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.