tonal memory model and observed interval sizes bealso Saldanha and Corso, 1964). The number of samples make minor adjustments based on auditory cues all the time. tween notes were not confounded to the initial tuning of the instrument. „It was not the purpose of the study to find out how a violinist intones in fast tempo. In that case, the frequency of a note depends completely on proprioceptive cues guiding finger placement, since the produced tone is so short that it can no longer be corrected. Neither did the present study want to give an answer to the question of whether the pitch of a tone emanates either from the memory for the “absolute” pitch of this tone or from its relation to the pitch of another tone produced before. However, due to the slow tempo, violinists were always able to rely upon their memory for absolute pitch (should they possess it). Therefore, it was desirable to allow the violinists to tune their instruments at their own discretion. In particular, possessors of absolute pitch can be annoyed by being forced into a pitch standard that is not their own (Risset, 1978). The fingering was left free because accuracy of intonation depends primarily on maximum sureness in the shifting from one hand position to another, and this, in turn, is determined by the personal habit of the performer. Furthermore, fingering is based on personal characteristics of the individual performer and on the peculiarities of his technical equipment (Yampolsky, 1967; Flesh, 1979). _ The performances took placé in a slightly damped room, and were recorded on magnetic tape (Agfa PER 525) by means of a high-quality unidirectional microphone (Peerless-MBC 548) held approximately three feet above the bridge of the instrument. Recordings were made at a tape speed of 15 in./s on a Nagra tape recorder, model III-B. At this speed the flutter (DIN.45507) was 0.07%. This value corresponds to the most unfavorable case, where the periodic faults of the play back, were added to those of the record527 - | J. Acoust. Soc. Am., Vol. 93, No. 1, January 1993 bow generates importantnoise components which prevent a reliable sinusoidal signal to be extracted (seein this context that were excluded varied between notes and usually corresponded to the first 50 ms of the tone. However, for every note played, 150 to 300 raw frequency counts were retained. Next, the arithmetic mean of those frequencies was computed. In. this paper, that mean is referred to as the observed 0 frequency of the note. measurement the of reliability the In order to evaluate procedure, the performance of two violinists (subjects 2 and 6) were analyzed twice. The greatest difference registeredin repeated measurements of the observed frequencies of a single note was 0.35, 0.51, and 0. 47 cents for, respectively, the lowest (C,-C,), the middle (C‚-Cs), and the highest (C,-C,) scale location. For notes near A, (440 Hz), As (880 Hz), and A, (1760 Hz), these differences correspond to a fundamental frequency change of about 0.09, 0.26, and 0.48 Hz, respectively. The observed frequencies were analyzed from two points of view. The first one (method 1) assumed that the pitch of each note was derived by the violinist from the pitch of the immediately preceding note. The second point of view ‘ (method 2) assumed that when a violinist is performing a piece of music in the key of C, only the sequence C to D gives information about the major second. F and G, even when played in succession, indicate the intonation of the fourth and fifth, respectively, since the musician refers all tones to the keynote and thusis really judging C to F and C to G. Method 1 corresponds to the approach adopted by Greene (1937a,b), whereas method 2 follows the procedure used by Nickerson (1949). Hence, the observed interval sizes between adjacent notes in method 1, and the observed interval sizes between each note and the keynote C in method 2, were both expressed in cents. These interval scores in cents constituted the basic data for further statistical analysis. Next, differences in cents were calculated between obFranz Loosen: Intonation of solo violin performance

TABLE I. Example of the computation of the mean absolute difference (MAD)in cents between observed and theoretical musical intervals. Method.A: Intervals between adjacent notes. Note and interval C, D, CD, . E, FE, D,-E, | G, E,-F, Ay F,-G, G,-A, B, Ay-B, : MAD in cents C, B,-C; Observed frequency in hertz Violinist 1—trial 1 261.88 Observed interval between adjacent ‘ notes in cents 295.18 207.21 331.00 198.30 . 349.68 392.42 95.05 199.64 442.99 497.76 209.85 201.79 524.57 . 90.85 Theoretical interval in cents Pythagorean tuning 203.91 203.91 90.22 203.91 203.91 203.91 90.22 Equal tuning 200.00 200.00 100.00 200.00 200.00 200,00 100.00 . Just intonation 203.91 182.40 111.73 203.91 182.40 203.91 111.73 Observed interval minus theoretical interval in cents Pythagoreantuning . \ j ‘ ‘ 3.30 — 5.61 4.82 — 4.27 Equal tuning 7.21 Just intonation 3.30 5.94 — 2.12 0.62 3.81 — 1.70 — 4.95 — 0.36 9.85 1.79 — 9.15 5.00 15.90 — 16.68 — 4.27. 27.45 — 2.12 + 20.88 12. 