[tuning-math] Three generator step scales
- From: Gene W Smith <genewardsmith@xxxxxxxx>
- To: tuning-math@xxxxxxxxxxxxxxx
- Date: Sat, 13 Jul 2002 13:08:07 -0700
I want to look a little more closely at some of the scales I discussed in
the previous article, focusing on the ones with three generator steps. By
this I mean the scales deriving from [-2 0 0]-[-1 0 0]-[0 0 0]-[1 0 0],
[0 -2 0]-[0 -1 0]-[0 0 0]-[0 1 0], and [0 0 -2]-[0 0 -1]-[0 0 0]-[0 0 1].
As JI scales, these are isomorphic sets of chords in the lattice, and
hence have isomorphic 7-limit graphs. We have:
From [0 0 1] as generator
[1, 15/14, 7/6, 5/4, 7/5, 3/2, 49/30, 7/4, 28/15, 15/8]
72-et version [0, 7, 16, 23, 35, 42, 51, 58, 65]
From [0 1 0] as generator
[1, 21/20, 25/21, 5/4, 21/16, 10/7, 3/2, 5/3, 7/4, 40/21]
68-et version [0, 5, 17, 22, 27, 35, 40, 50, 55, 63]
From [1 0 0] as generator
[1, 36/35, 35/32, 6/5, 5/4, 48/35, 35/24, 3/2, 12/7, 7/4]
84-et version [0, 3, 11, 22, 27, 38, 46, 49, 65, 68]
All of the JI scales are graph-isomorphic, each having 21 intervals, 16
triads and 4 tetrads in the 7-limit.
The first scale makes the least sense as a JI scale, having a step of
225/224; two notes are conflated in the 72-et
version, giving an excellent 9-note scale. The second scale makes the
most sense as a JI scale, and the least sense as a tempered scale, since
the tempered version is isomorphic to the non-tempered version. The last
scale is a little irregular so far as step size goes, but does gain from
tempering via Orwell, as the tempered version has four more intervals and
two more triads.
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