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)