94 served and corresponding theoretical interval sizes in Py- Since the violinists played adjacent scales without iriterthagorean, equally tempered, and just intonations. Finally, for each (N = 144) individual scale played, the arithmetic ruption (Fig. 1), and the data were analyzed per scale, the C’s on transition from one scale to another have always been mean of absolute differences between observed and theoretiincorporated into the MAD values of two scales. Thus the cal interval sizes was calculated as an “error” score for each of the three tuning models. The resulting means (in cents) frequency of C, was incorporated in the calculation of the MAD value of the C,-C, scale as well as in the calculation of are called mean absolute departures (MAD values) in this the MAD value of the C,-C, scale. paper. | Table I and Table IT present a summary of the computation of the MAD values according to method 1 and method 2, respectively. The same observed frequencies were ana- Il. RESULTS | The arithmetic mean of the fundamental frequencies to lyzed in both examples. In the last column of Tables Tand IE, three MAD values are tabulated: one for Pythagorean tunwhich the violinists tuned the A, string of their instrument at the beginning of the experiment, was 441.01 Hz (s.d. = 0.70 ing, one for equally tempered tuning, and oneforjustintonation. The smaller the MAD value, the better the performances correspond with the scale model. Hz). The range extended from 439.82-442.23 Hz (i. e., an interval of 9.46 cents). After the experiment the violinists were asked to dem- TABLE Il. Example of the computation of the mean absolute difference (MAD) in cents between observed and theoretical musical intervals. Method 2; Intervals from keynote. Note a MAD Observed frequency in hertz Violinist 1—trial 1 Cy D, E, F, G, A, B, Cs 261.88 295.18 331.00 349.68 392.42 442.99 497.76 §24.57 500.56 700.20 910.05 1111.84 1202.69 Observed interval from keynote in cents in cents \ tee 207.21 405.51 | ; . tee ne nee 203.91 200.00 203.91 407.82 400.00 386.31 498.04 500.00 498.04 701.95 700.00 701.95 905.86 900.00 884.35 1109.77 1100.00 1088.26 1200.00 1200.00 1200.00 theoretical interval in cents Pythagorean tuning vee 3.30 — 2.31 . 2.52 — 1.75 : 4.19 2.07 2.69 Si 2.69 Equal tuning Just intonation ce. wee 7.21 3.30 5.51 19.20 0.56 2.52 0.20 — 1.75 0.05 25.70 11.84 23.58 2.69 2.69 5.44 11. 25 : Theoretical interval in cents Pythagorean tuning Equal tuning Just intonation j Observed interval minus 528 J. Acoust. Soc. Am., Vol. 93, No. 1, January 1993

in cents of mean absolute departure TABLE II. Arithmetic means (M) over trials and standard errors (SE) from theoretical interval sizes in the Pythagore- For every violinist the lowest arithmetic mean within each octave interval is an (PT), the equally tempered (ET), and the just intonation (JI) scale systems. N = 24 for row totals, and N = 18 for column totals. were defined between adjacent notes (method 1). Note: N = 3 for cel entries, printed in italics. Intervals 4 5 6 7 8 Over all violinists Violinist 1 2 3 PTM s.e. ETM s.e. JIM s.e. 67 1.7 9.4 24 173 2.4 43 0.2 7.1 1.6 14.7 1.6 8.8 3.9 7.3 2.4 13.1 0.8 6.1 0.7 6.0 1.6 14.1 2.1 5.6 1.0 6.6 0.3 14.2 0.3 12,3 1.5 8.8 2.5 10.2 1.9 9.5 0.1 14.0 0.1 21.0 0.9 5.2 1.3 45 1.0 12.7 1.3 7.3 0.7 8.0 0.8 14.7 0.8 CC, PTM s.e. ET M s.e. IM s.e. 103 0.6 8.0 17 140 2.1 77 1.6 8.9 1.8 14.6 2.3 6.9 0.9 5.3 0.5 11.3 0.2 3.7 0.2 5.9 0.8 13.8 0.5 6.8 0.7 5.5 0.7 10.0 0.6 14.0 0.8 10.9 1.0 12.4 12 8.8 0.1 11.6 0.4 18.2 0.2 6.3 12 7.8 0.8 15.2 1.1 8.0 0.7 8.0 0.6 13.7 0.6 CC, PT M se. ET M s.e. JIM s.e. Ill 1.0 7.4 0.6 101 1.9 7.7 1.3 6.3 18 12,2 2.5 8.9 0.4 9.0 0.1 14.2 1.6 8.5 2.0 7.1 14 12.8 0.8 6.0 1.4 9.5 2.4 17.0 2.0 19.2 3.7 15.1 3.6 10.2 2.6 9.0 0.8 9.1 2.3 14.4 2.2 6.7 2.0 9.1 2.7 14.3 2.0 9.6 1.0 9.1 0.8 13.1 0.7 C.-C, PT M s.e. ETM s.e. JIM s.e. 8.7 2.1 9.1 1.9 182 2.0 6.2 0.4 5.7 0.9 12.2 1.7 10.1 1.6 9.4 1.8 13.6 0.9 6.1 0.6 6.8 0.5 15.3 0.5 10.1 0.4 71 1.8 13.4 1.6 13.0 1.2 9.7 0.6 10.3 1.5 5.2 0.8 7.6 1.6 15.6 2.7 4.0 02 6.4 1.5 13.7 1.3 7.9 0.7 7.7 0.5 14.0 0.6 C.-C; PT M s.e. ETM s.e. JIM s.e. 9.5 2.8 9.4 3.1 15.1 2.6 5.3 0.4 5.9 14 13.3 1.5 9.6 1.5 72 0.4 12.5 2.3 6.2 2.0 9.2 1.2 17.2 1.0 7.9 1.4 9.6 1.6 16.8 17 12.8 Li 11.3 0.4 14.1 1.0 48 0.3 7.2 0.5 16.2 1.1 6.0 0.5 8.1 0.5 13.9 0.6 7.8 0.7 8.5 0.5 14.9 0.6 CC, PT M s.e. ETM s.e. JIM s.e. 79 1.6 10.2 0.8 132 2.4 12.1 16 12.7 2.7 11.6 4.0 12.7 19 15.7 2.3 20.3 1.8 9.6 0.8 10.3 0.4 16.7 1.1 8.0 04 10.4 1.2 18.2 1.7 113 2.2 13.9 1.7 20.1 2.5 12.2 1.3 16.1 1.8 24.5 1.3 10.4 1.1 10.8 0.9 16.8 1.5 10,5 0.6 12.5 0.7 17.7 1.0 9.0 0.7 8.9 0.7 147 1.0 7.2 0.7 7.8 0.8 13.1 0.9 9,5 0.8 9.0 1.0 14.1 0.9 6.7 0.6 7,5 0.5 15.0 0.5 7.4 0.5 8.1 0.7 14.9 0.8 13.8 0.9 11.6 0.9 12.9 1.1 8.2 0.6 10.9 0.9 18.3 0.9 6.5 0.6 7.8 0.7 14.4 0.6 8.5 0.3 9.0 0.3 14.7 0.3 Ascending scales CC; Descending scales Over all scales PTM s.e. ET M s.e. JIM s.e. onstrate the fingering they had used. Without exception, they used the fingering indicated in Fig. 1. The main results of the present study are summarized in Table III and Table IV, in which the arithmetic means (M) of the MAD values over trials and the associated standard errors (s.e.) are tabulated. For example, the value 6.7 in the left upper corner of Table III is the arithmetic mean of the three Pythagorean MAD values yielded by the three trials of violinist 1 for the ascending scale C,-C,. Table IV is arranged in the same way, but here the intervals were measured from the keynote of the respective scales. The standard 529 J. Acoust. Soc. Am., Vol. 93, No. 1, January 1993 errors in the row margins of Tables III and IV refer to the variability in MAD values between violinists. Analogously, the standard errors in the cell entries and column margins refer to the variability within violinists. Visual inspection of Tables III and IV shows that violin performances always conform most closely with Pythagorean or equally tempered intonations, but for two exceptions in each table. From the analysis of the raw MAD values computed according to method 1, it appears that for 56.3% of the 144 scales played, the Pythagorean MAD value is smaller than the corresponding MAD values for the equally temFranz Loosen: intonation of solo violin performance

