Yahoo Tuning Groups Ultimate Backup tuning Pele and Parapyth in the gentle region

> If we assume 2 and 13/9 are eigenmonzos, one more eigenmonzo gives
> a tuning for the fifth, which varies in a linear fashion, as if we
> were looking at a linear temperament. For the extra eigenmonzos for
> pele/parapyth in the gentle region, we have this:

Dear Gene,

There's a humorous remark about the USA and UK being two peoples
divided by a common language, and here I suspect that ratios may
be such a common language for the two of us <grin>.

But actually we may agree on lots of the reference points you
give below, with a few comments from me maybe helpful in
clarifying a few points about my own outlook on optimizations,
which is rather definitely non-POTE. Understanding that each
viewpoint is valid, and that they sometimes coincide and
sometimes differ in their conclusions, is very important on all
sides.

For this gentle region, I do think that the terms "pele-compatible"
and "leapday-compatible" can do wonders. They are culturally
neutral, and describe what is there while letting the designer
of a tuning system define what the musical ethos or intent is.
Of course, if a designer chooses or accepts the pele or leapday
label, then that resolves the question of intent.

And I'd say that the GNU Public License or equivalent pretty much
implicitly applies: "Margo designed MET-24 for things other than
intentional 5-limit thirds, but I love the pele-compatiblity
so much that I've gone and expanded it to a MET-34 which I'm
using for unabashed pele!"

> 11/10: 703.500

Indeed, and this interval depends only on the size of the linear
generator, with 10 generators down producing 11/10.

Note that MET-24 is tempered about 0.21 cents higher, where 11/10
is about 2.1 cents narrow -- roughly the same amount as 14/13.

> 10/9, 13/10: 703.522

Again, MET-24 is a bit high for these, and has the real
limitation that 13/10 or the like doesn't come up, with 352/273
as the closest approach (not very close!).

> 8/7: 703.579

This is George Secor's generator, and one of his eigenmonzos, in
the HTT family. So it looks like we're agreed!

> 11/9, 13/11: 703.597

Yes, this is the tiny difference between Secor's (504/13)^(1/9) and
(44/13)^(1/3). In George's view, a very minute detuning of 13/11
and 11/9 might actually be a virtue.

> 7/6: 703.782

This one is interesting. I must admit that my immediate reaction
was: "Isn't this a bit low to use a pure 7/6 rather than 7/4 as
the basis for the spacing?" It's a matter of taste, but
explaining my reaction might help for you or others trying to
understand my optimization philosophy or mindset.

The complication for me is the effect that such a large spacing
has on things like 13/12, 12/11, and especially 11/8, 13/8, and
21/16, by comparison with MET-24. That isn't saying by any means
that this optimization is wrong, only that it is a somewhat
different shading than MET-24. And if we're aiming for or
especially welcome a 64/59, a la Safi al-Din al-Urmawi, this
might be just about ideal. It's actually hard to go wrong in
this part of the spectrum -- as might also be said for just
about anywhere else!

But on the compromises, one good test is to look at 4:6:7:9:11:13,
where 11/8 at 4.207 cents wide does stand out a bit.

> 9/7, 14/13: 704.043

Certainly we both see 14/13 as a vitally important point, and
here I would make 7/6 an eigenmonzo, where the consequences for
13/8 are more favorable than at 703.782 cents, because the
regular minor sixth is more like an 11/7. Around MET-24 or
a bit above, it's actually a slightly better 52/33, with
not so much of a distance to 13/8; so going for a just 7/4 treats
13/8 with a softer touch.

> 14/11: 704.377

Naturally: this stands out just like 1/4-comma meantone. And
George Secor used it in 1978 for the remote eight fifths of his
17-tone well-temperament; it wasn't until 2000 that I got into
the act, and then learned the next year that his 17-WT
definitely had precedence.

Best,

Margo

Pele and Parapyth in the gentle region

🔗genewardsmith <genewardsmith@...>

🔗Margo Schulter <mschulter@...>