217-EDO as adaptive-JI (was: 152-EDO as adaptive-JI)

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Thursday, January 31, 2002 2:29 AM
> Subject: Re: [tuning-math] 152-EDO as adaptive-JI
>
>
> So, since 19-EDO is a meantone, the lattice wraps into a cylinder.
>
> And since 1 step of 152-EDO is ~1/3-comma, you only need *3*
> chains of 19-EDO each separated by 1 step of 152-EDO, in order
> to represent the entire infinite JI lattice! And you don't
> get comma problems! *WAY* cool !!!!!!
>
> So the 57-tone subset of 152-EDO which is three 19-EDOs starting
> on the 1st, 2nd, and 3rd 152-EDO degrees is really some sort of
> magical tuning for us adaptive-JI fans!

And so, likewise, 217-EDO can perform this function for 31-EDO.

217 = 31 * 7. So 217-EDO is like 7 bicycle chains of 31-EDO.
1 step of 217-EDO is ~1/4-comma. 1/4-comma meantone, represented
by 31-EDO, bends the JI lattice into a cylinder in which 4 chains
of the meantone can cover the whole lattice.

So the 124-tone subset of 217-EDO which is four 31-EDOs starting
on the first four 217-EDO degrees also covers the whole meantone
lattice.

And of course, since 152- and 217- are both EDOs, they close
the meantone system and allow even further punning/bridging!

The 57-out-of-152 is much simpler, but my guess is that
the 124-out-of-217 would be even more appropriate for the
retuning of "common-practice" repertoire I'm interested in.

This is so cool, I need a coat.

-monz

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

> And so, likewise, 217-EDO can perform this function for 31-EDO.
>
> 217 = 31 * 7. So 217-EDO is like 7 bicycle chains of 31-EDO.
> 1 step of 217-EDO is ~1/4-comma. 1/4-comma meantone, represented
> by 31-EDO, bends the JI lattice into a cylinder in which 4 chains
> of the meantone can cover the whole lattice.
>
> So the 124-tone subset of 217-EDO which is four 31-EDOs starting
> on the first four 217-EDO degrees also covers the whole meantone
> lattice.

Again, your reasoning is slipping up somewhere. Each 31-tET covers
the whole meantone lattice, and the whole JI lattice. But depending
on the complexity of the chords you're trying to tune in JI, you may
need only two, or three, or perhaps five or six of the 31-EDOs.