TABLE IV. Arithmetic means (M) over trials and standard errors (s.e.) in cents of mean absolute departure from theoretical interval sizes in the Pythagore-: an (PT), the equally tempered (ET), and the just intonation (JI) scale systems. For every violinist the lowest arithmetic mean within each octave interval is printed in italics. Intervals were defined from keynote (method 2). Note. N = 3 for cel entries, N = 24 for row totals, and N = 18 for column totals. Violinist Over all 1 2 3 4 5 6 7 8 violinists Ascending scales C4-Cs PT M se. ET M s.e. H M s.e. 4.5 1.1 6.4 0.6 12.5 0.8 6.8 1.1 10.4 1.1 15.9 1.1 7.1 1.7 7.6 1.9 12.3 2.8 5.9 0.5 7.4 2.0 13.5 2.0 8.0 1.9 71 0.4 10.3 02 26.4 2.7 22.5 2.7 17.2 2.7 8.9 1.0 12.8 17 18.1 1.0 6.1 1.7 4.3 0.5 9.5 0.8 9.3 14 9.8 1.2 13.7 0.8 CC, PT M s.e. ETM s.e. JIM s.e, 7.3 0.4 9.5 1.6 15.0 1.7 7.3 2.2 10.8 2.5 15.9 2.6 6.4 0.8 48 1.4 8.7 1.2 5.1 0.3 4.7 0.2 10.6 0.5 7.1 0.8 6.5 2.0 11.0 2.5 11.7 1.9 12.6 2.1 16.6 2.0 14.4 3.3 18.3 5.7 23.6 3.3 6.2 1.4 8.3 1.7 13.8 2.1 8.2 0.8 9.4 Lt 14.4 1.1 C,-C, PT M se. ET M 8.9 2.7 79 10.8 1.2 95 112 1.1 11.4 10.2 0.6 7.4 5.1 1.2 73 20,9 8.4 21.9 11.8 Ll 14.3 6.7 2.4 7.6 10.7 1.3 10.9 — se. 1.2 0.6 1.7 0.5 1.1 10.0 3.4 0.9 1.5 JI M se. 9.6 2.2 12.6 2.1 15.6 2.4 8.1 0.8 13.0 0.9 25.0 11.2 19.3 2.2 11.6 0.7 14.3 1.7 C.-C, PT M s.e. ETM s.e. JIM s.e. 8.8 0.9 10.7 19 16.2 2.1 5.4 0.9 5.2 0.3 10.4 0.5 15.7 3.6 119 3.4 | 9.4 1.8 5.9 1.1 6.8 Ll 12.3 1.7 12.4 1.3 8.5 13 8.6 1.3 26.0 2.1 22.1 2.1 16.9 2.0 7.7 2.6 5.5 0.4 9.9 1.5 3.4 0.3 6.0 1.0 11.1 1.2 10.7 1.5 9.6 1.2 11.8 0.8 CC, PTM s.e. ET M s.e. JIM s.e. 108 2.8 12.6 4.0 18.1 42 9.5 2.6 11.7 3.5 17.6 32 13.2 2.3 10.2 1.7 8.1 1.3 5.4 1.3 6.7 0.9 12.6 1.0 5.3 1.0 6.9 0.6 12.4 0.6 12.3 2.3 10.8 1.2 13.1 1.3 11.0 0.4 5.4 0.6 9.1 1.9 7.2 1.4 11.1 1.3 16.4 1.4 93 1.0 9.4 0.8 13.4 1.0 c--C, PTM Se. ET M s.e. JIM s.e. 151 4.1 18.3 4.8 23.2 5.1 92 Ll 11.3 1.9 15.8 2.1 93 0.5 11.4 1.7 16.1 2.0 9,5 0.7 7.8 0.8 11.4 0.3 6.7 0.7 9.8 13 15.7 0.9 14.0 6.0 17.6 6.3 22.9 6.3 Id 0.7 13.6 0.6 17.3 1.9 11.6 1,9 14.7 2.7 20.3 2.4 10.8 1.0 12.9 1.2 17.8 1.2 Descending scales . Over all scales PT M 9.2 8.2 10.6 7.0 74 18.6 10.8 6.9 28 se. 1.1 0.7 1.0 0.6 0.7 2.2 1.0 08 0.5 ETM s.e. JI M s.e. 10.9 13 15.8 1.5 9.8 0.9 14.7 0.9 9.5 1.0 11.7 1.0 6.8 0.5 11.4 0.6 77 0.5 11.8 0.7 17.9 2.1 18.6 2.1 11.5 1.3 16.2 . 1.4 8.7 1.0 13.8 1.0 10.4 0.5 14.2 0.5 corresponding MAD values for the other scale models in Figure 2 shows the arithmetic means (over trials and subjects) of the MAD values for the three scale modéls. As can be seen, methods 1 and 2 yield analogous results. For 38.2% of the scales only. The 95% confidence intervals for both methods, the arithmetic means of the Pythagorean and those two proportions do not overlap. Hence, it may be statthe equally tempered MAD ed that more frequently the performed scales correspond better with Pythagorean than with equally tempered intonation (p <0.05). The analysis of the raw MAD values, com- (p <0.01) smaller than the arithmetic mean for the just intonation MAD values. The differences between the arithmepered and the just intonations, respectively. On the other hand, the equally tempered MAD value is smaller than the puted according to method 2 yielded a similar result. In this case the p value is even smaller (p < 0.001). 530 J. Acoust. Soc. Am., Vol. 93, No. 1, January 1993 values are significantly tic means of the Pythagorean and the equally tempered MAD values are not significant (p > 0.45). Figure 3 shows for each subject the number of scales

= METHOD 2 METHOD 1 & + Lm in QO iS ) Z 144+ = | SH 16} 7 < z 14} _— li FH it. Lu = un D ul a = oO 7) = u5 - ier <6 T O Œ LL Lul P u z 10 L | { | + = oO D < z di — 8 | | 1 i EQUALLY TEMPERED TUNING JUST INTONAT TON i PYTHAGGREAN TUNING —e— METHOD 1 —O-—— METHOD 2 by FIG. 2. Arithmetic mean of absolute departures from three scale systems reprepoint data Each major. C of scales diatonic playing eight violinists c sents the arithmetic mean of 144 MAD values expressing the arithmeti mean of absolute differences between performed and theoretical musical between intervals within a single scale. In method 1 intervals were defined adjacent notes; in method 2 intervals were defined from keynote. Vertical METHOD 1 4 10; | PEJ PEJ PES PEJ a 1 PES P = PYTHAGOREAN Pes 2 E = EQUALLY TEMPERED J = JUST INTONATION PES 3 PEJ PEJ 4 5 VIOLINIST | PES 6 P PEJ kee] METHOD 2 FIG. 3. Number of scales that best fit Pythagorean, equally tempered, and just intonation scales, respectively. For each individual scale played, the l arithmetic mean of absolute differences between observed and theoretica interval sizes was calculated as an “error” score for each of the three scale in systems. In method 1 intervals were defined between adjacent notes; method 2 intervals were defined from keynote. Each violinist played 18 diatonic scales in C major. 531 D L i A DIRECTION OF SCALE O——O @—® BM Pythagorean scale Equally tempered scale Just intonation scale Ascending Descending FIG. 4. Arithmetic mean of absolute departures from three scale systems by of the direceight violinists playing diatonic scales of C major as a function arithmetic the ts represen point data Each . sequence tion of the musical absolute differmean of 72 MAD values expressing the arithmetic mean of within a single ences between performed and theoretical musical intervals in method notes; adjacent between defined were s scale. In method 1 interval 2 intervals were defined from the keynote. played which best fit the respective intonation models. Apart from one exception (viz., violinist 6), single scales always fit just intonation most poorly. Figure 4 shows that descending scales always yield larger mean MAD values than ascending ones. Analogously, Fig. 5 shows upper scales always yield larger mean MAD values than lower and middle