Talk:Golden meantone

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131edo... and other notes

I am a little confused at how 131edo is "meantone." 77\131 seems to be the representation of the fifth, which is 1.3 cents sharp, so it doesn't temper out the syntonic comma. If you use 76\131 instead, the fifth is noticeably more out, but it sort of works. I think Kornerup also acknowledged the discrepancy in his book, if I'm not mistaken (I do not own a copy of his book).

The other note, is that I noticed an arithmetic series, as I'm sure almost every editor here has before, of (most of) the lower-numbered meantone edo tunings: 7, 12, 19, 31, 50, 81 - but, I don't see it mentioned how the fifth is represented in each of these systems also follows an arithmetic series (I'll get to why this is interesting to me in a minute): 4\7, 7\12, 11\19, 18\31, 29\50, 47\81.

The series is a1 = 4, a2 = 7, an = an-1 + an-2 ... 4, 7, 4+7 = 11, 7+11 = 18, 11+18 = 29, 18+29 = 47.

Going further through the series, after 131edo (50+81 = 131), there's 212edo, 343edo, 555edo, 898edo, etc., and most of these are not meantone the way I understand meantone (but I don't understand a lot of things). Representing the fifth in each of these, though...

29+47 = 76 (this is the more sour fifth in 131edo that makes a meantone tuning, at least the way I understand the terminology)

47+76 = 123, but the fifth in 212edo is represented by 124\212. So as disappointing as it is, I believe that the pattern of series breaks down beyond 81edo.

Beyond that, just for fun... 76+123 = 199, but 76+124 = 200. 201\343 is the better fifth, but 200\343 fit meantone better.

--Bozu (talk) 19:29, 30 January 2020 (UTC)

With 131edo, you can sort of squint at it just right and say that you get better overall consistency of harmonics 3 and 5 by using the second-best (flat) fifth, and you get some added utility by being able to support a 17L 2s scale that splits the 3/1 into 3 equal parts, and of having the diatonic and chromatic semitones in very close to the right ratio to each other. But yes, after a certain point, meantone (even golden) stops being the best use of the tonal resources you get from going to the trouble to support all those notes.

Unless given some reason as above to go further, I would normally say to stop at 50edo, which has amazing consistency up to high limits — just 1 pair inconsistent intervals in the 19-odd-limit — and no larger meantone maps both 9/8 and 10/9 consistently. And it sounds good, too.

Added: Lucius Chiaraviglio (talk) 08:02, 7 January 2026 (UTC)

Last modified: Lucius Chiaraviglio (talk) 08:13, 7 January 2026 (UTC)

OEIS

The OEIS sequence A001060 seems to represent the sequence of edos that approximate golden meantone better and better. Can this be added in? hotcrystal0 16:55, 6 January 2026 (UTC)