Does Miracle--11 count?

If the 11-note miracle scale counts in the quest for interesting 11-
note 7-limit scales, there are things like it. 1/11 is a
semiconvergent to 7/72, and similarly 3/11 is a semiconvergent to
15/58, for instance. We get 2,2,11,2,2,11,2,2,11,2,11 as the step
pattern.

Of course, for Paul's real quest, which is for interesting 7-limit
scales, we can go to more steps; 7/27 is the penultimate convergent
to 15/58, and gives us a 6*2, 3, 6*2, 3, 6*2, 3, 5*2, 3 pattern. The
point of 15/58, (or 17/46, or 24/65 or 56/135 in 2^(1/2), etc.) is
that it has a relatively low 7-limit complexity.

--- In tuning-math@y..., genewardsmith@j... wrote:
> If the 11-note miracle scale counts in the quest for interesting 11-
> note 7-limit scales,

Not really . . . there's not even one tetrad.

> Of course, for Paul's real quest, which is for interesting 7-limit
> scales, we can go to more steps; 7/27 is the penultimate convergent
> to 15/58, and gives us a 6*2, 3, 6*2, 3, 6*2, 3, 5*2, 3 pattern.
> The point of 15/58, (or 17/46,

This fraction doesn't seem to agree with the others.

> or 24/65 or 56/135 in 2^(1/2), etc.) is
> that it has a relatively low 7-limit complexity.

What is the complexity, and how low can we get the errors? Or did
this come up already?

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

> > Of course, for Paul's real quest, which is for interesting 7-
limit
> > scales, we can go to more steps; 7/27 is the penultimate
convergent
> > to 15/58, and gives us a 6*2, 3, 6*2, 3, 6*2, 3, 5*2, 3 pattern.
> > The point of 15/58, (or 17/46,

> This fraction doesn't seem to agree with the others.

17/46 is a completely different generator than 15/58, if that's
what's worrying you.

> > or 24/65 or 56/135 in 2^(1/2), etc.) is
> > that it has a relatively low 7-limit complexity.

> What is the complexity, and how low can we get the errors? Or did
> this come up already?

Graham is presumably the one to ask about what has come up
previously, but for the rest of it, we have:

3 10 1.49
5 9 6.79
7 7 3.59
5/3 -1 5.30
7/3 -3 2.09
7/5 -2 -3.20

The second column is the generator coordinate, and the third is
sharpness in cents. We have a complexity of 13 and a maximum error of
6.79 cents; of course that can be reduced if we go to a linear
temperament. A scale of 27 out of 58 would have a lot of complete
tetrads, and if you go out farther you find

11 25 7.30
13 -5 7.75

to add to the fun.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > > Of course, for Paul's real quest, which is for interesting 7-
> limit
> > > scales, we can go to more steps; 7/27 is the penultimate
> convergent
> > > to 15/58, and gives us a 6*2, 3, 6*2, 3, 6*2, 3, 5*2, 3 pattern.
> > > The point of 15/58, (or 17/46,
>
> > This fraction doesn't seem to agree with the others.
>
> 17/46 is a completely different generator than 15/58, if that's
> what's worrying you.
>
> > > or 24/65 or 56/135 in 2^(1/2), etc.) is
> > > that it has a relatively low 7-limit complexity.
>
> > What is the complexity, and how low can we get the errors? Or did
> > this come up already?
>
> Graham is presumably the one to ask about what has come up
> previously, but for the rest of it, we have:
>
> 3 10 1.49
> 5 9 6.79
> 7 7 3.59
> 5/3 -1 5.30
> 7/3 -3 2.09
> 7/5 -2 -3.20

OK -- you're talking about the seventh temperament in

http://x31eq.com/limit7.txt

i.e., "kleismic" (since the 5-limit unison vector that implies this
generator is the kleisma).

Gene -- the complexity is actually only 10, not 13 -- 11 notes in the
chain are enough to give a complete tetrad.

> A scale of 27 out of 58 would have a lot of complete
> tetrads,

Not significantly worse than 27 out of 31. Also, 11-, 15-, 19-, and
23-tone MOSs give tetrads.

>and if you go out farther you find
>
> 11 25 7.30

Notice the seventh temperament in
http://x31eq.com/limit11.txt

> 13 -5 7.75

Curiously, no kleismic temperament appears in

http://x31eq.com/limit13.txt
or
http://x31eq.com/limit15.txt

Graham found at least 10 ways to do "better" in 13-limit and 15-limit.

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

> Gene -- the complexity is actually only 10, not 13 -- 11 notes in
the
> chain are enough to give a complete tetrad.

Oops--so that's the definition people use?

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

> Curiously, no kleismic temperament appears in
>
> http://x31eq.com/limit13.txt
> or
> http://x31eq.com/limit15.txt

I didn't find the 17/46 I mentioned on the seven list; that turns out
not to be significantly different than the 19-27 linear termperament,
with a span for the tetrad of 13 (is that the complexity?) The 19 out
of 46 and 27 out of 46 scales with this generator might be reasonable
things for your quest to consider.

