Dept. for Speech, Music and Hearing Quarterly Progress and Status Report Statistical computer measurements of the tone-scale in played music Fransson, F. and Sundberg, J. and Tjernlund, P. journal: STL-QPSR volume: 11 number: 2-3 year: 1970 pages: 041-045 http://www.speech.kth.se/qpsr
STL-QPSR 2-3/1970 IV. MUSICAL ACOUSTICS A. STATISTICAL COMPUTER MEASUREMENTS OF THE TONE-SCALE IN PLAYED IvIUSIC F. Fransson, J. Sundberg, and PI Tjernlund An essential aspect of played music is the tone scale, i. e. the series of fundamental frequencies used in playing. In an earlier report a method for measuring this set of fundamentals was described( '). The method operates with a hardware fundamental frequency measuring equipment connected to a computer. statistically, The computer measures the period time and processes the data The results are presented in the form of fundamental fre- quency his tog ran is^ In this paper a study will be reported of the tone scale used by professional musicians playing melodies on different instruments. Problem Some years ago, one of the author discovered that the series of funda- mental frequencies blown on a flute is strongly dependent on the eventual presence of a tonal interpretation of the notated music. Thus, if a flutist plays after notes, he adopts one scale, and if he plays after successive dicta- tion of single notes, he adopts a different scale, that is rather closely related to the resonance frequencies of the flute(2), see Pig. IV-A- 1. Later on, similar results have been reported(3). This fact raises the question, which are the factors that determine the scale of a played melody? How do the scales of different musicians playing different melodies and instruments compare? Material Three professional first rate members of a symphony orchestra were used as subjects: a flutist, an oboist, and a violinist. They all played the same melody on their own instruments and, as far as the wood-wind instrumentalists were concerned, also on an instrument that was unknown to them. The melody was replayed and tape-recorded several times, and the subjects selected three versions for analysis. The melody was a couple of bars from a solo cadence of a flute concerto. It has an ambitus of 2 1/2 octaves. Finally, the subjects also improvized on their own instruments or played another melody that they choosed themselves.
STL-QPSR 2-3/1970 Analysis The tape -recordings of the se melodie s were analyzed with respect to the fundamental frequency distribution. Experiments showed that the intervals between the scale tones can be determined with an accuracy of f 5 cents, whereas the absolute frequency values of the toncs could be measured only within + 17 cents (1 $). Due to the large ambitus of the melodies and the spectrum properties of thc toncs, the cutoff frequency of the variable lowpass filter of the fun2amental frequency mcasuring equipment had to be varied synchronously with the fundamental frequency of the melodies. Results In Fig. IV-A-2, as in the following figures, the measured fundxnentzl frequencies are compared with those of the equal tempered scale and the deviations are expressed in the ccnt-unit. Moreover, in all cases the deviations were balanced with respect to this scale. This means that the influence on the results of an overall mistuning of the instrument as well as of faults in the absolute frequency measurements was eliminated. Fig. IV-A-2 shows, in the described way, the scale of the flute under various conditions. It is seen that the scale is not very dependent neither of the melody, nor of the instrument: all curves lie rather close together, and the distance between them is in most cases smaller than some 15 cents. Fig. IV-A-3 shows the corresponding curves for the oboist. In this case, the distances between the measuring points for a given tone?-re still smaller than for the flute. Thus we may conclude, that also in the case of the oboe, the melody and the type of instrument do not contribute very much to the determination of the scale. Fig. IV-A-4 shows the scales used by the violinist. Note that also in this case the played scale is rather stable. This is astonishing in view of the fact that the violin imposes no restrictions at all on the player as regards the choice of fundamental frcquencies. The rncasurements on the melody choosen by the violinist failed to give a complete series of fundamentals. Howevcr, on thc toncs that the melodies have in common the frequency valucs lie quitc close to each other.
CENT. 40 - f? I \ I \ P I \ / - Fig. IV-A-2.. Deviations from the equal tempered scale in flute playing mean values obtained from three recordings of a solo cadence played on the musician's own flute. mean values obtained from three recordings of the same solo cadence played on a flute unknown to the flutist. V values obtained when the flutist improvized on his own flute.
