From: mclaren
Subject: Wilson CPS scales - post 8 of 9
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Subatomic physics teaches us that most particles
come in pairs: particle and antiparticle. In this
respect the Wilson CPS scales prove eerily
reminiscent of such particles, since all Wilson
CPS tunings also come in pairs. In this case,
the subharmonic Wilson CPS and the harmonic
WIlson CPS.
To date these posts have dealt only with the
harmonic variety.
Let's take an example of a subharmonic CPS:
The subharmonic Wilson hexany 2,4 [1/1,1/3,1/5,.1/7]
is the mirror image of the harmonic Wilson hexany
2,4 [1,3,5,7].
The subharmonic hexany has pitches
16/15 111.7313 cents
The subharmonic hexany has pitches
16/15 111.7313 cents
8/7 231.1741 cents
4/3 498.0452 cents
32/21 729.2194 cents
8/5 813.6866 cents
64/35 1044.860 cents
while the harmonic hexany has pitches
35/32 155.1396 cents
5/4 386.3139 cents
21/16 470.7811 cents
3/2 701.955 cents
7/4 968.8264 cents
15/8 1088.269 cents
The connection between these two hexanies
is simple and self-evident. Flipping the ratios
of the first hexany produces the ratios of the
second: or, if you prefer, subtracting the cents
values of each interval of the first hexany from
second: or, if you prefer, subtracting the cents
values of each interval of the first hexany from
1200 produces the cents values of each interval
in the second hexany.
It is possible, however, to combine both subharmonic and
harmonic generators in a single hexany. Thus one could
also form the Wilson CPS 2,4 [1/1,1/3,1,3], along with
many other partially subharmonic and partially harmonic
hexanies.
To best visualize these mirror pairs of Wilson CPSs
requires ratio space. The harmonic Wilson hexany
can be viewed as points at the vertices of a tetrahedron
which occupies the octants of a 3-dimensional ratio
space--that is, three mutually orthogonal axes with
integer coordinates along the 3^X, 5^Y and 7^Z axes.
(Although the 2,4 [1,3,5,7] hexany nominally contains
4 generators the ratio space describing 1^X is unary
for all values of X so we can effectively omit the 1s
axis from the ratio space.)
for all values of X so we can effectively omit the 1s
axis from the ratio space.)
Then the subharmonic Wilson hexany can be viewed as
a symmetrical collection of points which describe the
vertices of a tetrahedron occupying the octants of the
same 3-dimensional ratio space but on the negative
axes rather than the positive axes.
Thus, the subharmonic WIlson hexany limns a tetrahedron
on the 3 axes 3^[-X],5^[-Y],7^[-Z].
One of the greatest virtues of the subharmonic or harmonic
Wilson CPS is that it constitutes the largest number of
unique harmonic possibilities within the smallest harmonic
distance from one another. Thus the Wilson CPS can be
thought of a particularly elegant solution of the "travelling
salesman" problem in ratio space.
Combining subharmonic and harmonic elements in a single
Wilson CPS eliminates this virtue. Such a CPS no longer
consituttes the largest possible number of unique harmonies
within the smallest harmonic distance from one another.
consituttes the largest possible number of unique harmonies
within the smallest harmonic distance from one another.
This is clear from the fact that one might well choose several
points from the top of the harmonic tetrahedron (in the case
of a hexany) and several points from the bottom of the
subharmonic tetrahedron. Such a construct will surely
require the traversal of more distance in ratio space than
either a fully subharmonic or a fully harmonic Wilson CPS.
However, the subharmonic Wilson CPS does have a distinctly
different "sound" from the harmonic Wilson CPS--just as
the subharmonic series has a different "sound" from the
harmonic Wilson CPS. It is also true that all s 2,4 [3,5,7,1/11] -> 2,4 [3,5,1/7,1/11]
-> 2,4 [3,1/5,1/7,1/11] -> 2,4 [1/3,1/5,1/7,1/11].
Again this is equivalent to following the edges of facets
of geoemtric figures in a 4-dimensional space. The
total colection of such paths describes a larger
geometric figure in which both the harmonic and
subharmonic Wilsons CPSs are embedded, and of
which both can be considered parts.
Next post, some extensions of the Wilson CPS
which both can be considered parts.
Next post, some extensions of the Wilson CPS
using embedded JI scales as generators.
--mclaren
