From: mclaren
Subject: Wilson CPSs--part 5 of 9
Continuing with a listing of some
combination product sets with unusual
of pitches, next up is the 35-pitch
Wilson CPS 3,7 [1,3,5,7,9,11,13]:
(First 7 odd numbers)
0: 1/1 00.00000
1: 65/64 26.84138
2: 33/32 53.27296
3: 135/128 92.17876
4: 273/256 111.3086
5: 35/32 155.1396
6: 143/128 191.8457
7: 585/512 230.7514
8: 297/256 257.1830
9: 77/64 320.1440
10: 39/32 342.4828
11: 315/256 359.0498
12: 1287/1024 395.7557
13: 165/128 439.5868
14: 21/16 470.7811
15: 693/512 524.0540
16: 351/256 546.3928
17: 715/512 578.1596
18: 45/32 590.2239
19: 91/64 609.3538
20: 189/128 674.6912
21: 385/256 706.4579
22: 195/128 728.7967
23: 99/64 755.2283
24: 819/512 813.2639
25: 105/64 857.0950
26: 429/256 893.8010
27: 27/16 905.8654
28: 55/32 937.6320
29: 455/256 995.6677
30: 231/128 1022.099
31: 117/64 1044.438
32: 15/8 1088.269
33: 495/256 1141.542
34: 1001/512 1160.672
35: 63/32 1172.736
Next, Wilson CPS 5, 10 [1,3,5,7,9,11,13,17,19].
There are only a few generators in this set
which are powers of other generators: thus this
scale is substantially larger than the 5 out
of [1,2,3,4,5,6,7,8,9,10] CPS.
1: 65637/65536 2.666017
2: 65835/65536 7.880584
3: 264537/262144 15.73200
4: 16575/16384 20.06550
5: 2079/2048 26.00886
6: 266475/262144 28.36881
7: 33345/32768 30.21940
8: 8415/8192 46.49708
9: 135135/131072 52.85025
10: 33915/32768 59.56307
11: 17017/16384 65.62695
12: 4275/4096 74.05048
13: 2145/2048 80.11435
14: 34425/32768 85.40287
15: 8645/8192 93.18034
16: 138567/131072 96.26907
17: 17325/16384 96.68132
18: 69615/65536 104.5327
19: 8721/8192 108.3334
20: 8775/8192 119.0201
21: 4389/4096 119.6119
22: 141075/131072 127.3234
23: 35343/32768 130.9643
24: 566865/524288 135.1748
25: 17765/16384 140.1001
26: 4455/4096 145.4517
27: 285285/262144 146.4533
28: 8925/8192 148.3638
29: 17955/16384 158.5177
30: 9009/8192 164.5815
31: 72675/65536 179.0059
32: 36465/32768 185.0698
33: 73359/65536 195.2237
34: 2295/2048 197.1342
35: 146965/131072 198.1358
36: 36855/32768 203.4873
37: 1155/1024 208.4126
38: 18525/16384 212.6232
39: 4641/4096 216.2640
40: 149175/131072 223.9756
41: 74613/65536 224.5673
42: 75075/65536 235.2540
43: 9405/8192 239.0547
44: 37791/32768 246.9062
45: 4725/4096 247.3184
46: 75735/65536 250.4071
47: 19019/16384 258.1846
48: 305235/262144 263.4731
49: 153153/131072 269.5370
50: 38475/32768 277.9605
51: 19305/16384 284.0244
52: 4845/4096 290.7373
53: 77805/65536 297.0904
54: 19635/16384 313.3681
55: 2457/2048 315.2187
56: 314925/262144 317.5786
57: 39501/32768 323.5220
58: 2475/2048 327.8555
59: 9945/8192 335.7069
60: 9975/8192 340.9215
61: 159885/131072 344.0102
62: 5005/4096 346.9853
63: 80325/65536 352.2739
64: 5049/4096 362.1385
65: 323323/262144 363.1401
66: 20349/16384 375.2045
67: 40755/32768 377.6275
68: 20475/16384 385.8911
69: 328185/262144 388.9799
70: 2565/2048 389.6919
71: 5187/4096 408.8217
72: 10395/8192 412.3227
73: 166725/131072 416.5333
74: 41769/32768 420.1741
75: 20995/16384 429.3100
76: 42075/32768 432.8110
77: 5265/4096 434.6615
78: 84645/65536 442.9648
79: 169575/131072 445.8769
80: 85085/65536 451.9408
81: 10659/8192 455.7415
82: 171171/131072 462.0947
