more on the relationship between maximal evenness and consonance

Joseph (and Dave),

First of all the ME 19-of-72 scale can have its lattice extended thus:

\ / \ / \
\ / \ / \
49--------19--------61
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
68--------38---------8
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
45--------15--------57--------27
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
64--------34---------4
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
11--------53--------23
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
30---------0--------42
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
49--------19--------61
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
68--------38---------8
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
45--------15--------57--------27
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
64--------34---------4
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
11--------53--------23
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
30---------0--------42
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
49--------19--------61
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \

And Dave's maximally even 31-out-of-72-tET scale has got to have more
consonant intervals (7-limit or 11-limit) than any other 31-tone
subset of 72-tET.

So sometimes maximal evenness and maximizing number of consonances go
hand-in-hand.

But it seems the maximally even m-out-of-n scale will only be
maximally consonant if both m-tET and (n-m)-tET are really good
approximations to the consonances in question.

For example, above we see that for 19-out-of-72, the ME scale seems
to be maximally consonant in the 5-limit and 19-tET and (72 - 19 =)
53-tET are both really good 5-limit scales.

For 31-out-of-72, the ME scale seems to be maximally consonant in the
7- or 11-limit, and 31-tET and 41-tET are both really good 7- and 11-
limit scales.

Hmm . . .

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_21880.html#21880

Well... this is quite some diagram... It almost looks like one of
those e-mails where you have to scoll down to the bottom to get
the "punch line..."

> And Dave's maximally even 31-out-of-72-tET scale has got to have
more consonant intervals (7-limit or 11-limit) than any other 31-tone
> subset of 72-tET.
>

This is the one with the strange accumulation of what look like 3-
dimensional hexanies, yes?

> So sometimes maximal evenness and maximizing number of consonances
go hand-in-hand.
>

Well, I've just noticed this... I may go "retro" for a moment and
return to some thinking about 19-tET shortly...

_______ ______ _____ ____
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> > And Dave's maximally even 31-out-of-72-tET scale has got to have
> more consonant intervals (7-limit or 11-limit) than any other 31-
tone
> > subset of 72-tET.
> >
>
> This is the one with the strange accumulation of what look like 3-
> dimensional hexanies, yes?
>
Actually, I don't think anyone's tried to draw a lattice of it yet,
but this MIRACLE scale would actually have SIXTEEN 1-3-5-7 hexanies!

And please don't go to 19-tET . . . it's not all that just-sounding
in the 5-limit, and positively foul in the 7-limit.

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_21880.html#21897

> >
> Actually, I don't think anyone's tried to draw a lattice of it yet,
> but this MIRACLE scale would actually have SIXTEEN 1-3-5-7 hexanies!
>

WELL, this is an imperative! And it should be in "groovy color" too!

If it's really a MIRACLE SCALE, we have to treat it with respect!

> And please don't go to 19-tET . . . it's not all that just-sounding
> in the 5-limit, and positively foul in the 7-limit.

I wouldn't do anything that "rash," but I do have a couple of
questions which will be posted shortly...

_________ ______ _____ _____
Joseph Pehrson