I started out looking at these as 7-limit 225/224 planar temperament scales, but decided it made more sense to check the 5 and 11 limits also, and to take them as 72-et scales; if they are ever used that is probably how they will be used. I think anyone interested in the

72-et should take a look at the top three, which are all 5-connected, and the top scale in particular, which is a clear winner. The "edges" number counts edges (consonant intervals) in the 5, 7, and 11 limits, and the connectivity is the edge-connectivity in the 5, 7 and 11 limits.

[0, 5, 12, 19, 28, 35, 42, 49, 58, 65]

[5, 7, 7, 9, 7, 7, 7, 9, 7, 7]

edges 15 27 35 connectivity 2 5 6

[0, 5, 12, 19, 28, 35, 42, 51, 58, 65]

[5, 7, 7, 9, 7, 7, 9, 7, 7, 7]

edges 14 25 35 connectivity 1 3 6

[0, 5, 12, 21, 28, 35, 42, 51, 58, 65]

[5, 7, 9, 7, 7, 7, 9, 7, 7, 7]

edges 13 25 35 connectivity 1 3 6

[0, 5, 12, 19, 26, 35, 42, 51, 58, 65]

[5, 7, 7, 7, 9, 7, 9, 7, 7, 7]

edges 11 21 35 connectivity 0 2 6

[0, 5, 12, 21, 28, 35, 42, 49, 58, 65]

[5, 7, 9, 7, 7, 7, 7, 9, 7, 7]

edges 12 25 33 connectivity 0 3 6

[0, 5, 14, 21, 28, 35, 42, 51, 58, 65]

[5, 9, 7, 7, 7, 7, 9, 7, 7, 7]

edges 10 24 33 connectivity 0 3 6

[0, 5, 14, 21, 28, 35, 44, 51, 58, 65]

[5, 9, 7, 7, 7, 9, 7, 7, 7, 7]

edges 10 23 33 connectivity 0 3 5

[0, 5, 12, 21, 28, 35, 44, 51, 58, 65]

[5, 7, 9, 7, 7, 9, 7, 7, 7, 7]

edges 11 22 33 connectivity 0 2 5

[0, 5, 12, 19, 28, 35, 44, 51, 58, 65]

[5, 7, 7, 9, 7, 9, 7, 7, 7, 7]

edges 10 20 33 connectivity 0 2 5

[0, 5, 12, 19, 26, 35, 44, 51, 58, 65]

[5, 7, 7, 7, 9, 9, 7, 7, 7, 7]

edges 7 15 32 connectivity 0 1 5

[0, 5, 14, 21, 28, 35, 42, 49, 58, 65]

[5, 9, 7, 7, 7, 7, 7, 9, 7, 7]

edges 9 23 31 connectivity 0 3 5

[0, 5, 12, 21, 28, 35, 42, 49, 56, 65]

[5, 7, 9, 7, 7, 7, 7, 7, 9, 7]

edges 9 22 31 connectivity 0 3 5

[0, 5, 14, 21, 28, 35, 42, 49, 56, 63]

[5, 9, 7, 7, 7, 7, 7, 7, 7, 9]

edges 7 22 31 connectivity 0 4 5

[0, 5, 14, 21, 28, 37, 44, 51, 58, 65]

[5, 9, 7, 7, 9, 7, 7, 7, 7, 7]

edges 8 21 31 connectivity 0 2 5

[0, 5, 14, 21, 28, 35, 42, 49, 56, 65]

[5, 9, 7, 7, 7, 7, 7, 7, 9, 7]

edges 7 21 31 connectivity 0 3 5

[0, 5, 14, 21, 30, 37, 44, 51, 58, 65]

[5, 9, 7, 9, 7, 7, 7, 7, 7, 7]

edges 6 19 31 connectivity 0 2 5

[0, 5, 12, 21, 28, 37, 44, 51, 58, 65]

[5, 7, 9, 7, 9, 7, 7, 7, 7, 7]

edges 7 18 31 connectivity 0 2 5

[0, 5, 14, 23, 30, 37, 44, 51, 58, 65]

[5, 9, 9, 7, 7, 7, 7, 7, 7, 7]

edges 6 18 30 connectivity 0 1 5

[0, 5, 12, 19, 28, 37, 44, 51, 58, 65]

[5, 7, 7, 9, 9, 7, 7, 7, 7, 7]

edges 6 15 30 connectivity 0 2 5

[0, 5, 12, 21, 30, 37, 44, 51, 58, 65]

[5, 7, 9, 9, 7, 7, 7, 7, 7, 7]

edges 5 15 30 connectivity 0 1 5

Gene,

Interested in calculating the 7-limit edge connectivity

of Paul's decatonic scales in 22-tET?

