Some 7-limit superparticular pentatonics

These are the ones which employ the two most proper possibilities,
(15/14)(8/7)(7/6)^2(6/5) and (15/14)(10/9)(7/6)(6/5)^2; both with a Blackwood index of 2.64 (largest over smallest scale step ratio.)

1--6/5--7/5--3/2--7/4
[6/5 7/6 15/14 7/6 8/7] c = 3

1--7/6--4/3--10/7--12/7
[7/6 8/7 15/14 6/5 7/6] c = 2

1--7/6--7/5--3/2--12/7
[7/6 6/5 15/14 8/7 7/6] c = 2

1--6/5--7/5--3/2--12/7
[6/5 7/6 15/14 8/7 7/6] c = 2

1--8/7--4/3--8/5--12/7
[8/7 7/6 6/5 15/14 7/6] c = 2

1--7/6--5/4--10/7--12/7
[7/6 15/14 8/7 6/5 7/6]

1--7/6--4/3--8/5--12/7
[7/6 8/7 6/5 15/14 7/6] c = 1

1--6/5--4/3--8/5--12/7
[6/5 10/9 6/5 15/14 7/6] c = 2

1--7/6--7/5--3/2--9/5
[7/6 6/5 15/14 6/5 10/9] c = 1

1--7/6--7/5--3/2--5/3
[7/6 6/5 15/14 10/9 6/5] c = 1

1--6/5--7/5--3/2--5/3
[6/5 7/6 15/14 10/9 6/5] c = 1

1--6/5--9/7--3/2--5/3
[6/5 15/14 7/6 10/9 6/5] c = 1

et__7-(odd)limit dyadic rms (cents)
22| 11
27| 8
37| 7
31| 4

Gene, are you allowing 9-limit edges?

> 1--6/5--7/5--3/2--7/4
> [6/5 7/6 15/14 7/6 8/7] c = 3

et__steps__________tetrachordality_
22| 0 6 11 13 18 _| 78$, 72% _____|
27| 0 7 13 16 22 _| 64$, 48% _____|
37| 0 10 18 22 30 | 74$, 57% _____|
31| 0 8 15 18 25 _| 71$, 61% _____|

> 1--6/5--4/3--8/5--12/7
> [6/5 10/9 6/5 15/14 7/6] c = 2

et__steps__________tetrachordality_
22| 0 6 9 15 17 __| 59$, 54% _____|
27| 0 7 11 18 21 _| 50$, 38% _____|
37| 0 10 15 25 29 | 52$, 40% _____|
31| 0 8 13 21 24 _| 55$, 47% _____|

Tomorrow, I'm going to try and tune these
up and see what they sound like.

I'm guessing a 10-cent difference between
every note any its 3:2 transposition isn't
as bad as a 50-cent difference between one
note and its transposition. Thus, the
next version of this software will offer
rms in addition to mad.

-Carl

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

> Gene, are you allowing 9-limit edges?

No, but it would be easy enough to do so. I'm already buried under the varied possibilities as it is, though, and also want to explore tempered scales using graph theory methods.

> Tomorrow, I'm going to try and tune these
> up and see what they sound like.

I was wondering also; I think I'll fiddle around on FTS with some of these.

>> Gene, are you allowing 9-limit edges?
>
> No, but it would be easy enough to do so.

Oh, it wasn't a request. I just wanted to make
sure we were on the same page, as I seem to remember
you using prime limits in the past.

-Carl

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> These are the ones which employ the two most proper possibilities,
> (15/14)(8/7)(7/6)^2(6/5) and (15/14)(10/9)(7/6)(6/5)^2; both with a
> Blackwood index of 2.64 (largest over smallest scale step ratio.)

This term "Blackwood index" should only be applied when there are
only two step sizes.