ones, respectively. In order to investigate whether the trends in Figs. 4 and sim - THE RESPECTIVE SCALE SYSTEMS NUMBER OF. SCALES WHICH BEST FITTED bars show the 99% confidence interval for the population mean. PEJ 1 L D | | PES A A = D= L 5= 8 al L WwW > a = Im Gi 4 au 12+ | > J. Acoust. Soc. Am., Vol. 93, No. 1, January 1993 5 are statistically significant, for each method of analysis, the | raw MAD values were subjected to three analyses of variance: one for the Pythagorean data, one for the equally tempered data, and one for the just intonation data. The model was a 2 (scale directions) X 3(scale locations) X 8-(subjects) mixed randomized block factorial design with three replications within each of the treatment combinations (Winer et al., 1991). Since there was no evidence of the existence of any interaction between violinists and direction of scale, or between violinists and scale location, and there was also no evidence of a triple interaction between direction of scale, scale location, and violinists, these three interaction terms were pooled with the experimental error, thus making the F test for treatment effects more powerful. Apart from one exception direction of scale never yields a significant (p < 0.01) main effect. Only the just intonation MAD values—for method 1—are significantly greater in descending scales than in ascending ones [F(1,131) = 9.12,

iu Taree METHOD 1 1964), greater weight must be attached to the second result than to the first one, because it relates to a measure that METHOD 2 utilizes more information which is relevant to the question. Thus, when restricting to the two best fitting models a 4 a zijl = 14 Ir | (the Pythagorean and the equally tempered) it can be concluded that: (1) ascending and descending scales fit both 8 à 12h - scales (C,-C,) fit both models less well than the middle = ul 16 u cn | — | ed 4D © 1) st - 2x z 8 < = L 1 CC, Cel, l 1 CEC, Cyl, L i CeC, CC SCALE LOCATION o—O Pythagorean scale @——-® Equally tempered scale @——8 Just intonation scale models equally well; (2) apart from one exception, the upper scales (C-C;); (3) lower(C,-C,) and middle scales are equally well approximated by both scale models; and (4) apart from one exception, there is no significant interaction between scale direction and scale location. However, analysis of violin performances can also be focused on the size of specific intervals. Table V shows arithmetic means of observed interval sizes between adjacent notes, and Table VI shows arithmetic means of interval sizes from the keynote. Since it appears from the previous analysis that performances are poorly modeled by just intonation, the FIG. 5. Arithmetic mean of absolute departures from three scale systems by eight violinists playing diatonic scales of C major as a function of scale location. Each data point represents the arithmetic mean of 48 MAD values expressing the arithmetic mean of absolute differences between performed and theoretical musical intervals within a single scale. In method1 intervals were defined between adjacent notes; in method 2 intervals were defined from the keynote. 8 & L'on w iS METHOD 1 EQUALLY TEMPERED SCALE PYTHAGOREAN SCALE 18p 4 Eo 16} 4 tú ez Q 14} Lu I ler We m - . 4 = p= 0.003]. Scale location, on the contrary, always yields a significant (p < 0.01) main effect, apart from two exceptions {the just intonation’ MAD values of method 1 [F(2,131) = 1.66,p = 0.195] and, the Pythagorean MAD 35 um 5 QM <x z ek {=e n MAD values of the upper scale location are significantly larger than the corresponding arithmetic means of the lower and middle scale locations (p <0.05). The interaction between scale direction and scale location (Fig. 6) is significant for the just intonation data of both methods and for the equally tempered data of method 1 [F(2,131) = 7.11, p=0.001, F(2,131) = 3.76, p=0.026, and F(2,131) = 5.39, p = 0.006, respectively]. In the above approach violin performances were analyzed, based on the assumption that a musical scale should be considered as an undivisible musical unit. The results of this analysis can be summarized as follows: (1) Violin performances approximate Pythagorean and equally tempered intonations better than just intonation; and (2) Pythagorean and equally tempered intonations are equally good models to describe performances. With reference to the second point, it could be objected that the scales played more frequently deviate less. from Pythagorean intonation than from equally tempered intonation. However, the arithmetic means of departures from these two scale models does not differ significantly. Hence, according to measurement theory (Coombs, 532 J. Acoust. Soc. Am., Vol. 93, No. 1, January 1993 . - ——? 4 L L. L L L L = W = A D A D A D 5 PYTHAGOREAN values of method 2 [F(2,131) = 2.16, p = 0.120]}. It appears from Tuckey’s studentized range tests that in all cases of a significant direction effect, the arithmetic means of the JUST INTONATION SCALE & Liaw w 5 ca METHOD 2 EQUALLY TEMPERED © JUST INTONATION SCALE SCALE SCALE 18f 4 E © 16} - at 4 Gz Wz Li we Wn 2 5 à OQ 9 D 5 = z < = a 14 tk Is HF 10 7 = A 4 4 === 4 sh Fe a A L i. L 1 L D A D A D DIRECTION OF SCALE o—O Cl A = Ascending eo——*® CC, D = Descending En CC; FIG. 6. Arithmetic mean of absolute departures from three scale systems by eight violinists playing diatonic scales of C major as a function of direction of musical sequence and scale location. Each data point represents the arithmetic mean of 24 MAD values expressing the arithmetic mean of absolute differences between performed and theoretical musical intervals within ‘a single scale. In method 1 intervals were defined between adjacent notes; in method 2 intervals were defined from the keynote.