83: 5355/4096 464.0052
84: 10725/8192 466.4282
85: 692835/524288 482.5829
86: 43605/32768 494.6473
87: 1365/1024 497.6225
88: 21879/16384 500.7112
89: 21945/16384 505.9258
90: 88179/65536 513.7772
91: 176715/131072 517.2781
92: 11115/8192 528.2646
93: 22275/16384 531.7656
94: 89505/65536 539.6170
95: 2805/2048 544.5422
96: 89775/65536 544.8316
97: 45045/32768 550.8954
98: 5643/4096 554.6962
99: 11305/8192 557.6082
100: 2835/2048 562.9598
101: 182325/131072 571.3836
102: 366795/262144 581.5376
103: 11475/8192 583.4481
104: 46189/32768 594.3143
105: 5775/4096 594.7265
106: 23205/16384 602.5780
107: 373065/262144 610.8812
108: 46683/32768 612.7318
109: 2925/2048 617.0653
110: 47025/32768 625.3687
111: 11781/8192 629.0095
112: 188955/131072 633.2200
113: 1485/1024 643.4969
114: 95095/65536 644.4985
115: 5967/4096 651.3483
116: 5985/4096 656.5629
117: 95931/65536 659.6516
118: 3003/2048 662.6268
119: 48195/32768 667.9153
120: 96525/65536 670.3383
121: 24225/16384 677.0511
122: 12155/8192 683.1150
123: 24453/16384 693.2689
124: 98175/65536 699.6820
125: 12285/8192 701.5325
126: 197505/131072 709.8359
127: 49725/32768 722.0208
128: 24871/16384 722.6126
129: 100035/65536 732.1747
130: 3135/2048 737.1000
131: 12597/8192 744.9514
132: 1575/1024 745.3637
133: 25245/16384 748.4524
134: 101745/65536 761.5184
135: 203775/131072 763.9414
136: 51051/32768 767.5822
137: 12825/8192 776.0057
138: 3213/2048 779.6466
139: 6435/4096 782.0697
140: 25935/16384 795.1356
141: 415701/262144 798.2244
142: 51975/32768 798.6366
143: 208845/131072 806.4880
144: 6545/4096 811.4133
145: 26325/16384 820.9754
146: 13167/8192 821.5672
147: 3315/2048 833.7521
148: 53295/32768 842.0554
149: 6669/4096 843.9061
150: 26775/16384 850.3191
151: 53865/32768 860.4730
152: 27027/16384 866.5369
153: 6783/4096 873.2497
154: 13585/8192 875.6727
155: 218025/131072 880.9613
156: 6825/4096 883.9364
157: 109395/65536 887.0251
158: 109725/65536 892.2397
159: 6885/4096 899.0895
160: 440895/262144 900.0911
161: 3465/2048 910.3679
162: 55575/32768 914.5785
163: 13923/8192 918.2194
164: 223839/131072 926.5227
165: 14025/8192 930.8562
166: 1755/1024 932.7067
167: 28215/16384 941.0101
168: 113373/65536 948.8615
169: 14175/8192 949.2737
170: 57057/32768 960.1400
171: 1785/1024 962.0505
172: 3591/2048 972.2044
173: 230945/131072 980.6282
174: 57915/32768 985.9797
175: 116025/65536 988.8919
176: 14535/8192 992.6926
177: 7293/4096 998.7565
178: 233415/131072 999.0457
179: 7315/4096 1003.971
180: 29393/16384 1011.822
181: 58905/32768 1015.323
182: 3705/2048 1026.309
183: 7425/4096 1029.810
184: 29835/16384 1037.662
185: 29925/16384 1042.876
186: 479655/262144 1045.965
187: 15015/8192 1048.940
188: 945/512 1061.005
189: 61047/32768 1077.159
190: 122265/65536 1079.582
191: 3825/2048 1081.493
192: 61425/32768 1087.846
193: 7695/4096 1091.647
194: 3861/2048 1097.711
195: 7735/4096 1100.623
196: 124355/65536 1108.926
197: 15561/8192 1110.777
198: 31185/16384 1114.278
199: 15675/8192 1123.413
200: 3927/2048 1127.054
201: 62985/32768 1131.265
202: 126225/65536 1134.766
203: 255255/131072 1153.896
204: 1995/1024 1154.608
205: 31977/16384 1157.697
206: 16065/8192 1165.960
