source file: mills2.txt Date: Tue, 3 Oct 1995 08:22:15 -0700 From: "John H. Chalmers" From: mclaren Subject: Tuning & psychoacoustics - post 10 of 25 --- So far we have seen the complexity of the ear's response to pitch; not a quantity linearly proportional to the logarithm of frequency, pitch appears to be influenced by many aspects of timbre--amplitude, masking tones, overtone harmonicity, and range of the note played. The ear's perception of complex sounds is equally complicated. "Both von Helmholtz and Wundt based the development of harmony and melody on the coinciding harmonics for consonant intervals." [Plomp, R. and Levelt, W.J.M., "Tonal Consonance and Critical Bandwidth," Journ. Acoust. Soc. Am., Vol. 6, No.1, April 1965, pg. 549] Simple experiments, conducted in the 1960s, showed this not to be the case. "On the basis of more recent and more sophisticated experiments (Plomp and Levelt, 1965) on consonance judgment involving pairs of pure tones and inharmonic complex tones, it became apparent that the beats between harmonics may not be the major detemrining factor in the perception of consonance. Two pure tones an octave or less apart were presented to a number of musically naive (untrained) subjects who were supposed to give a qualification as to the "consonance" or "pleasantness" of the superposition. A *continuous pattern* was obtained, that did not reveal preferences for any partuclar musical interval. Whenever pure tones are less than about a minor third apart, they were judged "dissonant" (except for the unison); intervals equal orlarger than minor third were judged as more or less consonant, irrespective of the actual frequency ratios. The shape of the curve really depends on the absolute frequency of the fixed tone." [Roederer, J., "The Physics and Psychophysics of Music," 1973, Pg. 142] Plomp and Levelt called this interval, somewhat smaller than a minor third, "the critical bandwidth." It changes size slightly at lower frequencies. "...maximal tonal dissonance is produced by intervals subtending 25% of the critical bandwidth, and maximal tonal consonance is reached for interval width of 100% of the critical bandwidth." [Plomp, R. and Levelt, W.J.M., "Tonal Consonance and Critical Bandwidth," Journ. Acoust. Soc. Am., Vol. 6, No.1, April 1965, pg. 549] "An interesting consequence of the significance of the critical bandwidth is that the degree of dissonance of a dyad depends on many factors other than the frequency ratio. If the tones have many strong overtones, the consonance quality is reduced. A consonant dyad becomes increasingly dissoannt when it is transposed downward on the frequency scale, just as for sine tones. For example, a major third sounds reasonably consonant around A4, but if played close to C2 it sounds quite dissonant on most instruments. The relation between consonance and frequency ratios is also entirely dependent on whether the tones have harmonic spectra. "Consonance is apparently a highly conditioned phenomenon. It is stimulating to realize that the dissonance/consonance concept in music theory would have been entire different if our musical instruments had not provided us with harmonic spectra!" [Sundberg, J., "The Science of Musical Sounds,"1992, pg. 85] This finding lends support to all three tunings (just, equal tempered, non- just non-equal) provided that the partials of the musical notes are changed so as to fit the tuning. As mentioned above, Plomp's and Levelt's findings (extended and refined by Kameoka and Kuriyagawa's formula for calculating the consonance of complex tones) also casts doubt on many conventional "rules" of harmony and melody--even if just intervals and perfectly harmonic overtones are used: "As an application of the consonance theory, effects of harmonic structure on the consonance characterisc are discussed. (...) (...) It became clear that the fifth was not always a consonant interval. A chord of two tones that consists of only odd harmonics, for example, shows muchworse cosnonance at the fifth (2:3) than at the major sixth (3:5) or some other frequency ratios. This was proved true by psychological experiments carried out in another institute (sensory Inspection Committe in the Japan Union of Scientists and Engineers) with a different method of scaling. Thus, the fact warns against making a mistake in applying the conventional theory of harmony to synthetic musical tones that can take variety in the harmonic structure." [Kameoka, A., and Kurigawa, M., "Consonance Theory Part