TABLE V. Arithmetic means (M) of interval sizes and standard errors (s.e.) in cents over violinists and trials. Method 1: Intervals between adjacent notes. Interval C-D D-E E-F F-G C,-C, M s.e. C.-C, M s.e. CC, M s.e. CC, M s.e. 198.0° 1.5 206.2 1.6 204.8 2.7 203.0 1.2 204.2 2.0 201.3 1.6 198.1 2.5 201.2 1.2 95.0 2.4 96.9 2.1 101.4 3.5 97.8 1.6 2014 1.6 200.7 1.8 201.7 2.3 201.3 1.1 C.-C, M s.e. CC; M s.e. CC, M s.e. CC, M 194.7 2.4 200.0 2.0 207.1 1.6 200.6 209.0 1.9 206.1 17 203.2 2.0 206.1 91.3 1.9 960 2.9 94.6 2.5 94.0 201.9 1.4 202.2 2.0 205.4 2.6 203.2, Ascending scales Descending scales . G-A A-B B-C 203.5 1.6 202.0 17 198.4 1.8 201.3 1.0 204.7 1.5 202.2 2.7 199.7 19 202.2 1.2 92.0 2.0 94.8 22 96.3 2.5 94.4 13 200.9 1.7 202.0 1.5 199.1 1.8 200.7 199.3 1.7 201.9 1.7 211.5 2.2 204.2 99,9 2.6 91.9 2.5 77,3 3.5 89.6 se. 13 1.1 1.4 12 1.0 13 2.0 Ascending and descending scalés CC: M s.e. C.-C, M se. C.-C, M s.e. C,-C, M s.e, 196.4 . 14 203.1 14 206.0 1.6 201.8 0.9 206.6 1.4 203.7 12 200.7 1.6 203.7 0.8 93.2 1.6 96.4 1.8 98.0 22 95.9 1.1 201.7 1.1 201.5 1.4 203.5 1.7 202.2 0.8 202.2 12 202.0 1.1 198.8 1.2 201.0 0.7 202.0 12 202.1 1.6 205.6 17 203.2 09 95.9 17 93.2 1.7 86.8 2.6 92.0 1.2 . = 72 for total note range C,-C;. For pooled ascending and descending * For ascending and descending scales separately: N = 24 for single scales, and N CC. range note total for 144 = N and scales, single for 48 scales: N = TABLE VI. Arithmetic means (M) of interval sizes and standard errors (s.e.) in cents over violinist and trials. Method 2: Intervals from keynote. Interval Ascending and descending scales C-G C-A C-B C-C 497.2 1.8 504.4 2.3 504,3 3.6 502.0 1.6 698.6 2.1 . 705.1 1.8 706.0 3.4 703.2 1.5 902.1 3.1 907.1 2.0 904.4 26 904.6 1.5 1106.8 3.4 1109.3 24 1104.1 2.5 1106.8 1.6 1198.8 28 1204.1 2.3 1200.4 3.1 1201.1 1.6 403.7 3.4 406.1 22 410.4 2.5 406.7 1.6 495.0 2.4 502.1 3.2 504.9 3.3 500.7 1.8 697.0 2.0 704.3 2.0 710.3 2.0 703.9 1.3 897.9 3.3 906.4 2.3 909.4 1.9 904.6 1.6 1097.2 2.9 1108.3 2.2 1120.9 3.0 1108.6 1.9 — 1197.1 2.3 1199,9 2.3 1198.2 2.6 1198.4 14 403.0 2.2 406.8 1.4 406.6 1.8 405.5 Ll 496.1 1.5 503.2 2.0 504.6 2.4 5013 1.2 697.8 1.5 704.7 1.3 708.1 2.0 703.5 1.0 900.0 2.2 906.7 1.5 906.9 1.6 904.6 1.1 1102.0 2.3 1108.8 1.6 1112.5 2.3 1107.8 | 1.3 1197.9 1.8 1202.0 1.6 1199.3 2.0 1199.7 1.1 C-E CC; M se. C.-C, M s.e. C.-C, M se. C.-C, M se. 198.0* 1.5 206.2 1.6 204.8 2.7 203.0 12 402.2 2.9 407.5 1.8 402.9 2.4 404.2 14 C.-C, M S.e. C.-C; M se. C,-C, M s.e. C;-C; M s.e. 194.7 2.4 200.0 2.0 207.1 16 200.6 13 : Cy-C, M $.e. C,-C, M se. CC, M s.e. C,-C,; M s.e. 196.4 14 203.1 14 206.0 1.6 201.8 0.9 Ascending scales Descending scales C-F C-D . | . for total note range C,-C;. For pooled ascending and descending 8 For ascending and descending scales separately: N = 24 for single scales, and N = 72 écales: N = 48 for single scales, and N = 144 for total note range C4-C). 533 J. Acoust. Soc: Am., Vol. 93, No. 1, January 1993

discussion will refer only to Pythagorean and equally tempered intonations from now on. The intervals that demand attention when contrasting 157 Pythagorean and equally tempered intonations are the miel (100 cents). Finally, over ascending and descending scales, arithmetic means of observed interval sizes for the minor second always lie between the normative interval sizes ı0r ist NON-C OCTAVES ann served interval sizes of minor seconds are, apart from one exception, always smaller than in the equally tempered mod- C OCTAVES PERCENTAGE nor second and the major third. Indeed, the size of those intervals differs considerably in both intonations: 9.8 cents for the minor second and 7.8 cents for the major third. Table V shows that, apart from one exception arithmetic means of observed interval sizes for minor seconds (E-F and B-C) are always larger than in the Pythagorean model (90.22 cents). Analogously, the arithmetic means of obof the Pythagorean and the equally tempered intonations, apart from one exception. The analysis of major seconds, - yielded similar results. Over all scale directions and scale locations, observed interval sizes of the major second always lie between the interval sizes of the Pythagorean and the equally tempered intonations. On the major third, this paper can only report indirectly. Indeed, in the diatonic scale of C major, the major third cannot be observed from two notes played immediately after each other. The size of the major third can only be calculated by summation of the sizes of two adjacent major seconds from the keynote C (or from the notes F or G). Table VI shows that computed mean interval sizes of the major third lay between the mean interval sizes of Pythagorean and equally tempered intonations. The results for the major thirds F-A and G-B are similar to those for the major third C-E. The last columnin the lower part of Table VI shows the arithmetic means of computed interval sizes for C octaves [ie., the octave intervals C,-C,,, (i= 4,5,6) and C,-C,_ , (i= 5,6,7)] over violinists and trials. As can be seen, computed interval sizes deviate only slightly from theoretical interval size (1200 cents). The upper part of Fig. 7 shows the distribution of differences between computed interval size and 1200 cents for all C octaves (N = 144). The arithmetic mean of the distribution is — 0.3 cent. The standard deviation (12.4 cents) reveals that the size of the computed interval sizes may vary considerably. The arithmetic -36 -24 -12 OY 12 24 . 