CENT 40 Fig. IV-A-3. Deviations from the equal tempered scale in oboe playing. ' on the musician's own flute. mean values obtained from three versions of the same cadence O played on an oboe unknown to the player. v values obtained when the oboist improvized on his own instrument. mean values obtained rrom three versions of a solo cadence played
CENT 40 Fig. IV-A-4. Deviations from the equal tempered scale in violin playing. mean values obtained from three recordings of a solo cadence. 0 values obtained when the violinist played the melody "Schon Rosemarin".
STL-QPSR 2-3/1970 44. contexts in the melody. Such an influence might perhaps be quite strong and it would appear mainly as a broadening of the peaks in the fundamcntal fre- quency histogram. The substantial differences in the values of the tone F4 for the violin do indeed point towards such a dcpendencc. However, this question falls outsidc thc scope of the present investigation. Our mcasurcments have shown that the mcan scales of a soloplaying flutist, oboist, and violinist are quite similar. They all employ not the equal tempered scale, but a stretched scale. What are the possible reasons for this stretch? From the acoustics of the wood-wind instruments we know that thc res- (8) onance frcqucncics of those instruments makc a stretched scale expectable. On the other hand, we know that the playcr cen play according to the equal tempered scale, if he wants, by adjusting his blowing technique(9). It is noteworthy that also thc violinist in our experiments seems to prefer a slightly stretched scale. This indicates that the musical ear for some un- known reason demands a stretched scale. Another interesting fact is that also the piano is tuned in a stretchcd scale ( 10). It has been shown that this probably is an acoustical consequence of the inharmonicity of the partials of the piano tone spectrum("). This stretch belongs to the criteria of a good piano (12). Thus, also in the case of the piano, the musical ear seems to demand a stretchcd scale. There is a difference bctwecn the octave perceived as pure and the octave corresponding to the frequency ratio i:2(i3). Tho subjective octave is a bigger interval than the mathematical octave. This has been shown by ex- periments with sine waves. However, in view of the results prcsented herc, it seems likely that also when complex tones arc concerned, the musical ear wants stretched octaves and strctchcd intervals of other sizes as well. Presently cxpe rilnents are carried out in order to shed some light over this problem. Refercnce s: (1) Sundberg, J. and Tjernlund, P. : "Computer Measurements of the Tone Scale in Performed IvIusic by Means of Frcquency Histograms", STL-QPSR 2-3/1969, p. 33. (2) Fransson, F. : "h~easurcments in Flutes from Different Periods", STL-QPSR 4/1963, p. 12. refs. continue on next page
CENT 40 Fig. IV-A-5. Mean deviations from the equal tempered scale in 0 flute playing. V oboe playing. violin playing.
CENT 40 Fig. IV-A-6. Mean deviations from the equal tempered scale in flute-, oboe-, and violin playing. The solid line shows a model scale consisting of pure tone steps (204 cents) and small semi tone steps (97.5 cents).
STL-QPSR 2-3/1970 Coltman, J. 77. : "Acoustics of the Flutett, Physics Today 2 1: 1 I (1968), p. 27. Sundberg, J. : "The ' Scale' of Musical Instrumentstt, Svensk Tidskrift for Musikforskning 49 - (1967), p. 126. Greene, P. C. : "Violin Intonation", J. Acoust. Soc. Am. 9 (1937), p. 43. Nickerson, J. F. : "Intonation of Solo and Ensemble Performance of the Same Melody", J.Acoust. Soc.Am. 2f (1949), p. 593. Lottermoser, TI. and Meyer, Fr. -J. : "Frequcnzmessungen an gcsungcnen Alclcordentt, Acustica 10 (1960)) p. 18 1. Bcnade, A. H. : "On the Mathematical Theory of Tloodwind Fingcr Holes", J.Acoust.Soc.Am. - 32:12 (1960)~ p. 1591. Meycr, J. : Akustik der Holzblasinstrumente in Einzeldar stellungen (~rankfurt a. M., 1966). Schuclc, 0. H. and Young, R. TI. : "Observation on thc Vibrations of Piano Strings", J. licoust. Soc.Am. - 15:1 (1943), p. 1. Young, R. TI. : "Inharmonicity of Plain Wire Piano Strings", J. Acoust.Soc.Am. - 24:3 (1952), p. 267. Martin, D. TI. and Vard, TT. D. : "Subjective Evaluation of Musical Scale Temperament in Pianos", J. Acoust. Soc. Am. - 33:5 (1961), p. 5G2. VJard, 17. D. : Itsubjective Musical Pitch", J. Acoust. Soc. Am. - 26:3 (1954), p. 369.