Just so I'm straight, this is the least number of

connections, over every pitch in the scale, that the

given pitch has with any other pitch in the scale, right?

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> Gene,

>

> Interested in calculating the 7-limit edge connectivity

> of Paul's decatonic scales in 22-tET?

That was the first example I did, but it might be interesting to do something with three step sizes in 22-et like the stuff I've been working out in 72-et.

> Just so I'm straight, this is the least number of

> connections, over every pitch in the scale, that the

> given pitch has with any other pitch in the scale, right?

It is how many edges (representing consonant intervals) would need to be removed in order to render the scale disconnected; very often this will be the same.

>That was the first example I did,

Found it. I don't see any other scales c=6 in the 7-limit,

and only the 225:224 stuff has been up to c=5.

>It is how many edges (representing consonant intervals) would

>need to be removed in order to render the scale disconnected;

>very often this will be the same.

Cool.

-C.

In-Reply-To: <a0thh3+terr@eGroups.com>

Gene wrote:

> I started out looking at these as 7-limit 225/224 planar temperament

> scales, but decided it made more sense to check the 5 and 11 limits

> also, and to take them as 72-et scales; if they are ever used that is

> probably how they will be used. I think anyone interested in the 72-et

> should take a look at the top three, which are all 5-connected, and the

> top scale in particular, which is a clear winner. The "edges" number

> counts edges (consonant intervals) in the 5, 7, and 11 limits, and the

> connectivity is the edge-connectivity in the 5, 7 and 11 limits.

Well, more than 72-equal, they're all Miracle consistent, aren't they? In

which case, they're also all Blackjack subsets. But the decimal MOS isn't

one of them. Interesting.

Graham

--- In tuning-math@y..., graham@m... wrote:

> Well, more than 72-equal, they're all Miracle consistent, aren't they? In

> which case, they're also all Blackjack subsets. But the decimal MOS isn't

> one of them. Interesting.

The decimal MOS has two step sizes, and so belongs to a linear rather than a planar temperament, but perhaps I should run it through the evaluation process for the sake of comparison.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

Here by way of comparison is the 10-note Miracle MOS; it is relatively undistinguished in this company.

> [0, 5, 12, 19, 28, 35, 42, 49, 58, 65]

> [5, 7, 7, 9, 7, 7, 7, 9, 7, 7]

> edges 15 27 35 connectivity 2 5 6

>

> [0, 5, 12, 19, 28, 35, 42, 51, 58, 65]

> [5, 7, 7, 9, 7, 7, 9, 7, 7, 7]

> edges 14 25 35 connectivity 1 3 6

>

> [0, 5, 12, 21, 28, 35, 42, 51, 58, 65]

> [5, 7, 9, 7, 7, 7, 9, 7, 7, 7]

> edges 13 25 35 connectivity 1 3 6

>

> [0, 5, 12, 19, 26, 35, 42, 51, 58, 65]

> [5, 7, 7, 7, 9, 7, 9, 7, 7, 7]

> edges 11 21 35 connectivity 0 2 6

>

> [0, 5, 12, 21, 28, 35, 42, 49, 58, 65]

> [5, 7, 9, 7, 7, 7, 7, 9, 7, 7]

> edges 12 25 33 connectivity 0 3 6

>

> [0, 5, 14, 21, 28, 35, 42, 51, 58, 65]

> [5, 9, 7, 7, 7, 7, 9, 7, 7, 7]

> edges 10 24 33 connectivity 0 3 6

>

> [0, 5, 14, 21, 28, 35, 44, 51, 58, 65]

> [5, 9, 7, 7, 7, 9, 7, 7, 7, 7]

> edges 10 23 33 connectivity 0 3 5

>

> [0, 5, 12, 21, 28, 35, 44, 51, 58, 65]

> [5, 7, 9, 7, 7, 9, 7, 7, 7, 7]

> edges 11 22 33 connectivity 0 2 5

>

> [0, 5, 12, 19, 28, 35, 44, 51, 58, 65]

> [5, 7, 7, 9, 7, 9, 7, 7, 7, 7]

> edges 10 20 33 connectivity 0 2 5

>

> [0, 5, 12, 19, 26, 35, 44, 51, 58, 65]

