FW: prime-factoring of meantones

Joe Monzo sent me some lattices and asked me to forward my comments to the
list.

>Comments are appreciated. In particular, are these lattices
>showing anything of value?

Honestly, not that I can see. As you know, I prefer to draw lattices where
the consonant intervals get the shortest distances (possibly in a
multi-dimensional space), regardless of whether the consonances are pure or
tempered. If anything, tempering can give you reasons to add more dimensions
in your lattice, so that scales can fold and twist in order to portray the
vanishing commas of temperament.

>In the sheet in that file named 'comparative', you can also
>see my musings on the usual 2^(x*7/12) 12-EDO cycle of '5ths'.
>I noticed that 2^(x/12) 12-EDO appears to be exactly the same
>as 1/11-comma meantone. Has anyone else ever pointed that out?

Yes -- I even mention that fact in my paper, and the footnote cites Barbour.

>Is there any kind of significance to that? More directly, my
>question is: can you show me mathematically how the 1/11-comma
>formula is the same as 2^(x/12)?

It's only approximately true, and it's due to the fact that the syntonic
comma is approximately 11/12 the size of the Pythagorean comma.

Joe, I won't be able to check my e-mail for a week, so if you need to reach
me use the list (as long as it concerns tuning, of course).

-Paul

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