From: mclaren
Subject: Wilson CPS scales - post 2 of 9
The list of possible Wilson CPS scales in
the prior post was incomplete because our
assumption was that all generators were
mutually prime. This need not be so.
In fact, there are many advantages to
using generators some of which are factors
of other generators.
Erv Wilson himself prefers such CPS scales.
Using degenerate sets of generators (i.e.,
sets of numbers some of which are multiples
of others) simplifies the relationships between
some of the notes and cuts down on the total
number of tones in the scale by forcing
some of the pitches to overlap one another
in ratio space.
As it turns out, for highly degenerate
sets of generators the Wilson CPS
procedure generates numbers of tones
previously thought unobtainable.
It was long thought, for instance, that
a 17-tone Wilson CPS scale was
impossible without throwing tones
away or adding them promiscuously
(or using the trivial dodge of taking
16 out of 17 generators).
However, the 3 out of 13 Wilson CPS
[1,2,3,4,5,6,7,8,9,10,11,12,13] produces
only 17 tones as its output. This is an
almost completely degenerate set,
since virtually all of the generators are
multiples of one another: only 7, 11 and
13 are relatively prime to all the other
generators. Every other generator is a
factor of some other generator.
Consequently, the range of possible
tones in a full Wilson CPS is much larger
than previously suspected. Intermediate
and supposedly unobtainable numbers of
tones (17, 19, 22, 27, 30, 69, etc.) can
be produced in a full Wilson CPS scale
without adding or subtracting tones,
simply by using highly degenerate sets
as generators.
Example:
- Wilson CPS of 10 generators taken 5 at a time:
5 out of [1,2,3,4,5,6,7,8,9,10] This is a
highly redundant scale because 2 is a factor
of 4, 3 is a factor of 6 and 9, 4 is a factor
of 8 and 5 is a factor of 10. The result is
a much smaller gamut than the theoretical
maximum of 252 pitches:
- SCALE RATIO CENTS FROM 1/1
- DEGREE
- 0: 1/1 00.00000
- 1: 525/512 43.40835
- 2: 135/128 92.17876
- 3: 35/32 155.1396
- 4: 567/512 176.6459
- 5: 9/8 203.9100
- 6: 4725/4096 247.3184
- 7: 75/64 274.5825
- 8: 315/256 359.0498
- 9: 5/4 386.3139
- 10: 81/64 407.8201
- 11: 21/16 470.7811
- 12: 675/512 478.4926
- 13: 175/128 541.4535
- 14: 2835/2048 562.9598
- 15: 45/32 590.2239
- 16: 189/128 674.6912
- 17: 3/2 701.9553
- 18: 1575/1024 745.3637
- 19: 25/16 772.6278
- 20: 405/256 794.1340
- 21: 105/64 857.0950
- 22: 27/16 905.8654
- 23: 7/4 968.8264
- 24: 225/128 976.5379
- 25: 945/512 1061.005
- 26: 15/8 1088.269
- 27: 63/32 1172.736
- 28: 2025/1024 1180.447
- Wilson CPS 2,12 [1,2,3,4,5,6,8,9,10,12,14,15]:
- 0: 1/1 00.00000
- 1: 135/128 92.17876a
- 2: 35/32 155.1396
- 3: 9/8 203.9100
- 4: 75/64 274.5825
- 5: 5/4 386.3139
- 6: 21/16 470.7811
- 7: 45/32 590.2239
- 8: 3/2 701.9553
- 9: 25/16 772.6278
- 10: 105/64 857.0950
- 11: 27/16 905.8654
- 12: 7/4 968.8264
- 13: 15/8 1088.269
- 14: 63/32 1172.736
Can we construct 16-pitch Wilson CPS
scales using generators which are mostly
factors of one another?
Yes, easily:
- Wilson CPS 3,9 [1,2,3,4,5,6,7,8,9] -- a
16-note CPS. Only a few of the 84
theoretically possible pitches occur
because most of the generators are
factors of one another.
- 0: 1/1 00.00000
- 1: 135/128 92.17876
- 2: 35/32 155.1396
- 3: 9/8 203.9100
- 4: 315/256 359.0498
- 5: 5/4 386.3139
- 6: 81/64 407.8201
- 7: 21/16 470.7811
- 8: 45/32 590.2239
- 9: 189/128 674.6912
- 10: 3/2 701.9553
- 11: 105/64 857.0950
- 12: 27/16 905.8654
- 13: 7/4 968.8264
- 14: 15/8 1088.269
- 15: 63/32 1172.736
Here's another 16-note CPS. This one
comes from 3,9 [1,2,3,4,5,6,8,9,11]:
- 0: 1/1
- 1: 32.32
- 2: 135/128
- 3: 9/8
- 4: 297/256
- 5: 5/4
- 6: 81/64
- 7: 165/128
- 8: 11/8
- 9: 45/32
- 10: 3/2
- 11: 99/64
- 12: 27/16
- 13: 55/32
- 14: 15/8
- 15: 495/256
Here's a 4-out-of-11 CPS with a
surprising number of pitches.
- Wilson CPS 4,11 [1,2,3,4,5,6,8,9,10,12,14]:
- 0: 1/1 00.00000
- 1: 525/512 43.40835
- 2: 135/128 92.17876
- 3: 35/32 155.1396
- 4: 567/512 176.6459
- 5: 9/8 203.9100
- 6: 75/64 274.5825
- 7: 315/256 359.0498
- 8: 5/4 386.3139
- 9: 81/64 407.8201
- 10: 21/16 470.7811
- 11: 675/512 478.4926
- 12: 175/128 541.4535
- 13: 45/32 590.2239
- 14: 189/128 674.6912
- 15: 3/2 701.9553
- 16: 1575/1024 745.3637
- 17: 25/16 772.6278
- 18: 405/256 794.1340
- 19: 105/64 857.0950
- 20: 27/16 905.8654
- 21: 7/4 968.8264
- 22: 225/128 976.5379
- 23: 945/512 1061.005
- 24: 15/8 1088.269
- 25: 243/128 1109.775
- 26: 63/32 1172.736
So far, we've seen only the inky-dinky
little tiny CPSs. Next post, a listing of
some robust CPSs with > 100 pitches.
--mclaren