> [5, 7, 7, 7, 9, 9, 7, 7, 7, 7]

> edges 7 15 32 connectivity 0 1 5

>

> [0, 5, 14, 21, 28, 35, 42, 49, 58, 65]

> [5, 9, 7, 7, 7, 7, 7, 9, 7, 7]

> edges 9 23 31 connectivity 0 3 5

>

> [0, 5, 12, 21, 28, 35, 42, 49, 56, 65]

> [5, 7, 9, 7, 7, 7, 7, 7, 9, 7]

> edges 9 22 31 connectivity 0 3 5

>

> [0, 5, 14, 21, 28, 35, 42, 49, 56, 63]

> [5, 9, 7, 7, 7, 7, 7, 7, 7, 9]

> edges 7 22 31 connectivity 0 4 5

>

> [0, 5, 14, 21, 28, 37, 44, 51, 58, 65]

> [5, 9, 7, 7, 9, 7, 7, 7, 7, 7]

> edges 8 21 31 connectivity 0 2 5

>

> [0, 5, 14, 21, 28, 35, 42, 49, 56, 65]

> [5, 9, 7, 7, 7, 7, 7, 7, 9, 7]

> edges 7 21 31 connectivity 0 3 5

>

> [0, 5, 14, 21, 30, 37, 44, 51, 58, 65]

> [5, 9, 7, 9, 7, 7, 7, 7, 7, 7]

> edges 6 19 31 connectivity 0 2 5

>

> [0, 5, 12, 21, 28, 37, 44, 51, 58, 65]

> [5, 7, 9, 7, 9, 7, 7, 7, 7, 7]

> edges 7 18 31 connectivity 0 2 5

[0, 7, 14, 21, 28, 35, 42, 49, 56, 63]

[7, 7, 7, 7, 7, 7, 7, 7, 7, 9]

edges 7 22 30 connectivity 0 3 5

> [0, 5, 14, 23, 30, 37, 44, 51, 58, 65]

> [5, 9, 9, 7, 7, 7, 7, 7, 7, 7]

> edges 6 18 30 connectivity 0 1 5

>

> [0, 5, 12, 19, 28, 37, 44, 51, 58, 65]

> [5, 7, 7, 9, 9, 7, 7, 7, 7, 7]

> edges 6 15 30 connectivity 0 2 5

>

> [0, 5, 12, 21, 30, 37, 44, 51, 58, 65]

> [5, 7, 9, 9, 7, 7, 7, 7, 7, 7]

> edges 5 15 30 connectivity 0 1 5

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> I started out looking at these as 7-limit 225/224 planar

temperament scales, but decided it made more sense to check the 5 and

11 limits also, and to take them as 72-et scales; if they are ever

used that is probably how they will be used. I think anyone

interested in the

> 72-et should take a look at the top three, which are all 5-

connected, and the top scale in particular, which is a clear winner.

The "edges" number counts edges (consonant intervals) in the 5, 7,

and 11 limits, and the connectivity is the edge-connectivity in the

5, 7 and 11 limits.

>

> [0, 5, 12, 19, 28, 35, 42, 49, 58, 65]

> [5, 7, 7, 9, 7, 7, 7, 9, 7, 7]

Was this Dave Keenan's 72-tET version of my Pentachordal Decatonic?

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

> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> > I started out looking at these as 7-limit 225/224 planar

> temperament scales, but decided it made more sense to check the 5

and

> 11 limits also, and to take them as 72-et scales; if they are ever

> used that is probably how they will be used. I think anyone

> interested in the

> > 72-et should take a look at the top three, which are all 5-

> connected, and the top scale in particular, which is a clear winner.

> The "edges" number counts edges (consonant intervals) in the 5, 7,

> and 11 limits, and the connectivity is the edge-connectivity in the

> 5, 7 and 11 limits.

> >

> > [0, 5, 12, 19, 28, 35, 42, 49, 58, 65]

> > [5, 7, 7, 9, 7, 7, 7, 9, 7, 7]

>

> Was this Dave Keenan's 72-tET version of my Pentachordal Decatonic?

No. The 72-tET version of your Pentachordal Decatonic is this one

(pentachordal in two ways).

7 9 7 7|5 7|7 9 7 7

7 7 9 7|7 5|7 7 9 7

But the one you give above, is listed along with it in

/tuning/topicId_27221.html#27221?expand=1