Magic Miracle Lumma Fokker?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> Looks OK, though I'm not blown away by it . . . looks closely
related
> to the Lumma-Fokker scale . . .

Is that

1-15/14-9/8-7/6-5/4-4/3-45/32-3/2-45/28-5/3-7/4-15/8-(2) ?

I wouldn't call that closely related.

> What 11-limit chords does it have?

We have a few 11-limit intervals:

12/7 385/384 = 5/4 11/8
8/5 385/384 = 7/6 11/8
7/6 = 385/384 16/15 12/11
15/8 = 385/384 12/7 12/11

> In general, though, I think what you're saying is basically that if
> you take a block, and leave two, rather than only one, of the
unison
> vectors untempered, you get a good non-MOS scale.

That's one way of looking at it. Or you could say, one val gives an
et, two vals a linear temperament, and three vals a planar
temperament. Or, combine two good generators and try for synergy.

--- In tuning@y..., genewardsmith@j... wrote:

> > In general, though, I think what you're saying is basically that
if
> > you take a block, and leave two, rather than only one, of the
> unison
> > vectors untempered, you get a good non-MOS scale.
>
> That's one way of looking at it. Or you could say, one val gives an
> et, two vals a linear temperament, and three vals a planar
> temperament.

So it's what I was referring to as hyper-MOS on the tuning-math list
a few months ago. If you look at the scale as a cycle of steps, it's

sMssLssMs

Now if you map these three categories to two families in any
reasonable way, you get an MOS pattern:

s->a
M,L->B
aBaaBaaBa : MOS

s,M->a
L->B
aaaaBaaaa : MOS

The same thing should work if you view the scale as a cycle
of "thirds", "fourths", etc. Does it?

>Or, combine two good generators and try for synergy.

Does doing so, using your "square" approach or anything similar, have
any advantages over the seemingly more generic "ways of looking at
it" described above? Even if it's merely equivalent, it seems quite
novel and you should try to go more into it in a language that non-
mathematicians might be able to understand.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> Now if you map these three categories to two families in any
> reasonable way, you get an MOS pattern:
>
> s->a
> M,L->B
> aBaaBaaBa : MOS
>
The three steps are roughly 16/15, 35/32 and 9/8, and the above sets
both 36/35~1 and 225/224~1, making the 12-et a sensible MOS et; the 9
out of 12 pattern being as above.

> s,M->a
> L->B
> aaaaBaaaa : MOS

This is 49/48~1 and 225/224~1, and now the 19-et seems a good choice;
this is the 10+9 system, with generator 2/19.

Of course we could run the Osmium process farther and then make
identifications leading to a MOS.

> Does doing so, using your "square" approach or anything similar,
have
> any advantages over the seemingly more generic "ways of looking at
> it" described above? Even if it's merely equivalent, it seems quite
> novel and you should try to go more into it in a language that non-
> mathematicians might be able to understand.

It seems to be equivalent.