In-Reply-To: <9q2hsr+gnl9@eGroups.com>
Paul wrote:

> Curiously, no kleismic temperament appears in
>
> http://x31eq.com/limit13.txt
> or
> http://x31eq.com/limit15.txt
>
> Graham found at least 10 ways to do "better" in 13-limit and 15-limit.

27-equal isn't consistent in the 11-limit. So I don't think this
temperament is even being considered in that program. It can be
calculated like this:

>>> import temper
>>> et27 = temper.PrimeET(27, temper.primes[:5])
>>> et58 = temper.PrimeET(58, temper.primes[:5])
>>> et27.basis[4]=94
>>> et27+et58

22/85, 310.3 cent generator

basis:
(1.0, 0.25859397023437097)

mapping by period and generator:
[(1, 0), (-1, 10), (0, 9), (1, 7), (-3, 25), (5, -5)]

mapping by steps:
[(58, 27), (92, 43), (135, 63), (163, 76), (201, 94), (215, 100)]

unison vectors:
[[-1, 5, 0, 0, -2, 0], [9, -1, 0, 0, 0, -2], [5, 2, -5, 0, 1, 0], [0, -2,
0, 5,
-1, -2]]

highest interval width: 25
complexity measure: 25 (27 for smallest MOS)
highest error: 0.006590 (7.909 cents)

I think it might have made the 13-limit list if it had been included. It
should certainly be in the MOS list, as it's worse than this:

complexity measure: 23 (29 for smallest MOS)
highest error: 0.008301 (9.961 cents)

in both measures.

Graham

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Gene -- the complexity is actually only 10, not 13 -- 11 notes in
> the
> > chain are enough to give a complete tetrad.
>
> Oops--so that's the definition people use?

Well, that's Graham's complexity.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Curiously, no kleismic temperament appears in
> >
> > http://x31eq.com/limit13.txt
> > or
> > http://x31eq.com/limit15.txt
>
> I didn't find the 17/46 I mentioned on the seven list; that turns
out
> not to be significantly different than the 19-27 linear
termperament,
> with a span for the tetrad of 13 (is that the complexity?)

Yup. This temperament is #16 in my recent list.

> The 19 out
> of 46 and 27 out of 46 scales with this generator might be
reasonable
> things for your quest to consider.

Yes -- if you look back at my recent comments, you'll see that I
thought 27-tET is far from ideal for this 19-tone MOS, but I may be
willing to live with it, since a 27-tET guitar fingerboard will allow
so many other possibilities as well, and 46-tET, while attractive for
other reasons, is too tight on a guitar for me.

Graham wrote,

> I think it might have made the 13-limit list if it had been
included.

Hmm . . . are there things missing from the 5-, 7-, 9-, and 11-limit
lists for similar reasons?

> It
> should certainly be in the MOS list, as it's worse than this:
>
> complexity measure: 23 (29 for smallest MOS)
> highest error: 0.008301 (9.961 cents)
>
> in both measures.

What's your MOS list? I'm very confused now.

Paul wrote:

> Graham wrote,
>
> > I think it might have made the 13-limit list if it had been
> included.
>
> Hmm . . . are there things missing from the 5-, 7-, 9-, and 11-limit
> lists for similar reasons?

Meantone-31 is missing from the 11-limit list, because there are no other
ETs it's consistent with. It's inaccurate and not unique, but might make
the MOS list as it fits into 19.

Ah, it can be generated from 31 and 43

31/74, 503.3 cent generator

basis:
(1.0, 0.419405836425)

mapping by period and generator:
[(1, 0), (2, -1), (4, -4), (7, -10), (11, -18)]

mapping by steps:
[(43, 31), (68, 49), (100, 72), (121, 87), (149, 107)]

unison vectors:
[[-4, 4, -1, 0, 0], [1, 2, -3, 1, 0], [4, 0, -2, -1, 1]]

highest interval width: 18
complexity measure: 18 (19 for smallest MOS)
highest error: 0.009185 (11.022 cents)

Temperaments without two consistent ETs aren't likely to be that accurate
or unique, but they can be simple. I might write a script that can
generate temperaments from all the possible generators for an ET, then run
it for all consistent ETs. Hopefully all linear temperaments of note will
have at least *one* consistent equal representative.

> What's your MOS list? I'm very confused now.

The smallest MOS is taken as the complexity measure, instead of the number
of generators for a complete chord. Dave Keenan asked for it a while
back, and the script keeps churning them out so I keep uploading them.
They're a .mos suffix instead of .txt and there are also .key keyboard
mappings and .micro microtemperaments. The template is limitN.whatever
where N now goes as odd numbers from 5 to 21.

Graham