Interesting 46-et, 8-tone scale

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....)

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