"Euclidean reduced" 8-tone scales

By "Euclidean reduced" I mean I choose the least distance from the unison in the symmetric lattice of octave classes; if this is a tie, I try to use odd Tenney height (the Tenney height of the odd part) to break it, and if that doesn't work, Tenney height. If my thinking is right, the last should be good at tie-breaking.

It would also be possible to take the distance from the middle of a fifth, or a triad, etc.

h8, 5-limit

1, 10/9, 6/5, 4/3, 25/18, 3/2, 5/3, 9/5

h8, 7-limit (not epimorphic)

1, 7/6, 8/7, 4/3, 35/24, 3/2, 7/4, 12/7

h8, 9-limit (not epimorphic)

1, 7/6, 6/5, 4/3, 9/7, 3/2, 5/3, 12/7

h8+v7, 7-limit

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

h8+v7, 9-limit (same as 7-limit)

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

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

> h8, 5-limit
>
> 1, 10/9, 6/5, 4/3, 25/18, 3/2, 5/3, 9/5

If we look at commas for 5-limit h8, we find 16/15, 648/625 and
250/243, suggesting fourth-thirds, diminished and porcupine as possible temperaments. The above scale is in the class defined by
(10/9)^4 (27/25)^3 (25/24); diminished equates 27/25~25/24, leading to L^4 s^4 systems, whereas porcupine equates 10/9~27/25, leading to L^7 s systems.

> h8+v7, 7-limit
>
> 1, 8/7, 6/5, 4/3, 7/5, 3/2, 5/3, 7/4

If we look at h8+v7, we add to our previous list of commas, in particular with 28/27, 36/35, 50/49, 126/125 and 245/243. Temperaments we get from these are 36/35^50/49 = [4,4,4,-2,5,-3], diminished; 50/49^245/243 = [6,10,10,-5,1,2], the "9/7" system of
22-et, with an 8/22 generator and a half-octave period; and
126/125^245/243 = [7,9,14,5,-1,-2], which if it ever had a name is one I've forgotten, but which has been mentioned in connection with
17/46 by both me and Graham, and which I find interesting.

The steps for this scale are(8/7)^2 (10/9)^2 (15/14) (21/20)^3; this becomes 42322324 in 22-et or 93735739 in 46-et; permutations of this are of course entirely possible.