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
