ASA 31 - Tones And Tuning With Stretched Partials

Description

Most tonal musical instruments used in Western culture have spectra that are exactly or nearly harmonic. Tone scales used in Western music are also based on intervals or frequency ratios of simple integers, such as the octave (2:1), the fifth (3:2), and the major third (5:4). When several instruments play a consonant chord such as a major triad (do-mi-so), harmonics will match so that no beats will occur. If one changes the melodic scale to, for instance, a scale of equal temperament with 13 tones in an octave, a normally consonant triad may cause an unpleasant sensation of beats because some harmonics will have nearly but not exactly the same frequency. Similarly, instruments with a nonharmonic overtone structure, such as conventional carillon bells, can create unpleasant beat sensations even if their pitches are tuned to a natural scale. Beats will not occur, however, if the melodic scale of an instrument's tones matches the overtone structure of those tones.

First, a four-part chorale ("Als der gutige Gott") by J.S. Bach is played on a synthesized piano-like instrument whose tones have s exactly harmonic partials with amplitudes inversely proportional to harmonic number, and with exponential time decay. The melodic scale used is equally tempered, with semitone frequency ratios of the 12th root of 2.

In the second example, the same piece is played with equally stretched harmonic as well as melodic frequency ratios. The harmonics of each tone have been uniformly stretched on a log-frequency scale such that the second harmonic is 2.1 times the fundamental frequency, the 4th harmonic 4.41 times the fundamental, etc. The melodic scale is similarly tuned in such a way that each "semitone" step represents a frequency ratio of the 12th root of 2.1. The music is, in a sense, not dissonant because no beats occur. Nevertheless the harmonies may sound less consonant than they did in the first demonstration. This suggests that the presence or absence of beats is not the only criterion for consonance. The listener may also find it difficult to tell how many voices or parts the chorale has, since notes seem to have lost their gestalt due to the inharmonicity of their partials.

In the third example, the tones are made exactly harmonic again, but the melodic scale remains stretched to an "octave" ratio of 2.1. Disturbing beats are heard, but the four voices have regained their gestalt. The piece sounds as if it is played on an out-of-tune instrument.

In the final example, the harmonics of all tones are stretched, as was done in example 2, but the melodic scale is one of equal temperament based on an octave ratio of 2.0. Again there are annoying beats. This time, however, it is again very difficult to hear how many voices the chorale has.

References

E.Cohen (1984), "Some effects of inharmonic partials on interval perception," Music Perc. 1, 323-349.

H.F.I. von Helmholtz (1877), On the Sensations of Tone, 4th ed., Transl. A.J.Ellis, (Dover, New York, 1954).

M.V.Mathews and J.R.Pierce (1980), "Harmony and nonharmonic partials," J. Acoust. Soc. Am. 68, 1252-1257.

F.H.Slaymaker (1970), "Chords from tones having stretched partials," J. Acoust. Soc. Am. 47, 1569-1571.

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Transcript

You will hear a 4-part Bach chorale played with tones having 9 harmonic partials.

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Now the same piece is played with both melodic and harmonic scales stretched logarithmically in such a way that the octave ratio is 2.1 to l.

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In the next presentation you hear the same piece with only the melodic scale stretched.

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In the final presentation only the partials of each voice are stretched.