meantone puzzles

Hi folks,

I've been immersed in the world of meantone all
day today, first because I'm trying to understand
what Telemann wrote on microtonality, and secondly
because I wrote a short piece on my broken 19-tone
guitar, and only realized after it was written that
the two lowest strings were out of tune (more than
~60 cents sharp), and I've been going crazy trying to
figure out what I played that sounded so good.

Anyway, been making lots of calculations on various
meantones, and have some puzzling observations that
I'd like to understand better.

The EDOs which approximate basic common meantones
have L/s (Large/small) relationships as follows:

EDO degrees
meantone EDO L s

1/3 19 3 2
1/4 31 5 3
1/5 43 7 4
1/6 55 9 5

In all cases I name the notes according to a
meantone cycle with the "5th" a few cents narrow,
and use L to represent the diatonic whole-tone,
and s for the diatonic semitone.

There are linear progressions in all four of these
columns:

- the denominator of the fraction of a comma increases by 1,
- the number of EDO degrees in the 8ve increases by 12,
- the number of EDO degrees in L increases by 2,
- and the number of EDO degrees in s increases by 1.

I'm sure these relationships have been noticed before, but
I'm rather inexperienced with meantones, most of my research
having been done on JIs. Can anyone explain this?

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> EDO degrees
> meantone EDO L s
>
> 1/3 19 3 2
> 1/4 31 5 3
> 1/5 43 7 4
> 1/6 55 9 5

> I'm sure these relationships have been noticed before, but
> I'm rather inexperienced with meantones, most of my research
> having been done on JIs. Can anyone explain this?

The 7 et has a fifth 9.47 relative cents flat, and the 12 et is 1.96
rc flat. Your sequence of ets is 7+12n, leading to fifths which are
12(9.47+1.96n)/(7+12n) cents flat. Taking the reciprocal, converting
to commas, and expanding as a power series gives
1.32322 + 1.99683 n - 0.412064 n^2 + 0.0850336 n^3 - ... comma
temperament, so it is not really linear, but it is linear to a first
approximation.

The 12 et has L 2 and s 1, and the 7 et has L 1 and s 1. Hence
h7 + n h12 has L 2n+1 and s n+1.

That's both parts of your question.