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While looking for 31TET music I stumbled across this recording of "flow my tears" my John Dowland

The video description and the album artwork indicate that it's being played on the fokker organ, a 31TET instrument. However the uploader made a comment that it's actually playing in quarter comma meantone.

My ears lead me to think it's an unequal temperament (i.e. meantone), but I'm not at all confident.

Some_Guy
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Since 31 is a prime number, selecting either 7 or 12 pitches from 31 to give a diatonic or chromatic scale will inevitably produce an unequal temperament.

Selecting the appropriate pitches from 31TET gives a very good approximation to quarter-comma meantone. The major thirds are only 0.8 cents different. In a pipe organ, there is often acoustic coupling between notes that are playing simultaneously, and that can be enough to "bend" the pitches and give a just intonation (equal to a 1/4 comma meantone) major third. There is no coupling in an electronic simulation, of course, unless you explicitly program it into the simulator.

The difference between minor thirds in the two temperaments is about 0.6 cents, though both these thirds are 5 or 6 cents different from a just intonation minor third.

An intuitive explanation of why this "works out right" is because 31TET divides a (diatonic) whole tone into five parts, not two as in 12TET or four if you use quarter-tones. Splitting the five parts into 2+3 gives a pretty good approximation to the two different sizes of semitones in meantone temperament.

See https://en.wikipedia.org/wiki/31_equal_temperament, and the references on that web page.

  • Do you have any reference to the pitch coupling in pipe organs, or other instruments? this is new (and interesting) to me – Dave Oct 15 '17 at 19:45
  • I can't think of a good reference off hand, but it's a well known practical problem for pipe organ tuning. Some types of pipe are more sensitive than others. The physical layout of the ranks of pipes on the windchest is also relevant. Most of the sound from a pipe actually comes from the pipe mouth where it is blown from, not the open end, and if one pipe is directly facing another, the sound from one can disturb the wind being blown into the other one. –  Oct 15 '17 at 23:24
  • Thanks for such a great explanation. That is such a close approximation as to be practically indistinguishable! So really 31TET can be used to essentially construct a meantone in a way that "squares the circle" i.e. allows us to eventually end up back home after 31 fifths (or any interval). And if you stop the circle of fifths early, to only produce 12 notes, then the "wolf fifth" is exactly 1/31 out . I was originally interested in 31TET because of the possibility of an equal tempered scale including microtonal intervals for interesting scales, but this is a real surprise!! – Some_Guy Oct 16 '17 at 01:48
  • @Some_Guy to put 0.8 cents in perspective, C♯4 is 275 Hz in 5-limit just intonation, assuming A3=220 Hz, whereas in 31 EDO it is 275.1244 Hz. If you play these frequencies simultaneously, the beats should have a frequency of about 1/8 per second, which is about 7.5 beats _per minute._ It would not be detectable except in very long notes. But note that 11 of your fifths are 3 cents narrower than equal, so 5 cents narrower than pure, and the 12th is about 35 cents wider than equal. But you also have 8 major thirds that are 10/31 of an octave (387.1 cents) and four that are 11/31 (425.8 cents). – phoog Nov 28 '21 at 19:58