I took the Euclidean reduced 8-tone scale

1--8/7--6/5--4/3--7/5--3/2--5/3--7/4

looked at its 46-et version in all 1680 permutations of its steps, and reduced this to 108 under equivalence under mode and inversion. Of these 108 classes, one stood out using counts of edges and 3-note

chords, being vastly superior in the 3-note chord department. Moreover, it wasn't one that was particularly promising in its JI version!

It is easy to find 3-note chords from the characteristic polynomial or the adjacency matrix, and it may be that looking at them will help a great deal in sorting out the gold; it certainly did in this case.

Here it is:

39375739, plus all its modal forms and their inversions. It has 21

7-limit edges, and 20 7-limit 3-note chords. Its characteristic polynomial (the characteristic polynomial of the adjacency matrix, which has a 1 if two nodes are connected, and a 0 otherwise) is

x^8-21*x^6-40*x^5+12*x^4+48*x^3; the -21*x^6 term means it has 21

7-limit intervals, and the -40*x^5 term means it has 20 7-limit

three-note chords. The x^2, x, and constant term are all zero which means it has multiple zero eigenvalues, but I don't know what *that* means, at least as yet.

The closest competition had only 14 3-note chords!

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

> Here it is:

>

> 39375739, plus all its modal forms and their inversions. It has 21

> 7-limit edges, and 20 7-limit 3-note chords. Its characteristic

polynomial (the characteristic polynomial of the adjacency matrix,

which has a 1 if two nodes are connected, and a 0 otherwise) is

> x^8-21*x^6-40*x^5+12*x^4+48*x^3; the -21*x^6 term means it has 21

> 7-limit intervals, and the -40*x^5 term means it has 20 7-limit

> three-note chords. The x^2, x, and constant term are all zero which

means it has multiple zero eigenvalues, but I don't know what *that*

means, at least as yet.

>

> The closest competition had only 14 3-note chords!

This does look like a good 8-note 7-limit scale.

It looks good melodically too (from what little I know about that).

It is very close to being a subset of Herman Miller's 12-tone Starling

tuning, from which Herman has used several 7-note subsets. So

"Starling-8" might be a good name for it.

See miller_12.scl and miller_12a.scl in the Scala archive

Starling temperament is essentially a 7-limit planar temperament where

the septimal semicomma 125:126 vanishes. Narrowing the octave by about

1.4 cents improves things. See

http://www.io.com/~hmiller/music/marrgarrel.html

http://dkeenan.com/Music/DistibutingCommas.htm

I'm guessing it should work well in 31-tET too as

2 6 2 5 3 5 2 6

On Mon, 11 Mar 2002 01:10:32 -0000, "dkeenanuqnetau" <d.keenan@uq.net.au>

wrote:

>It looks good melodically too (from what little I know about that).

>It is very close to being a subset of Herman Miller's 12-tone Starling

>tuning, from which Herman has used several 7-note subsets. So

>"Starling-8" might be a good name for it.

It does seem to have some resemblance (due to the 125:126 and the chromatic

semitones), although Starling as I've used it has slightly _narrow_ fifths,

while 46-ET has slightly wide fifths. Still, the 125:126 is really the

defining feature of Starling temperament, and the size of the fifth is more

incidental. It also looks similar to the octatonic scale:

A#

F# C#

A E

C G

Eb

Substitute a Gb for the F#, and you have the octatonic scale in diminished

temperament. (Of course, due to the 648/625, these notes represent the same

pitch in diminished temperament.)

This also suggests a possibly useful 12-note subset of 46-ET, by analogy:

A#

F# C#

D A E

F C G

Ab Eb

Cb

>I'm guessing it should work well in 31-tET too as

>2 6 2 5 3 5 2 6

Or in 34 as

2 7 2 5 4 5 2 7

Probably any temperament in the general area with a 125:126 would work. I

wonder if it works in one of the more extreme ones, like 28, 15, or even

16? (Well, in 16 it'd be identical to the octatonic scale, if it works at

all....)

--

see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--

hmiller (Herman Miller) "If all Printers were determin'd not to print any

@io.com email password: thing till they were sure it would offend no body,

\ "Subject: teamouse" / there would be very little printed." -Ben Franklin