36 48 DEPARTURE FROM 1200 CENTS IN CENTS FIG. 7. Distribution of deviations from 1200 cents in C-octave intervals (i.e., intervals between two C’s) computed from the performances of eight violinists playing diatonic scales of C major. The distribution in the upper part of the figure relies upon 144 C-octave intervals in the note range C,-C,, The lower part distribution relies upon 576 non-C-octave intervals.in the same note range. Cases of octaves contracted occur at the left of the O departure, cases of stretched octaves, at the right. The arithmetic mean of the distribution is 5.0 cents (s.d. = 11.8 cents) and the arithmetic means of departures of non-C-octave intervals from 1200 cents for the eight violinists are: 2.0, 13.1, 5.8, 0.1, 1.4, 8.2, 5.3, and 4.1, respectively. An analysis of variance revealed that the difference bemeans of the C-octave departures from 1200 cents for the tween the mean stretch for ascending (3.3 cents) and descending (6.7 cents) non-C octaves is significant [F(1,565) = 13.76, p = 0.0002]. However, the difference between the mean stretch for non-C octaves localized beeight violinists are: 2.9, 10.5, — 6.7, 1.3, — 3.8, — 3.5, — 3.1, and — 0.8 cents, respectively. tween D, and B, is not significant [F(1,565) = 2.86, tween D, and B; and mean stretch for non-C octaves bejects )-mixed randomized block factorial analysis of variance of the sizes of the C-octave departures from 1200 cents rep = 0.091]. Thus a shift in scale location is not accompanied by a change in physical octave extent. The interaction between direction of musical scale and location of scale is not vealed that the mean departure score for ascending scales significant [F(1,565) = 1.89,p = 0.170]. (1.1 cents) did not differ significantly from the mean departure score for descending scales (1.6 cents) [F(1,131) = 2.23, p = 0.138]. Furthermore, mean depar- Finally, intra-individual variability in interval size between adjacent notes was analyzed. For each violinist and for each of the 42 intervals between adjacent notes played on ture scores did not differ significantly for different scale loca- [F(2,131) = 0.80, p = 0.453]. ' The distribution of departure scores of non-C-octave ineach trial, the total range (in cents) of the interval sizes across the three trials was computed. In Table VII the arithmetic means of those total ranges across subjects are tabulated. For example; the value 8.5 in the left upper corner of the table was obtained as follows. First, the interval-size of tervals from 1200 cents is shownin the lower part of Fig. 7. C,-D; was computed for each subject on each of the three A 2(scale directions) X3(scale locations) x 8(subtions [F(2,131) = 1.66, p = 0.194]. Neither was the interaction between scale direction and scale location significant 534 J. Acoust. Soc. Am., Vol. 93, No. 1, January 1993

TABLE VIL. Arithmetic mean (M) of total ranges and standard errors (s.e.) times by eight violinists. Descending scales C-D D-E E-F Interval F-G 8.5 0.4 10.5 0.5 18.3 1.5 13.1 0.8 11.3 0.7 12.9 0.6 14.2 0.7 8.1 0.4 13.3 0.6 10.8 04 13.3 0.6 13.7 0.4 16.6 0.9 18.8 1.5 14,7 8.5 0.6 15.4 1.1 145 three A-B B-C 9.1 0.9 8.3 0.7 10.0 0.4 134 0.5 9.9 0.5 12.8 0.8 17.9 0.7 11.7 0.5 15.5 0.6 9.6 0.4 8.3 0.6 9.0 12.9 0.7 12.3 0.5 13.5 CA - C,-C, M s.e. C.-C, M s.e. CC, M s.e. C.-C, M s.e. C.-C; M se. C;-Cs M se. 92 0.5 7.9 0.4 14.0 0.4 11.5 0.5 10.8 0.5 13.3 0.6 trials. Next, the total range of the three observed interval sizes was computed for each of the eight subjects. The arithmetic mean of these eight total ranges is 8.5 cents. The raw set of 336 total ranges was subjected to a 2(scale directions) x 3(scale locations) X 7 (musical intervals) x 8 (subjects)mixed randomized block factorial design. Direction of scale and scale location yielded no significant effect. Neither was any interaction significant. On the contrary, the arithmetic means of total ranges for different musical intervals differ significantly [F(6,299) = 3.74, p= 0.001]. The arithmetic means of the total ranges for the minor second E-F and B-C are significantly larger than the arithmetic means of the total ranges for the major seconds (Tuckey’s test, p<0.001). There is also a significant difference between the mean total 0.6 . 1.0 . 0.6 16.0 11 19.3 1.1 17,5 0.5 0.7 terval information is stored in memory as an abstract entity h and and not as particular notes (Deutsch, 1978; Deutsc Roll, 1974). The conformity in the fingering is hardly surprising, considering the fact that the diatonic scale of C major is a standard exercise, and knowing that all subjects were educated in the same pedagogical tradition. The results of the data analysis that makes use of individual scales as units of analysis, are in accordance with the unacprevious literature. Indeed, one time it was found that by imated companied violin performances were well approx 1949), Pythagorean intonation (Green, 1937a,b; Nickerson, and another time that performances corresponded fairly closely to equally tempered intonation (Ward, 1970). Anal- The arithmetic mean of the starting frequencies of the the : ogously, Cornu and Mercadier (1871, 1873) found that orean intonation of violinists did well approximate Pythag all. intonation, and was not modeled by just intonation at Mawith The findings of the present study are also in line total range of the starting A, frequencies over violinists (9.5 cents) corresponds almost perfectly with the arithmetic viated least from Pythagorean intonation, whereas the other deviated least from equally tempered intonation. Analoion. gously, the greatest departure occurred from just intonat ranges for the eight violinists [F(7,299) = 4.08, p< 0.001]. IN. DISCUSSION open A, string (441.01 Hz) illustrates the actual custom to tune to a higher frequency than the current international standard tuning frequency for Western music (440 Hz). The mean of the individual total ranges for the played A, during the experiment, namely, 9.8 cents (s.e. = 2,1 cents; N= 8). Corso (1954b), who examined a highly trained violinist obtained a root mean square of “error” scores (i.e, the intervals in cents between the frequency of a tuned violin tone and the frequency of a reference tone) of 6.13 cents. Analogously, Ward (1954) found that, for pure tones, octave judgments of trained musicians have a standard deviation averaging about 10 cents, and intersubject variability is two to five times as great. Furthermore, variability in starting A, frequencies shows that, in daily performance, a scale is not a sequence of fixed frequencies. Indeed, musical scales can be continuously slid up and down in pitch without distorting relative interval sizes (Siegel and Siegel, 1977; Burns and Ward, 1978). There is even experimental evidence that in535 J. Acoust. Soc. Am., Vol. 93, No. 1, January 1993 son’s (1960) research on the solo performances of the members of two woodwind quintets. One of the quintets de- The results of the present study are also congruent with the setfinding of Moran and Pratt ( 1926) that average interval tings were more in agreement with interval sizes as defined in equally tempered intonation than with interval sizes as defined in just intonation. Clearly, the present study fails to support Willemze’s (1973) claim that descending scales are p. played according to just intonation, and Benade’s (1976, time when 295) assertion that just intonation is preferred and circumstances permit. The result that departures from Pythagorean and the equally tempered intonations are not significantly different for ascending and descending scales corresponds also with the literature on wind instrumentalists, where no difference in intonation accuracy was found between ascending and descending directions (Mason, 1960; Duke, 1985). Franz Loosen: Intonation of solo violin performance 535 neee Ascending scales in cents for sizes of musical intervals observed in scales of C major played

