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Comment by worik | Hacker News Reader

I'm interested in the details of 31 divisions of the octave

Presumably it is an equal ratio like a piano keyboard, but with the 31st root of 2 as the ratio between successive frequencies instead of the 12th root of 2. Simple frequency ratios like 3/2 tend to sound good (this is commonly attributed to Pythagoras), so tunings are chosen to include them or close approximations.

The relative frequencies for a piano octave are:

  >>> (2.**(1/12))**np.linspace(0., 12., 13)
  array([1.        , 1.05946309, 1.12246205, 1.18920712, 1.25992105,
         1.33483985, 1.41421356, 1.49830708, 1.58740105, 1.68179283,
         1.78179744, 1.88774863, 2.        ])

Notice how there is one very close to 1.5 = 3/2, one very close to 1.3333... = 4/3, and one sort of close to 1.25 = 5/4. These intervals are the 5th, 4th, and major 3rd, and sound good. That's the main reason we use 12 semitones per octave.

Relative frequencies for 31 divisions are:

  >>> (2.**(1/31))**np.linspace(0., 31., 32)
  array([1.        , 1.02261144, 1.04573415, 1.0693797 , 1.09355991,
         1.11828687, 1.14357294, 1.16943077, 1.19587327, 1.22291369,
         1.25056552, 1.2788426 , 1.30775907, 1.33732938, 1.36756832,
         1.398491  , 1.43011289, 1.46244979, 1.49551788, 1.52933369,
         1.56391412, 1.59927646, 1.6354384 , 1.67241801, 1.71023378,
         1.74890462, 1.78844987, 1.82888929, 1.8702431 , 1.91253198,
         1.95577707, 2.        ])

There is still one very close to 1.5 = 3/2, one kind of close to 1.3333... = 4/3, one close to 1.25 = 5/4, and additionally one close to 1.2 = 6/5, one pretty close to 1.1666... = 7/6, one close to 1.1428... = 8/7, and one close to 1.111... = 10/9. More consonant intervals are possible with this keyboard than with standard notes, but they will sound strange and unfamiliar.

More info here: https://en.wikipedia.org/wiki/31_equal_temperament and https://en.wikipedia.org/wiki/Regular_temperament

Personally, I don't find 31-TET compelling since it gives worse ratios for perfect fifths 3/2 (1.495 instead of 1.498) and thus perfect fourths 4/3 (1.337 instead of 1.335).

Anyway, back to 12-TET. Here's a useful formatting:

    1.00 C     = 1/1 unison, [diminished second]
    1.06 C#/Db       (augmented unison), minor second
    1.12 D     ~ 9/8 major second, [diminished third]
    1.19 D#/Eb ~ 6/5 (augmented second), minor third
    1.26 E     ~ 5/4 major third, [diminished fourth]
    1.33 F     ~ 4/3 perfect fourth, [augmented third]
    1.41 F#/Gb !!!!! satanic tritone (augmented fourth, diminished fifth)
    1.50 G     ~ 3/2 perfect fifth, [diminished sixth]
    1.59 G#/Ab ~ 8/5 (augmented fifth), minor sixth
    1.69 A     ~ 5/3 major sixth, [diminished seventh]
    1.78 A#/Bb ~16/9 (augmented sixth), minor seventh
    1.89 B           major seventh, [diminished octave]
    2.00 C     = 2/1 octave, [augmented seventh]

Note that intervals are named based on the natural version (i.e. considering only letter half), then modified for the accidentals (sharps and flats). The names in (parentheses) are somewhat less likely to be used of each pair (That is, C to D# is an augmented second, but C to Eb is the minor third and much more common). The names in [brackets] would also refer to the same interval ratio but require either a double accidental on one end, or a sharp and flat in different directions, neither of which I wrote out here. "Doubly/triply/quadruply diminished/augmented" are also possible prefixes but increasingly rare; I don't think further is possible when you're limited to double accidentals.

Yeah, 31 is pretty good but the fifths being about 5 cents off kind of throws everything a little out of tune. 5/4 being less than a cent off is great though.

If I had to choose an EDO I think it's hard to go wrong with 41. 3/2, 5/4, 6/5, and 7/4 are all closer to just intonation in 41 EDO than 12 EDO.

(I've been casually involved with a local group[1] that's trying to popularize a form of 41-EDO guitar that only has every other fret and uses an odd interval tuning so notes not on one string will be on the next. It works surprisingly well, due to some convenient mathematical coincidences that place all the notes you're likely to commonly use right next to each other.)

[1] https://kiteguitar.com/

Here's Mike Battaglia playing a cover of "House of the Rising Sun" on a 31 EDO synth setup which I always loved: https://www.youtube.com/watch?v=IlZv13YZzSM

For a really deep dive into these sorts of tuning systems: https://en.xen.wiki

For examples of people playing them, search for Lumatone on YouTube, or “microtonal”, or “xenharmonic”.

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