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From: mclaren

Subject: Wilson CPS tunings - post 1 of 9

Here, to start off as simply as possible, is a list of the number of tones (< 300) in Wilson CPSs for which all generators are mutually prime:

2 out of 3 generators: 3 tones

2 out of 4 generators: 6 tones

2 out of 5 generators: 10 tones

2 out of 6 generators: 15 tones

2 out of 7 generators: 21 tones

2 out of 8 generators: 28 tones

2 out of 9 generators: 36 tones

2 out of 10 generators: 45 tones

2 out of 11 generators: 55 tones

2 out of 12 generators: 66 tones

2 out of 13 generators: 78 tones

2 out of 14 generators: 91 tones

2 out of 15 generators: 105 tones

2 out of 16 generators: 120 tones

2 out of 17 generators: 136 tones

2 out of 18 generators: 153 tones

2 out of 19 generators: 171 tones

2 out of 20 generators: 190 tones

2 out of 21 generators: 210 tones

3 out of 4 generators: 4 tones

3 out of 5 generators: 10 tones

3 out of 6 generators: 20 tones

3 out of 7 generators: 35 tones

3 out of 8 generators: 56 tones

3 out of 9 generators: 84 tones

3 out of 10 generators: 210 tones

4 out of 5 generators: 5 tones

4 out of 6 generators: 15 tones

4 out of 7 generators: 35 tones

4 out of 8 generators: 70 tones

4 out of 9 generators: 126 tones

4 out of 11 generators: 165 tones

4 out of 12 generators: 220 tones

5 out of 6 generators: 6 tones

5 out of 7 generators: 21 tones

5 out of 8 generators: 56 tones

5 out of 9 generators: 126 tones

5 out of 10 generators: 210 tones

6 out of 7 generators: 7 tones

6 out of 8 generators: 28 tones

6 out of 9 generators: 84 tones

6 out of 10 generators: 210 tones

7 out of 8 generators: 8 tones

7 out of 9 generators: 36 tones

7 out of 10 generators: 120 tones

8 out of 9 generators: 9 tones

8 out of 10 generators: 45 tones

8 out of 11 generators: 165 tones

9 out of 10 generators: 10 tones

9 out of 11 generators: 55 tones

9 out of 12 generators: 220 tones

10 out of 11 generators: 11 tones

10 out of 12 generators: 66 tones

10 out of 13 generators: 286 tones

11 out of 12 generators: 12 tones

11 out of 13 generators: 78 tones

12 out of 13 generators: 13 tones

12 out of 14 generators: 91 tones

13 out of 14 generators: 14 tones

13 out of 15 generators: 105 tones

14 out of 15 generators: 15 tones

14 out of 16 generators: 120 tones

15 out of 16 generators: 16 tones

15 out of 17 generators: 136 tones

16 out of 17 gnerators: 17 tones

16 out of 18 generators: 153 tones

17 out of 18 generators: 18 tones

17 out of 19 generators: 171 tones

18 out of 19 generators: 19 tones

18 out of 20 generators: 190 tones

19 out of 20 generators: 20 tones

19 out of 21 generators: 210 tones

20 out of 21 generators: 21 tones

20 out of 22 generators: 231 trones

21 out of 22 generators: 22 tones

21 out of 23 generators: 253 tones

22 out of 23 generators: 23 tones

22 out of 24 generators: 276 tones

All other Wilson CPS scales in which *all* generators are mutually prime have 300 or more tones, and are left as an exercise for the interested forum subscriber to calculate.

Several rules of thumb for mutually- prime-generator Wilson CPS scales:

Unless the Wilson CPS contains generators 1, 3 or a transposition thereof, there won't be a 3/2 in the scale. Thus, as a Wilson CPS scale climbs higher and higher in the prime series of generators, the gap where 3/2 used to be becomes ever more densely surrounded with exotic ratios. For highly gapped prime generators, the Wilson CPS has NO familiar intervals.
Example: for the Wilson CPS 2,4 and generator [1,3,5,7] the 3/2 interval appears.

But for the Wilson CPS 2,4 and generator [1,5,11,17] there is no 3/2, only a 5/4.

However, for the Wilson CPS 2, 4 and generator [1,11, 19, 31] no familiar intervals appear. This Wilson Hexany exhibits no familiar 6/5, 5/4, 3/2 or other standard JI intervals. (The notation X,Y indicates "X out of Y generators." [A,B,C,D] indicates a set of Y generators from which X at a time are taken to form ratios to the product of A*B*C*D.)
As the number of tones in the Wilson CPS grows larger, the smallest intervals in the scale become smaller and the largest intervals become larger. Moreover, the scale tends to "scrunch up" toward the middle--that is, the largest gaps in the scale are increasingly found near 1/1 and 2/1 and the region twixt 3/2 and 4/3 becomes ever more densely populated with small intervals as the number of tones in the CPS increases.
As the prime integer generators grow individually larger, the smallest gaps in the scale grow proportionately smaller.
Thus the smallest intervals in a Wilson CPS 2,4 [1,3,5,7] will be much larger than the smallest intervals in a Wilson CPS 2,4 [1, 31, 79, 137].
This is easily demonstrated. The 2,4 [1,3,5,7] scales has intervals:
```
1       155.1396 cents
2        386.3139 cents
3        470.7811 cents
4        701.9563 cents
5        968.8264 cents
6        1088.269 cents
Smallest interval = 102.47 cents
```
while the 2,4 [1,31,79,137] scale has intervals:
```
1       62.6741 cents
2       117.6385 cents
3       309.5776 cents
4       364.5370 cents
5       482.1756 cents
6       1145.036 cents
Smallest interval = 54.95 cents.
```
All Wilson CPS scales of order M contain sub-CPSs of the next lowest order. Example: A 20-tone Wilson 2,5 scale contains 6 different 2,4 scales inside itself.
Thus the 70-tone hebdomekontany contains 20 different dodekanies within itself, etc. This radically increases the possibilities for modulation since sub-CPSs can modulate as well as individual notes.
If the generators are prime and of the same relative magnitude except for one which spikes up to a much higher prime, the net result is that all the intervals are "knocked askew" by a few cents from the continuation of the series without the higher prime.
Thus, the hexany 2,4 [1,3,5,7] will have intervals close to--but systematically off by a few cents from--the hexany 2, 4 [1,3,5,97].
Moreover, the *higher* the single prime that "spikes up," the *smaller* the number of cents by which the intervals of the CPS will be "knocked askew." Thus the hexany 2,4 [1,3,5,101] will be farther off the hexany 2,4 [1,3,5,7] than the much higher-spiking hexany 2,4 [1,3,5,1367]. The [1,3,5,101] hexany is knocked about 10 cents/note off the [1,3,5,7] values while the [1,3,5,1367] hexany is perturbed by only about 5 cents/note.

At this point it would seem that the number of tones in a Wilson CPS is strictly constrained by the combination equation which defines the number of combinations of n out of m things. Thus there is obviously no such thing as a Wilson 17-tone scale, unless one chooses the trivial option of taking 16 out of a 17 generators at a time.

Naturally, since it's obvious that this must be true, it's not so...as will be seen in the next post.

--mclaren