The most equal superparticular scales

There are some of these in the Scala archive. How does one
calculate/find these or is it just brute force? I'm especially
interested in scales of higher cardinality (like 22 and 31) because
they might be great as rationally intoned well-temperaments.

Kalle

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:

> There are some of these in the Scala archive. How does one
> calculate/find these or is it just brute force? I'm especially
> interested in scales of higher cardinality (like 22 and 31) because
> they might be great as rationally intoned well-temperaments.

I'd start by defining what it is I am trying to find. Is every
interval superparticular, and what is your criterion for evenness?

Of course if it doesn't need to be most even, it gets easier;
(25/24)^5 (28/27)^4 (36/35)^10 (49/48)^3 = 2 can be found simply by
inverting the matrix of monzos for these intervals; the same is true of
(25/24)^2 (28/27)^7 (33/32)^3 (36/35)^7 (45/44)^3 = 2 and numerous
higher limit examples. The trouble with this approach is that it
assumes the ratios are all indepentdent.

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:

> There are some of these in the Scala archive. How does one
> calculate/find these or is it just brute force? I'm especially
> interested in scales of higher cardinality (like 22 and 31) because
> they might be great as rationally intoned well-temperaments.

One way to get rational well-temperaments is via Fokker blocks with
small commas; for 22 one might for instance use 225/224, 6144/6125, and
10976/10935. The smaller the commas one uses, the smaller the
deviation from equal temperament. I could give some of these if there
were any interest.

The above method does not generally give superparticular scale steps.
A 22-note 7-limit scale with superparticular steps I think is
interesting is the one I give below. It is a modification of the scale
I called hahn22.scl in

/tuning-math/message/10822

This has all superparticular steps, but one of the steps is 126/125.
If we replace 25/18 with 48/35 we get a scale which is much more
regular, with step sizes 25/24, 28/27, 36/35 and 49/48, and enough
harmonic possibilities to keep you occupied.

! ha22.scl
Modified Hahn reduced 22-note scale
22
!
25/24
15/14
10/9
8/7
7/6
6/5
5/4
9/7
4/3
48/35
7/5
35/24
3/2
14/9
8/5
5/3
12/7
7/4
9/5
15/8
35/18
2

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
>
> > There are some of these in the Scala archive. How does one
> > calculate/find these or is it just brute force? I'm especially
> > interested in scales of higher cardinality (like 22 and 31)
because
> > they might be great as rationally intoned well-temperaments.
>
> I'd start by defining what it is I am trying to find. Is every
> interval superparticular, and what is your criterion for evenness?

Every scale step must be a superparticular ratio.

I'm not sure what evenness criterion those Scala archive scales
(super_5, super_6 etc.) fulfill. Maybe Manuel knows. I guess it's
just as close to equal tuning as possible while still being
superparticular.

What comes to the exact order of the scale steps I would choose the
scale which is as low in the harmonic series as possible.

And there is no restriction which primes can be used.

I found this for 22 but I don't know if it is the most even one:

1/1
33/32
17/16
35/32
9/8
7/6
29/24
5/4
31/24
4/3
11/8
17/12
35/24
3/2
31/20
8/5
33/20
17/10
7/4
29/16
15/8
31/16
2/1

Kalle wrote 25-7:
>There are some of these in the Scala archive. How does one
>calculate/find these or is it just brute force? I'm especially
>interested in scales of higher cardinality (like 22 and 31) because
>they might be great as rationally intoned well-temperaments.

I used the brute force method. All one-step intervals are
superparticular and the standard deviation from equal tempered
(as printed by SHOW DATA) is the lowest possible.
Larger scales took too much time, and I don't remember where
I've put the code, if I still have it.

Manuel

I see it's easy to find superparticular scales
which are more even than the ones in the archive,
so please forget that claim. I'll come back for it
if I can find out what I did then.

Manuel

--- In tuning@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> Kalle wrote 25-7:
> >There are some of these in the Scala archive. How does one
> >calculate/find these or is it just brute force? I'm especially
> >interested in scales of higher cardinality (like 22 and 31) because
> >they might be great as rationally intoned well-temperaments.
>
> I used the brute force method. All one-step intervals are
> superparticular and the standard deviation from equal tempered
> (as printed by SHOW DATA) is the lowest possible.
> Larger scales took too much time, and I don't remember where
> I've put the code, if I still have it.

What about a chain of 45/44s? While all one-step intervals are
trivially superparticular this produces 31-tone equal temperament
with octave of 1206.07897027 cents. So must the octave be just?