Pentachordal scales in EDOs

George D. Secor wrote on Wed May 17, 2006, re 34-et
in the 7-limit:

[I'm snipping much *musically* interesting stuff about 34]

> I assume that you're including the pentachordal form of Paul's
> decatonic scale in your mention of 'pajara'. It's unfortunate that
> Paul didn't mention anywhere in his 22-tone paper that it's possible
> to use his decatonic scales in 34. The interval sequence ssLsssLsss
> (the pentachordal major form) has L=3 and s=2 degrees in 22, but in
> 34 it would be L=5 and s=3.
>

Hi George,

Just taking up your last sentence, and exploring
what numbers support "the pentachordal major
form" (but making no pretence as to knowing how
useful any of the results would be musically, nor
restricting the results to any particular limit):

Pentachordal major scales in EDOs
----------------------------------------

George D Secor notes that the pentachordal major
scale has the interval sequence:
ssLsssLsss

This comprises 8 small and 2 Large steps.

Below I tabulate the EDOs corresponding to some
different sizes of small step s and large step L,
where L ranges from s+1 to 4*s:

..... s ....... L ..... EDO
..... 1 ....... 2 ...... 12
..... 1 ....... 3 ...... 14
..... 1 ....... 4 ...... 16
..... 2 ....... 3 ...... 22
..... 2 ....... 5 ...... 26
..... 2 ....... 7 ...... 30
..... 3 ....... 4 ...... 32
..... 3 ....... 5 ...... 34
..... 3 ....... 7 ...... 38
..... 3 ....... 8 ...... 40
..... 4 ....... 5 ...... 42
..... 3 ...... 10 ...... 44
..... 3 ...... 11 ...... 46
..... 4 ....... 7 ...... 46
..... 4 ....... 9 ...... 50
..... 5 ....... 6 ...... 52
..... 4 ...... 11 ...... 54
..... 5 ....... 7 ...... 54
..... 5 ....... 8 ...... 56
..... 4 ...... 13 ...... 58
..... 5 ....... 9 ...... 58
..... 4 ...... 15 ...... 62
..... 5 ...... 11 ...... 62
..... 5 ...... 12 ...... 64
..... 5 ...... 13 ...... 66
..... 5 ...... 14 ...... 68
..... 5 ...... 16 ...... 72
..... 5 ...... 17 ...... 74
..... 5 ...... 18 ...... 76
..... 5 ...... 19 ...... 78

I have sorted the entries by increasing EDO.
Note that 46-EDO is the first EDO that supports
two different ratios of small to Large step,
namely 3/11 and 4/7, and thus two different
pentachordal scales.

Other EDOs with this property are 54, 58 and 62.

If one extended this table to larger values of s,
I would expect there to be many more such EDOs.
Possibly some EDO supports 3 or more different
step size ratios, and thus pentachordal scales.

Regards,
Yahya

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--- In MakeMicroMusic@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@...>
wrote:
>
>
> George D. Secor wrote on Wed May 17, 2006, re 34-et
> in the 7-limit:
>
> [I'm snipping much *musically* interesting stuff about 34]
>
> > I assume that you're including the pentachordal form of Paul's
> > decatonic scale in your mention of 'pajara'. It's unfortunate
that
> > Paul didn't mention anywhere in his 22-tone paper that it's
possible
> > to use his decatonic scales in 34. The interval sequence
ssLsssLsss
> > (the pentachordal major form) has L=3 and s=2 degrees in 22, but
in
> > 34 it would be L=5 and s=3.
>
> Hi George,
>
> Just taking up your last sentence, and exploring
> what numbers support "the pentachordal major
> form" (but making no pretence as to knowing how
> useful any of the results would be musically, nor
> restricting the results to any particular limit):
>
> ...
> Pentachordal major scales in EDOs
> ...
> scale has the interval sequence:
> ssLsssLsss
>
> This comprises 8 small and 2 Large steps.
>
> Below I tabulate the EDOs corresponding to some
> different sizes of small step s and large step L,
> where L ranges from s+1 to 4*s:
>
> ..... s ....... L ..... EDO
> ..... 1 ....... 2 ...... 12
> ..... 1 ....... 3 ...... 14
> ..... 1 ....... 4 ...... 16
> ..... 2 ....... 3 ...... 22
> ..... 2 ....... 5 ...... 26
> ..... 2 ....... 7 ...... 30
> ..... 3 ....... 4 ...... 32
> ..... 3 ....... 5 ...... 34
> ..... 3 ....... 7 ...... 38
> ..... 3 ....... 8 ...... 40
> ...
>
> I have sorted the entries by increasing EDO.
> Note that 46-EDO is the first EDO that supports
> two different ratios of small to Large step,
> namely 3/11 and 4/7, and thus two different
> pentachordal scales.
>
> Other EDOs with this property are 54, 58 and 62.
>
> If one extended this table to larger values of s,
> I would expect there to be many more such EDOs.
> Possibly some EDO supports 3 or more different
> step size ratios, and thus pentachordal scales.

If you stray too far from the 22-ET s:L ratio of 2:3, you'll get a
scale with distorted harmonies, so many of the entries in your table
are of questionable value (or possibly something quite different from
pajara).

--George

On 5/19/06, George D. Secor <gdsecor@yahoo.com> wrote:
> If you stray too far from the 22-ET s:L ratio of 2:3, you'll get a
> scale with distorted harmonies, so many of the entries in your table
> are of questionable value (or possibly something quite different from
> pajara).

Right, some of them (14 and 26 in particular) are injera instead of pajara.

Keenan