Consonantly generated difference diamonds

Looking at 5-limit 7-note difference diamonds seems to confirm the
idea that a generating set consisting of p-limit consonances
(including unity) is a good way to get interesting p-limit difference
diamonds. The [9/8, 5/4, 3/2] difference diamond (the Ptolemy/Al
Farabi diatonic diamond) can also be derived from [1, 3/2, 5/3].
Another interesting difference diamond derives from the consonances
[5/4, 4/3, 5/3]; this gives the scale Scala has listed as follows:

! helmholtz.scl
!
Helmholtz's Chromatic scale and Gipsy major from Slovakia

7
!
16/15
5/4
4/3
3/2
8/5
15/8
2/1

This scale is not only symmetrical around two axes, it is congruent to
the Ptolemy/Al Farabi diamond as well. We have, therefore, two 5-limit
scales which are epimorphic and which transform isometrically to
themselves and each other in various ways. Reduced to meantone, the
Ptolemy diatonic is just the standard diatonic scale, but the Gipsy
major gives instead a modmos, -5,-4,-1,0,1,4,5.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

The [9/8, 5/4, 3/2] difference diamond (the Ptolemy/Al
> Farabi diatonic diamond) can also be derived from [1, 3/2, 5/3].
> Another interesting difference diamond derives from the consonances
> [5/4, 4/3, 5/3]; this gives the scale Scala has listed as follows:

If I apply the transformation 5/4==>6/5, 3/2==>8/5 to [1, 3/2, 5/3] I
get [1, 8/5, 3/2], which gives the Gipsy Major. Applying it again
gives [1, 5/3, 8/5], which also gives a difference diamond, this time
<7 12 17|-epimorphic rather than <7 11 16|-epimorphic. This scale

1, 25/24, 6/5, 5/4, 8/5, 5/3, 48/25

is unknown to Scala, but it has the same character as the other two in
some ways, being epimorphic, symmetrical, and possessing two major and
two minor triads. Scala analyzes it as permutation epimorphic for
<7 11 17|, which is true, but I don't know why <7 12 17| is
overlooked; though it does mark it down as Constant Structure.

We can apply the 120 degree rotations in the lattice directly to the
scales themselves; this scrambles the ordering. Under 5/4==>6/5,
3/2==>8/5 we get

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

sent to

1, 15/16, 4/3, 5/4, 8/5, 3/2, 32/15

which is then sent to

1, 24/25, 5/4, 6/5, 5/3, 8/5, 25/12

and then, of course, back where we started.