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Page 202 ï~~The Relevance of Beating Partials for Musical Intonation Douglas F. Keislar CCRMA, Dept. of Music Stanford University Stanford, CA 94305, USA email: DougKeislar@NeXT.com Abstract: Helmholtz believed that a critical factor in the intonation of consonant harmonic intervals was the beating of partials that almost coincide in frequency. This study tested Helmholtz' idea by using synthesized stimuli in which the beat rate was controlled independently of the interval tuning. The results of three psychoacoustic experiments suggest that musicians base judgments of an interval's intonation on its frequency ratio rather than on its beat rate. Furthermore, subjects did not prefer intervals of just intonation to equal-tempered ones. These conclusions tend to strengthen the argument for a cultural basis for musical intonation, as opposed to the acoustical basis set forth by Helmholtz. One implication is that computer music composers need not be overly concerned with beats when choosing a tuning system. Digital sound synthesis allows composers complete freedom in their choice of tuning system, since any arbitrary set of frequencies can be specified. It is interesting to explore whether there are natural acoustical constraints on this choice. For example, Helmholtz (1877) emphasized the importance of beating for intonation. When a consonant harmonic interval is mistuned slightly from its "just" tuning (in which the two notes' frequencies form a ratio of small integers), there are usually several pairs of beating harmonics. In each of these pairs, a harmonic of one tone almost coincides in frequency with another harmonic of the other tone, creating an interference pattern within the ear that is audible as beating. Helmholtz' influential theory of music holds that this phenomenon not only explains consonance and dissonance, but also makes the intervals of just intonation sound the most in tune. Several authors (e.g., Vos , Hall and Hess ) have presented a twocomponent model of intonation perception, in which the two components are beat rate and interval size. There is some experimental evidence that beating affects perceived intonation. For example, Vos (1986) found that subjects rated fifths and major thirds as more pure when the beating partials were removed. However, the contribution of beat rate vis-1-vis interval size has not been thoroughly examined, because no previous studies have controlled these two factors independently. With traditional musical instruments, the beat rate can only be changed by changing the interval tuning. With computer-generated sound, on the other hand, one can make the interval tuning independent of the beat rate by introducing another ICMC 202
Page 203 ï~~technique of controlling the beat rate-for example, frequency-shifting the beating partials. With this technique, instead of moving all the partials of a tone in parallel to change the beat rate (as would be the case in a normal instrument), only one partial of each beating pair is moved. The present study used this technique to examine the relative importance of beat rate and interval tuning for perceived intonation. Three psychoacoustic experiments were conducted, using musically experienced subjects and two harmonic intervals: the perfect fifth F4-C5 and the major third F4-A4. The beat rate was controlled by two different methods: (1) the traditional method of simply retuning the interval, and (2) frequency-shifting one partial of each pair of beating partials without changing the overall interval tuning. (The second method does have the side effect of introducing inharmonicity, which is why the first method was included as a check.) In addition, two levels of beat amplitude were introduced by using either a complete spectrum of 16 equal-amplitude partials per note, or a spectrum that minimized beating by omitting one partial from each pair of beating partials. The data are summarized in the graphs below, which require some explanation. "Projected beat rate" refers to the expected beat rate in maximally beating stimuli, or the beat rate that would be expected were the beating partials not removed in the minimally beating stimuli. The physical variable that is varied to generate the different values of projected beat rate is either the ratio of fundamental frequencies (in the case of the "Retuned Interval" stimuli) or the amount of frequency shift of the inharmonic partials (in the case of the "Shifted Partials" stimuli). The "Retuned Interval" stimuli were chosen such that the interval got progressively smaller than the equal-tempered version as the projected beat rate got higher. (At a projected beat rate of 25 Hz, the fifth was 661.1 cents and the third was 361.9 cents.) For the major thirds, two different tunings were used for the "Shifted Partials" stimuli: equal temperament (400.0 cents) and just intonation (386.3 cents). On the other hand, the "Shifted Partials" fifths were all equal-tempered; just intonation was omitted because the just fifth is so close in