The finding that the intonation on the highest scale loca- METHOD tion mostly deviates more from scale models than the intonation on the lower scale locations remains unexplained. The fact that it is harder to place fingers on the finger-board accurately in higher hand positions than in lower ones does not satisfy as an explanation. Indeed, the violinists played so w E ia Oo 6 6 4 4 5 ew 0 0 ral “5. 3 slowly that they were able to make tone corrections at any moment, and obviously, those adjustments only influence S24 ES 6 De < & =| a DE og 4 -6 8 the variability and not the arithmetic mean of raw frequency data. The greater deviation from scale models in the highest scale location cannot be explained by a higher differential threshold in that frequency range either. The power to discriminate between differences in frequency is quasiconstant 2 2 ES 5 an CD OF A FG EF GA ii AB BC _ [se a 5 © WU 2 z OTE, (a) The results of the analysis of separate musical intervals, not considering the context of the scales in which they were played, are consistent with Rakowski (1976, 1985), Abraham (1923), and Ward’s (1970, p. 421) conclusion “that, on the average, the internal scale of musical pitch used by musicians corresponds fairly closely to equal temperament, but with a slight stretch that results in a tendency toward sharpening all tones relative to the tonic” Moreover, the present study allows one to specify further that the size of intervals played lies between interval sizes as defined in Pythagorean and equally tempered intonations. In the upper part of Fig. 8(a) intervals between adjacent notes in the diatonic scale of C major C.-C, are plotted on the abscissa, deviation in cents above and below theoretical interval sizes is plotted on the ordinate. The two curves in the upper part of Fig. 8(a) show the departures from Pythagorean and equal tempered scales. Intervals contracted relative to these scale systems, occur under the horizontal zero line, the expanded ones above it. If the intervals had been played in accordance with the scale models, all points would have fallen on the SE CO ER PE ES RE A, \ SE PE DE EF FG GA AB MUSICAL INTERVAL IN NOTE RANGE GrC, Be (b) Ser © te CF EG CA CB CC fi CD A CE L CF fi CG N CA fi CB A CC dons N uf N mn CD A CE A CF. Ri CG mn CA 1 CB i CC 1 CD CE N CF 1 CG fi CA ‘ CB 1 CC on (1937a,b). OBC METHOD 2 MEAN DEPARTURE FROM THEORETICAL INTERVAL SIZE IN CENTS ( ; ROMA AOS nom» The finding that direction of scale was not associated with goodness of fit for scale models is in line with Greene <=" -2 , cian, the differential threshold for a tone of 261: Hz (C,) is flat A,) it is about 2.76 Hz (i.e, 2.74 cents). Apparently, 4 ii DE EF FG GA AB Of Se about 0.43 Hz (i.e., 2.85 cents) and for a tone of 1740 Hz (a traditional scale models as such are less adequate to describe violin intonation in higher frequency ranges. CO > En L at TO DE EF FG GA AB BC MUSICAL INTERVAL IN NOTE RANGE CzC, in the interval 262-2986 Hz (Scharf and Buus, 1986). More specifically, Boring (1940) found that for a trained musi- Va MUSICAL INTERVAL IN NOTE RANGE C5C, MUSICAL INTERVAL IN NOTE RANGE GC, Ce) Cd) O0 es m Pythagorean scale Equally tempered scale Pooled scale FIG. 8. (a) Upper plot. Arithmetic mean of departures from Pythagoréan and equally tempered intonations by eight violinists playing diatonic scales of C major. Labels on the x axis represent ascending and descending intervals between adjacent notes in the note range C,—C,. Each data. point indicates the arithmetic mean of 48 departure measurements. Lower plot. As upper plot, but departures are computed from the arithmetic mean of the corresponding interval sizes in Pythagorean and equally tempered intonations. (b) Same as (a) but labels on the x axis represent ascending and descending intervalsin the note range C,-C,. Each data point indicates the arithmetic mean of 144 departure measurements. (c) Same as (a) for inter: val sizes defined from the keynote C. (d) Same as (b) but for interval sizes defined from the keynote C. horizontal zero line, except for chance deviations. As can be seen, the departures from Pythagorean and the equally tempered tuning almost mirror each other. The lower plot of Fig. 8(a) shows mean departures of observed interval sizes from the arithmetic mean of the corresponding interval sizes in Pythagorean and equally tempered intonations. The largest mean departure from the mean interval size is smaller than 2 cents. Figure 8(b)is analogous to Fig. 8(a), but now the figure applies to the whole note range (C,—C,) that was examined. In Fig.8(c) and (d) musical intervals are defined relative to the tonic. scale was deliberately stretched by about 3.5 cents per octave. The stretching of octave intervals has been known for a long time (Burns, 1974; Corso, 1954a; Railsback, 1938). Ward (1954) found that, through the midrange of up. to frequencies of about 900 Hz, the frequency ratio of a subjective octave is about 2.02/1 (i.e., about 17.2 cents). This value is considerably larger than the 3.9 cents foundin the presof all octave intervals that could be computed from the scales ent study. However, such a divergenceis not exceptional. | For example, Martin and Ward (1961) found that the mean played shows a significant mean stretch of 3.9 cents (N = 720). It is interesting to note that Kolinski (1959) proposed a new system of temperament in which the musical smaller than that for subjective pure--tone octaves. Analo| gously, Sundberg and Lindqvist (1973) found a mean. . Finally, the octave intervals should be discussed. The set 536 J. Acoust. Soc. Am., Vol. 93, No. 1, January 1993 stretchin the middle range of the piano scaleis considerably 7