II: Consonance of Complex Tones and Its Calculation Method," Journ. Aoucst. Soc. Am., Vol. 45, No. 6, 1969, pg. 1460] Because consonance and dissonance depend not on the harmonicity of two complex tones, but on the coincidence (or lack thereof) of their component partials within the critical bandwidth for that frequency range, "...these examples refer to a vast domain opened up by digital synthesis, namely that of inharmonic tones. Most sustained instrumental tones are equasi- periodic, and their frequency components are harmonically related, which stresses certain intervals like the octave and the fifth. With the freedom of constructing tones from arbtirary frequency components, one can break the relationship between consonance-dissonance aspects and fixed, privileged intervals (Pierce 1966). In his piece Stria (1977), Chowning has thus been able to make rich textures permeate each other without dissonance or roughness, by controlling the frequencies constituting these textures. This is also a case where spectra not only play a coloristic role (see Roads 1985) but actually peform a quasi-harmonic function." [Risset, J.C., "Digital Techniques and Sound Structure in Music," in "The Music Machine," ed. Curtis Roads, 1985, pg. 122] "By using a digital computer, musical tones with an arbitrary distribution of partials can be generated. Experience shows that, in accord with Plomp's and Levelt's experiments with pairs of sinusoidal tones, when no two successive partials are too close toegher such tones are consonant rather than dissonant, even though the partials are harmonics of the fundamental. For such tones, the conditions for consnance of two tones will not in general be the traditional ratios of the frequencies of the fundamentals. (...) It appears that, by providing music with tones that have accurately specified but nonharmonic partial structures, the digital computer can release music from the tyranny of 12 tones without throwing consonance overbarod." [Pierce, J.R., Journ. Acoust. Soc. Am., Vol. 6, No. 12, 1966, pg. 249] "I suggest that the nonharmonic domain of frequency relationships may in some way contain a necessary system of hierarchical structural functions." [Dashow, J., "Spectra As Chords," Computer Music Journal, 1980]" "The hypothesis has been made that perceived effects similar to the consonance and dissonance experienced with harmonic tones should exist for inharmonic tones. Clearly, it cannot be claimed that the perceptions are exactly the same, since inharmonic and harmonic tones themselves sound different to the ear. However, the experiments do establish a similarity between the consonance dissonance phenomenon in harmonic and inharmonic sounds." [Geary, J.M., "Consonance and Dissonance of Pairs of Inharmonic Tones," J.Acoust. Soc. Am, 67 (5), May 1980] "The chords sounded smooth and nondissonant but strange and somewhat eerie. The effect was so different from the tempered scale that there was no tendency to judge in-tuneness or out-of-tuneness. It seemed like a peek into a new and unfamiliar musical world, in which none of the old rules applied, and the new ones, if any, were yet undiscovered." [Slaymaker, F. H, "Chords From Tones Having Stretched Partials," J. Acoust. Soc. Am., Vol. 47, pp. 1469-1471, 1970] "We have to compose real music of many kinds within all and any of our new tuning schemes, if this work is to have any lasting value at all, or be taken seriously by the music community..."[Carlos, W., "Tuning: At the Crossroads," Computer Music Journal, 1987] In short, "Experiments with inharmonic partials (Slaymaker, 1970; Pierce, 1966) have shown that consonance or dissonace is indeed dependent on the coincidence of partials and not necessarily on the simple frequency ratio between the fundamnetal frequencies..." [Rasch, R.A. and Plomp, R., "The Perception of Musical Tones," in "The Psychology of Music," ed. Diana Deutsch, 1982, pg. 21]. Thus all three tuning systems appear equally viable on the basis of the evidence considered in this post, given a digital or acoustic instrument whose partials are matched to the tuning system in question. The next post will discuss the important phenomenon of categorical perception, and its implications for tuning and music. --mclaren Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 3 Oct 1995 19:30 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id KAA04880; Tue, 3 Oct 1995 10:30:18 -0700 Date: Tue, 3 Oct 1995 10:30:18 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu