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Relative complexity revisited

🔗Gene Ward Smith <gwsmith@svpal.org>

5/25/2005 11:31:58 PM

Long long ago and far far away in a distant galaxy, I came up wwith
the notion of relative complexity. Relative complexity of a comma,
with respect to a temperament, is found by wedging the comma with a
comma basis for the temperament, and applying a complexity measure to
the result. I was using geometric complexity, but we can, for
instance, use Graham complexity when it is applicable. Below I order
intervals within an octave (this measure is octave equivalent)
according to their Graham relative complexity with respect to
2401/2400, with the results 2401/2400 reduced. This gives us shells
and balls and scales again, if we want them. Below are the first few;
I note that the pentatonic scale from ball 2 and the nine-note scale
from ball 3 are epimorphic. With the 17 notes of ball 4 we finally get
a complete diamond, and the very same convex closure of the tonality
diamond I very recently presented.

0
{1}
[1]

1
{}
[1]

2
{49/30, 49/40, 7/5, 10/7}
[1, 49/40, 7/5, 10/7, 49/30]

3
{8/7, 7/6, 7/4, 12/7}
[1, 8/7, 7/6, 49/40, 7/5, 10/7, 49/30, 12/7, 7/4]

4
{49/25, 8/5, 49/48, 3/2, 5/3, 6/5, 5/4, 4/3}
[1, 49/48, 8/7, 7/6, 6/5, 49/40, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 49/30,
5/3, 12/7, 7/4, 49/25]

5
{40/21, 15/14, 28/15, 21/20}
[1, 49/48, 21/20, 15/14, 8/7, 7/6, 6/5, 49/40, 5/4, 4/3, 7/5, 10/7,
3/2, 8/5, 49/30, 5/3, 12/7, 7/4, 28/15, 40/21, 49/25]

6
{72/49, 28/25, 35/24, 90/49, 25/21, 64/49, 49/32, 42/25, 48/35, 49/36,
25/14, 49/45}
[1, 49/48, 21/20, 15/14, 49/45, 28/25, 8/7, 7/6, 25/21, 6/5, 49/40,
5/4, 64/49, 4/3, 49/36, 48/35, 7/5, 10/7, 35/24, 72/49, 3/2, 49/32,
8/5, 49/30, 5/3, 42/25, 12/7, 7/4, 25/14, 90/49, 28/15, 40/21, 49/25]