207: 32175/16384 1168.383
208: 2025/1024 1180.447
209: 129675/65536 1181.449
210: 8151/4096 1191.314
211: 130815/65536 1196.602
212: 4095/2048 1199.577
Wilson CPS 5 out of 10 generators:
5 out of [1,2,3,5,6,7,9,10,11,13]
This is another highly redunant set:
1: 2079/2048 26.00886
2: 525/512 43.40835
3: 1053/1024 48.34768
4: 2145/2048 80.11435
5: 135/128 92.17876
6: 17325/16384 96.68132
7: 273/256 111.3086
8: 8775/8192 119.0201
9: 275/256 123.9454
10: 4455/4096 145.4517
11: 9009/8192 164.5815
12: 567/512 176.6459
13: 2275/2048 181.9810
14: 1155/1024 208.4126
15: 585/512 230.7514
16: 4725/4096 247.3184
17: 297/256 257.1830
18: 75/64 274.5825
19: 19305/16384 284.0244
20: 2457/2048 315.2187
21: 2475/2048 327.8555
22: 5005/4096 346.9853
23: 315/256 359.0498
24: 20475/16384 385.8911
25: 1287/1024 395.7557
26: 81/64 407.8201
27: 10395/8192 412.3227
28: 325/256 413.1552
29: 5265/4096 434.6615
30: 165/128 439.5868
31: 10725/8192 466.4282
32: 675/512 478.4926
33: 1365/1024 497.6225
34: 693/512 524.0540
35: 175/128 541.4535
36: 351/256 546.3928
37: 45045/32768 550.8954
38: 2835/2048 562.9598
39: 715/512 578.1596
40: 45/32 590.2239
41: 5775/4096 594.7265
42: 11583/8192 599.6658
43: 2925/2048 617.0653
44: 1485/1024 643.4969
45: 3003/2048 662.6268
46: 189/128 674.6912
47: 12285/8192 701.5325
48: 385/256 706.4579
49: 6237/4096 727.9642
50: 195/128 728.7967
51: 25025/16384 733.2993
52: 1575/1024 745.3637
53: 99/64 755.2283
54: 6435/4096 782.0697
55: 405/256 794.1340
56: 819/512 813.2639
57: 825/512 825.9007
58: 105/64 857.0950
59: 27027/16384 866.5369
60: 6825/4096 883.9364
61: 429/256 893.8010
62: 3465/2048 910.3679
63: 1755/1024 932.7067
64: 891/512 959.1383
65: 3575/2048 964.4735
66: 225/128 976.5379
67: 455/256 995.6677
68: 7371/4096 1017.174
69: 231/128 1022.099
70: 7425/4096 1029.810
71: 117/64 1044.438
72: 15015/8192 1048.940
73: 945/512 1061.005
74: 1925/1024 1092.771
75: 3861/2048 1097.711
76: 975/512 1115.110
77: 495/256 1141.542
78: 1001/512 1160.672
79: 32175/16384 1168.383
80: 63/32 1172.736
81: 2025/1024 1180.447
82: 4095/2048 1199.577
This concludes the list of simple Wilson CPS
scales to be posted. Such a list, however,
barely scratches the surface. It is far
from being exhaustive . In fact, quite the
contrary--it may have deceived some of
you into imagining these are the only or the
most interesting Wilson CPS scales. Not so.
There are infinitely many Wilson CPS scales,
some similar-sounding to others, yet other
Wilson CPSs sounding entirely alien to
near neighbors.
For example, the simple Wilson hexany
2,4 [17 19,23,29] is my favorite hexany
so far--yet it sounds nothing like the
classic Wilson hexany 2,4 [1,3,5,7] (which
is usually described as a set of intertwined
just 7th tetrads).
The final post in this mini-series deals
with musical uses of Wilson CPSs and extensions
of the CPS idea as specified in my 1990 article
"General Methods of Generating Musical Scales"
in Xenharmonikon 13...an article which none of
you will ever bother to peruse, in a journal for which
most of you have nothing but contempt and disdain
(since Xenharmonikon is published and vetted entirely
outside of academia).
--mclaren