size (702.0 cents) to the equal-tempered version (700.0 cents). A number of statistical analyses were performed upon the data, including correlation of responses to experimental variables, consistency of subject responses, and analysis of variance (ANOVA). The numerical results are omitted here, due to space limitations. Some of the major results can be inferred by inspecting the graphs. To summarize the analytical results, all three experiments indicated that beating does not contribute significantly to the percept of "out-of-tuneness" for these stimuli, because it made no difference statistically whether the beat amplitude was maximal or minimal. In contrast, mistuning the interval was highly significant: tunings that were more distant from the equal-tempered version were heard as more out of tune. The major third of just intonation was not preferred to the equal-tempered third. Subjects were less consistent in their judgments of thirds than of fifths. ICMC 203
Page 204 ï~~o Maximum Beat Amp, Retuned Interval o Maximum Beat Amp, Shifted Partials A Minimum Beat Amp, Retuned Interval * Minimum Beat Amp, Shifted Partials 04. o Maximum Beat Amp, Retuned Interval V Max Beat Amp, Shifted Partials (Just) O Max Beat Amp, Shifted Partials (ET) * Minimum Beat Amp, Retuned Interval f Min Beat Amp, Shifted Partials (Just) Min Beat Amp, Shifted Partials (ET) 0 O 4.) C" '0 0 CU 0 Ono 0 WA f/! -b -5 0 5 10 15 20 25 30 Projected Beat Rate (Hz) Judgments of intonation of fifths. Each of the four combinations of "beat amplitude" and "method of controlling beat rate" is fitted with a second-order polynomial regression curve. -15 -10 -5 0 5 10 15 20 25 30 Projected Beat Rate (Hz) Judgments of intonation of major thirds. The various combinations of beat amplitude, method of controlling beat rate, and tuning are fitted with second-order polynomial regression curves. To tell which curve belongs with each symbol, see the points at 25 Hz. Additionally, it appears that inharmonicity can affect the perceived intonation. For the fifths, frequency-shifting the appropriate partials to more inharmonic positions had almost as negative an effect on the perceived intonation as had mistuning the interval. This is illustrated by the steep downward slope of the "Shifted Partials" curves in the graph of the fifths. For thirds, on the other hand, this effect was weaker; the "Shifted Partials" curves are almost flat, whereas the "Retuned Interval" curves exhibit an inverse correlation between projected beat rate and goodness of intonation, as with the fifths. Presumably the inharmonicity was less important for thirds because there were fewer inharmonic partials and they were higher in the harmonic series. Since it is unlikely that beats would be more audible in real musical situations than under these laboratory conditions, the results of these experiments suggest that the perception of intonation in music is dependent on the actual interval tuning rather than the concomitant beat rate. If so, this strengthens the argument for a cultural basis for musical intonation, as opposed to the acoustical basis set forth by Helmholtz and others. The present ICMC 204
Page 205 ï~~results do not disprove Helmholtz' theory that beating partials were responsible for the historical origin of the consonances and dissonances, but they indicate that such factors are not necessarily paramount for contemporary listeners' perception of intonation. Subjects apparently made reference to learned interval sizes instead of relying on acoustic cues such as beating, which suggests that cognitive processes play a crucial role in intonation judgments. One implication of these conclusions is that computer music composers need not be overly concerned with minimizing beats when choosing a tuning system from the infinite possibilities available to them. Some composers have cited acoustical factors like beating as a reason to restrict themselves to just intonation, for example; but if these factors are not necessarily very relevant perceptually, it behooves such composers to do what great composers have always done and follow the dictates of their aural imagination, unfettered by supposedly scientific theoretical systems. References Hall, D. E., and J. T. Hess (1984). Perception of musical interval tuning. Music Perception 2:166-195. Helmholtz, Hermann von (1877). Die Lehre von den Tonempfindungen als physiologische Grundlageftir die Theorie der Musik. 6th ed. (Braunschweig: Vieweg, 1913) translated as On the Sensations of Tone as a Physiological Basis for the Theory of Music by Alexander J. Ellis (1885). Rpt. New York: Dover, 1954. Keislar, Douglas (1991). Psychoacoustic Factors in Musical Intonation: Beats, Interval Tuning, and Inharmonicity. Ph.D. Dissertation, Stanford University. Vos, Joos (1982). The perception of pure and mistuned musical fifths and major thirds: Thresholds for discrimination, beats, and identification. Perception and Psychophysics 32:297-313. Vos, Joos (1986). Purity ratings of tempered fifths and major thirds. Music Perception 3:251-257. ICMC 205