From: mclaren
Subject: Wilson CPS scales - post 9 of 9
So far a number of the properties
of the Wilson combination product
set scales have been explored. Further
subtleties remain.
Because the Wilson CPS scale-generation
method is insensitive to the exact
character of the generators as long
as they're mutually prime, it's possible
to embed entire just tunings or entire
equal temperaments in Wilson CPSs.
Example:
- Kathleen Schlesinger's Dorian Enharmonic
tuning can be used as a set of 12 generators.
- Taking 2,12 [1/1,16/15,8/7,64/55,128/109,
4/3,32/22,64/43,128/85,32/21,16/9]
produces a valid Wilson CPS:
- 2,11 [kathleen schlesinger's dorian enharmonic]:
- 1: 256/255 6.775878
- 2: 64/63 27.26410
- 3: 1024/981 74.26884
- 4: 104/99 85.29975
- 5: 16/15 111.7313
- 6: 512/473 137.1644
- 7: 1024/935 157.4130
- 8: 256/231 177.9012
- 9: 4096/3655 197.2132
- 10: 1024/903 217.7014
- 11: 8/7 231.1741
- 12: 2048/1785 237.9500
- 13: 128/109 278.1789
- 14: 13/11 289.2098
- 15: 32/27 294.1351
- 16: 128/105 342.9055
- 17: 2048/1635 389.9102
- 16: 128/105 342.9055
- 17: 2048/1635 389.9102
- 18: 208/165 400.9411
- 19: 128/99 444.7722
- 20: 512/387 484.5725
- 21: 4/3 498.0452
- 22: 1024/765 504.8211
- 23: 1024/763 509.3531
- 24: 104/77 520.3840
- 25: 256/189 525.3093
- 26: 1664/1199 567.3887
- 27: 64/45 609.7766
- 28: 16/11 648.6823
- 29: 64/43 688.4826
- 30: 128/85 708.7312
- 31: 32/21 729.2194
- 32: 256/165 760.4136
- 33: 512/327 776.2241
- 34: 52/33 787.2550
- 33: 512/327 776.2241
- 34: 52/33 787.2550
- 35: 1024/645 800.2139
- 36: 2048/1275 820.4625
- 37: 512/315 840.9508
- 38: 128/77 879.8565
- 39: 512/301 919.6567
- 40: 2048/1199 926.8612
- 41: 208/121 937.8922
- 42: 1024/595 939.9053
- 43: 256/147 960.3936
- 44: 8192/4687 966.6616
- 45: 832/473 977.6925
- 46: 16384/9265 986.9101
- 47: 16/9 996.0905
- 48: 1664/935 997.9411
- 49: 4096/2289 1007.398
- 50: 416/231 1018.429
- 51: 256/135 1107.821
- 50: 416/231 1018.429
- 51: 256/135 1107.821
- 52: 64/33 1146.727
- 53: 256/129 1186.527
Schlesinger's entire scale is not embedded
in this Wilson CPS superset; however, most
of the pitches are present. Only the 64/55
and the 32/22 are missing. For practical
purposes, however, the other Wilson CPS
pitches fall so close to these that any
differences would be inaudible--so in
practical terms the Schlesinger scale
is present inside the Wilson CPS
superset. This raises the possibility of
generating superset tunings based on
attractive just arrays; and since most
just arrays are symmetrical, only 1/2
the pitches would be required as
generators.
Equal-tempered tunings can be embedded
in the same way. A 17-tone equal-tempered
scale can be approximated by the ratios
1/1 776/745 166/153 304/269
206/175 244/199 576/451
294/221 503/363 433/311 721/147
The scale is symmetrical about 721/147;
by inverting the previous ratios the complete
17-TET scale is obtained to within 0.01 cents
per scale-step.
This opens the possibility of generating a
Wilson CPS from 2,10 [1/1, 776/745,166/153
304/269,206/175,244/199,576/451,
294/221,503/363,433/311,721/147].
Again, the entire 17-TET scale would not be
present but those pitches out of tune would
be detuned by such a minuscule amount as
present but those pitches out of tune would
be detuned by such a minuscule amount as
to be effectively members of 17.
Thus any given equal temperament or just
intonation tuning can serve as the nucleus
of a Wilson CPS.
This offers a novel and heretofore unexplored
method of expanding tunings to obtain a superset
of the given scale. Subsets (AKA modes) are
more familiar--but this approach, which uses
any given equal temperament or just array as
a mode of the superset tuning, promises to
prove equally useful.
Many more properties of Wilson CPS tunings
remain to be discussed, but as always there's
no time.
--mclaren
