Mixed timbre

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A mixed timbre is a type of musical timbre that is intermediate between a perfectly harmonic (mode-locked) timbre and an inharmonic one. These types of timbres introduce a slight amount of inharmonicity, with partials being bent up or down by a few cents, while also maintaining the same "clear" quality that distinguishes harmonic timbres from the more "opaque" or bell-like inharmonic ones. The term mixed timbre was coined by Mason Green, although the concept may have been proposed before.

Mixed timbres are ideal for use with tunings that almost, but not quite, provide satisfactory approximations for certain intervals. For example, 22edo is a fairly good 11-limit system, and it also allows for a great deal of new modal and tonal possibilities, which have been written about by Paul Erlich among others. Some of these scales sound much better if the 600-cent tritone is treated as a consonance rather than a dissonance. Similarly, it might be a good idea to treat the tempered whole tone (~218 cents) as a consonance as well, since it occurs in inversions of the harmonic seventh chord.

However, with most harmonic timbres, neither of these intervals sounds particularly consonant. The tritone is too far apart from both 7:5 and 10:7 to sound like either. By introducing some inharmonicity, though, it is possible to make these intervals sound much sweeter, as consonances should. That way, 22edo will not sound nearly as "out of tune" as it otherwise would.

Mixed timbres are one way to do this. A mixed timbre contains mixed partials, which are partials consisting of two or more closely-spaced frequency components (they must be close enough that they are perceived as one "beating" partial rather than separate entities). One of the components is a perfect harmonic multiple of the fundamental, and the other(s) are slightly higher or lower.

For example, we could imagine a mixed partial consisting of one component at 3f, where f is the fundamental frequency, and a second component with half the amplitude at 3.03f. Because of the harmonic component, this partial maintains some semblance of (imperfect) mode-locking with the fundamental, which results in a more stable timbre. Interference between the two components causes this partial to "beat", but beats on overtones are generally much less noticeable than beats on the fundamental, and because the inharmonic component is smaller to begin with, the beating will not be very noticeable and the timbre will sound very close to a perfectly harmonic one.

This mixed overtone will have an effective pitch which is a weighted average of its two components; in this case, about 5-6 cents above 3f. Thus, we can say that the pitch of this overtone has been "bent up" through the addition of the inharmonic component. Assuming octave equivalence and that the 2nd, 4th, 8th, etc. are all perfectly harmonic, it might be inferred that the ideal width of a fifth, when using this timbre, will be somewhere around ~707 cents. This may make a huge difference if we are using a tuning where the fifths are already very sharp to begin with (for example, 22edo or 27edo).

If other non-octave-equivalent timbres are similarly modified, it will be possible to construct a timbre around a particular temperament. Using 22edo, we should bend the overtones corresponding to the 3rd, 7th and 9th harmonics up, and those corresponding to the 5th and 11th harmonics down. We can also omit the 13th harmonic altogether (since 22edo does not deal with this harmonic well). By adapting the timbre to the temperament in this way, the tritone will sound much more consonant and will function more credibly as both 7:5 and 10:7; other intervals including the whole tone, thirds, and fifths will also sound much better.

Another tuning that would benefit a great deal from this is 17edo, which is superb for melody (as documented by George Secor), but whose harmonies beyond the 3-limit are rather lacking. By bending the 7th, 11th, and 13th harmonic partials down, while bending the 5th harmonic partial way up (assuming the sharp val for 5), we can compensate for 17edo's deficiencies and effectively convert it into a full 13-limit system, while still using timbres that don't sound too inharmonic. Using the flat val for 5 will result in a different timbre with a more xenharmonic effect, since it maps both the "5:4" and "5:6" onto a neutral third.

Timbres like this are easy to synthesize on a computer, but it may also be possible to achieve this effect in physical instruments by coupling a harmonic or nearly harmonic oscillator (such as a bowed string), to a more inharmonic one such as an idiophone. String resonance (like that which naturally occurs in a piano when the damper pedal is pressed) is another option.

In fact, it's quite likely that many if not most of the "perfectly harmonic" timbres in nature, including the human voice, are in fact mixed timbres. Few if any physical objects are completely free of inharmonicity, and mode-locking is not always perfect (injection pulling can occur as often as injection locking). Because of this, a mixed timbre may in fact sound more organic, or less artificial, than a perfectly harmonic one.

It's not just xenharmonic tunings that can benefit from using mixed timbres; even plain old 12edo can, as well (if you use a timbre with the 5th and 7th partials bent up).