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7-limit 32805/32768-planar orthogonal projection scales

🔗Gene Ward Smith <gwsmith@svpal.org>

3/8/2004 2:32:18 PM

These are what you get from balls around the unison from the
orthogonal projection of 32805/32768 onto the unison of the
symmetrical lattice. The 32805/32768-planar tuning I used was just 3's
and 7's, for three reasons--one, this gives a JI scale for JI
affecianados, two it is easy to retune a scale given this way to the
tuning you prefer, and three the tuning works just as it is. These
scales or ones like them would be very suitable for Michael Harrison
style music. I give first the radius and corresponding shell, and
second the number of notes and scale, up to a 35 note scale.

Shells

3
[4/3, 3/2]

6
[9/8, 16/9]

7
[32/27, 27/16]

9
[81/64, 128/81]

10
[256/243, 243/128]

11
[1024/729, 729/512]

13
[2187/2048, 4096/2187]

14
[8192/6561, 6561/4096]

15
[64/63, 63/32]

17
[256/189, 189/128]

18
[21/16, 32/21]

19
[567/512, 1024/567]

21
[8/7, 7/4]

22
[2048/1701, 1701/1024]

23
[7/6, 12/7]

25
[5103/4096, 8192/5103]

26
[19683/16384, 32768/19683]

Scales

3
[1, 4/3, 3/2]

5
[1, 9/8, 4/3, 3/2, 16/9]

7
[1, 9/8, 32/27, 4/3, 3/2, 27/16, 16/9]

9
[1, 9/8, 32/27, 81/64, 4/3, 3/2, 128/81, 27/16, 16/9]

11
[1, 256/243, 9/8, 32/27, 81/64, 4/3, 3/2, 128/81, 27/16, 16/9, 243/128]

13
[1, 256/243, 9/8, 32/27, 81/64, 4/3, 1024/729, 729/512, 3/2, 128/81,
27/16, 16/9, 243/128]

15
[1, 256/243, 2187/2048, 9/8, 32/27, 81/64, 4/3, 1024/729, 729/512,
3/2, 128/81, 27/16, 16/9, 4096/2187, 243/128]

17
[1, 256/243, 2187/2048, 9/8, 32/27, 8192/6561, 81/64, 4/3, 1024/729,
729/512, 3/2, 128/81, 6561/4096, 27/16, 16/9, 4096/2187, 243/128]

19
[1, 64/63, 256/243, 2187/2048, 9/8, 32/27, 8192/6561, 81/64, 4/3,
1024/729, 729/512, 3/2, 128/81, 6561/4096, 27/16, 16/9, 4096/2187,
243/128, 63/32]

21
[1, 64/63, 256/243, 2187/2048, 9/8, 32/27, 8192/6561, 81/64, 4/3,
256/189, 1024/729, 729/512, 189/128, 3/2, 128/81, 6561/4096, 27/16,
16/9, 4096/2187, 243/128, 63/32]

23
[1, 64/63, 256/243, 2187/2048, 9/8, 32/27, 8192/6561, 81/64, 21/16,
4/3, 256/189, 1024/729, 729/512, 189/128, 3/2, 32/21, 128/81,
6561/4096, 27/16, 16/9, 4096/2187, 243/128, 63/32]

25
[1, 64/63, 256/243, 2187/2048, 567/512, 9/8, 32/27, 8192/6561, 81/64,
21/16, 4/3, 256/189, 1024/729, 729/512, 189/128, 3/2, 32/21, 128/81,
6561/4096, 27/16, 16/9, 1024/567, 4096/2187, 243/128, 63/32]

27
[1, 64/63, 256/243, 2187/2048, 567/512, 9/8, 8/7, 32/27, 8192/6561,
81/64, 21/16, 4/3, 256/189, 1024/729, 729/512, 189/128, 3/2, 32/21,
128/81, 6561/4096, 27/16, 7/4, 16/9, 1024/567, 4096/2187, 243/128, 63/32]

29
[1, 64/63, 256/243, 2187/2048, 567/512, 9/8, 8/7, 32/27, 2048/1701,
8192/6561, 81/64, 21/16, 4/3, 256/189, 1024/729, 729/512, 189/128,
3/2, 32/21, 128/81, 6561/4096, 1701/1024, 27/16, 7/4, 16/9, 1024/567,
4096/2187, 243/128, 63/32]

31
[1, 64/63, 256/243, 2187/2048, 567/512, 9/8, 8/7, 7/6, 32/27,
2048/1701, 8192/6561, 81/64, 21/16, 4/3, 256/189, 1024/729, 729/512,
189/128, 3/2, 32/21, 128/81, 6561/4096, 1701/1024, 27/16, 12/7, 7/4,
16/9, 1024/567, 4096/2187, 243/128, 63/32]

33
[1, 64/63, 256/243, 2187/2048, 567/512, 9/8, 8/7, 7/6, 32/27,
2048/1701, 5103/4096, 8192/6561, 81/64, 21/16, 4/3, 256/189, 1024/729,
729/512, 189/128, 3/2, 32/21, 128/81, 6561/4096, 8192/5103, 1701/1024,
27/16, 12/7, 7/4, 16/9, 1024/567, 4096/2187, 243/128, 63/32]

35
[1, 64/63, 256/243, 2187/2048, 567/512, 9/8, 8/7, 7/6, 32/27,
19683/16384, 2048/1701, 5103/4096, 8192/6561, 81/64, 21/16, 4/3,
256/189, 1024/729, 729/512, 189/128, 3/2, 32/21, 128/81, 6561/4096,
8192/5103, 1701/1024, 32768/19683, 27/16, 12/7, 7/4, 16/9, 1024/567,
4096/2187, 243/128, 63/32]