exceptions to the rule

Having finished one book on tuning systems, I am now on Tuning, Timbre, Spectrum, Scale by William Sethares. Overall, it explains how consonance and dissonance work, and how if you’re interested in consonance, you have to match your tuning/scale to your timbre or vice versa.

The quick version:

Most instruments in Western music have a mostly harmonic spectrum, because of the physics. Strings are held in place at both ends and can’t vibrate, so any waves must be exactly that strength length or must divide the length equally (so 1/2, 1/3, 1/4 etc.) Air columns in a wind/brass instrument, organ pipes or the human voice behave similarly. With any acoustic instrument there is generally some additional mechanical noise, and factors like string tension which can warp things a bit, but overall it’s pretty close.

The harmonic spectrum is why small ratios for intervals work so well — going up to 2/1 or 3/2 or 4/3 or 5/4 etc., you’ve got partials that match up with the root’s partials. The higher the ratios, the higher up the spectrum (and thus weaker) those matches are — and more importantly, some of the non-matching partials wind up in a zone where they clash with each other.

But percussive instruments are inherently inharmonic. You can carve a wooden arch in the back of a xylophone bar, shave material off a church bell in strategic locations etc. to try to to reduce the dissonance either by weakening some vibration modes or bending them into better frequencies, but you can’t completely fix it. And of course synths can create inharmonic tones using FM, additive synthesis, or a frequency shifter.

The book points out that these nice small ratios, even octaves, can be dissonant when the timbre is inharmonic. But you can find a tuning/scale that fits the spectrum. This is what Java and Bali do with their gamelan scales, tailored to fit the spectra of the individual instruments.

Conversely, you can find spectra that will complement exotic scales such as 10edo, which is kind of a problem for harmonic instruments.

With the majority of my 18edo-based album already recorded, I wonder if I can find a Tension setting for Odessa that results in consonant neutral thirds. Mostly I’ve just played what felt right and interesting, but I do tend toward minimal harmony and not following the theoretical rules about chord progressions, voice leading etc. anyway.

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