stretch of only 4.5 and 5.2 cents for octave matching by a violinist for reference tones of 1050 and 2000 Hz (SPL of 65 dB). Moreover, for reference tones of 330 and 600 Hz, they found mean octave stretch values of — 0.4and — 0.6 cents, respectively. Similarly, Corso (1954a) found that the size of the octave stretch by the violinist he observed did not fluctuate strongly and irregularly from one scale position to another, and was also around zero cents for more than one octave position. The lack of octave stretching in C-octave intervals can be explained in the context of the literature on perceived tonal organization and absolute pitch (Deutsch, 1980; Deutsch and Feroe, 1981; Dowling, 1978, 1991; Dowling and Harwood, 1986; Harris and Siegel, 1975; Krumhansl and Kessler, 1982). This literature stresses that subjects use cognitive reference points based on their knowledge of the notes of the musical scale. Thus, in the present investigation in which diatonic scales of C major were played by highly trained musicians, it can be supposed that the keynote C was more important than the other notes of the scale. Of course, this speculative hypothesis should be investigated further for other keys. Moreover, in the present research, octave intervals were calculated by summing the sizes of adjacent intervals, whereas traditional studies on octave intervals always deal with situations in which the two notes delineating the octave interval are played immediately after each other, and without any musical context. In the present study, the mean stretch for the lowest non-C-octave intervals was not significantly smaller than _ the mean stretch for the highest non-C-octave intervals. For violin performances also Corso (1954a) found that the magnitude of physical octaves did not increase with upward shifts in the location of octaves on the frequency continuum. The phenomenon that non-C-octave intervals yield a significantly greater stretching in descending scales than in ascending scales is somewhat surprising. From the subjective experience of violinists one would rather have expected the opposite. are presented for comparison in one of both tunings. Yet, in daily musical performance, those abilities are rather irrelevant, because violinists never play in exactly one tuning or the other, and because listeners pay attention to more important aspects of music. In addition, the total range of interval sizes for the same musical interval can easily reach 10 cents, which is larger than the greatest difference in interval size between adjacent notes in Pythagorean and equally tempered scales. Also, Ward ( 1970) showed that in real performances there is much variance in the frequency of notes being played, but that it is not usually noticed by listeners. Furthermore, it has been shown that a given interval may retain its character and identity in spite of considerable “mistuning” (Mursell, 1937). Finally, the acoustical properties of violin tones ensure that beats between their partials are almost inaudible, thus permitting the player great latitude in his choice of pitch ( Benade, 1976, p. 299). It is generally said, that the vibrato reduces the demands on accuracy in violin playing even more. However, experimental evi- _ dence shows quite clearly that the accuracy with which the pitch of a vibrato tone is perceived is not affected to any appreciable extent by the vibrato (Sundberg, 1982). Apparently, violin performances can be described equally well from intervals between the adjacent notes as from intervals from the keynote. An intriguing finding of the present research is the lack of any stretching in C octaves within scales of C major. It would be of interest to know whether this phenomena can be generalized to other musical keys. Indeed, analogous results for other musical keys can provide further evidence for recent theories of tonal organization in music perception and more specifically for theories stressing the influence of musion musical perception (see, i.e., Brown, 1988; cal knowledge Butler, 1983, 1989; Cross et al., 1991; Krumhansl, 1990; Meyer, 1956, 1973; Narmour, 1990; Rosner and Meyer, 1986). In the present study only scales in C major were played and these scales were performed very slowly without vibrato. Consequently, until further investigations are undertaken, it remains an open question whether the conclusions IV. CONCLUSIONS The results of this study indicate that, when individual scales of C major are analyzed as a whole, (i) violin performances clearly fit Pythagorean and equally tempered intonations more precisely than the just intonation, and (ii) performances fit the Pythagorean and the equally tempered model almost equally well. When the analysis focuses on the size of the intervals between notes, not considering the context of the individual scales in which they were played, interval size is halfway between the interval sizes in Pythagorean and equally tempered intonations. It does not seem meaningful to ponder any further about the question whether violin performances are more precisely approximated by Pythagorean intonation than by the equally tempered one. In special laboratory conditions, highly trained musicians may be able to discriminate with accuracy between isolated Pythagorean and equally tempered intervals synthesized on a computer. Experienced listeners can also have clear-cut preferences for musical sequences that 537 J. Acoust. Soc. Am., Vol. 93, No. 1, January 1993 obtained for the diatonic scale of C major can be legitimately extended to cover other situations. ACKNOWLEDGMENTS The author is indebted to Noel Bovens and Rik Delabastita for technical support in measuring the frequency of violin tones, to Christel Snels for her assistance in drawing the figures, and to Godelieve Feyaerts for helping with collecting the data. The author would also like to thank Dr. Reinier Plomp, Dr. Robert W. Young, and three anonymous reviewers for their valuable suggestions and comments on an earlier draft of this paper. Finally, the author express special thanks to Monique Lioen for her significant support in getting this paper into its final shape. as a variant of ‘Burns and Ward (1982, p. 242) categorized this intonation as Ptointervals same the contains diatonic ’ Didymus _just intonation. the major lemy’s diatonic syntonon, but with the minor tone (10/9) below tone (9/8) instead of the reverse.

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