From Brian McLaren

From: mclaren
Subject: Erv Wilson's CPS tunings
--
John Chalmers and Paul Erlich have
perceptively pointed out some issues
in regard to Erv Wilson's CPS scales.
Those of you not familiar with such
tunings would be well advised either
to go back and look at Topics 1, 2, 3 and
4 from Tuning Digest 30, Topic 2
from Tuning Digest number 31, or order
MusicWorks 60 from Gayle Young
and read Paul Rapoport's article "Just
Shape, Nothing Central." Paul Rapoport
is to date one of the keenest minds and
best writers to have explained these
types of tunings in detail, along with
the inestimable John Chalmers. Kraig Grady's
1989 1/1 article "Erv Wilson's Hexany" deals
with a subset of the Wilson CPS scales
and does so in some detail, but does not
concern itself with many of the implications
of Wilson's CPS tunings, nor with the many
operations which can be performed on
Wilson CPS tunings (stellation, cross-product
sets, etc.). Paul Rapoport has explained these
ideas with admirable clarity, and John Chalmers
has also explained them very well. Warren
Burt's Topic 2 of Digest 31 is the clearest
explanation of cross-product sets written
to date.
--
Paul Erlich pointed out that my lists of Wilson
CPS tunings sometimes added a "confusing and
extraneous "1/1. It is probably true that adding
the implied 1/1 was confusing--so let me take
a stab at explaining why Erv Wilson doesn't,
and why I sometimes do.
The Wilson CPS tunings can be highly tonal
but never have a 1/1 in the form Erv Wilson
generates them. Erv typically starts with 1
and takes some mutually prime set of
numbers and generates an abstract set of
points in ratio space by combinatorial
methods. (See Tuning Digest 30 if you're not
familiar with this process.) This abstract
set of points in ratio space is not strictly
speaking a tuning, but a framework within
which individual tunings can "collapse" down
into acoustic space.
What I'm talking about here is akin conceptually
to the process by which an equal-tempered
chromatic scale generates a melodic mode.
Depending on which note of the chromatic scale
one starts on, quite different-sounding modes
can be generated. Thus, phrygian sounds entirely
different lydian or dorian--yet all are theoretically
contained within the 12-TET chromatic scale.
Similarly, the process of getting individual tunings
(as on a guitar or a synthesizer or a set of
tubulongs) from abstract points in ratio space
is a process of projection from a higher-
dimensional ratio space into a lower-dimensional
acoustic space. This process is familiar to
students of analytic geometry: the family of
conic curves is usually derived by explaining
them as projections of 3 dimensional lines
inscribed on the surface of a 3-D cone down onto
the two-dimensional space of graph paper.
Depending on the angle of the cone to the
2-D graph paper, one can obtain an hyperbola,
a parabola, etc. A classic example of projective
geometry--one of the more fascinating byways
of mathematics.
Erv Wilson's CPS structures in ratio space
exhibit a number of paradoxical properties.
If low prime numbers are used as generators,
the tunings will strongly imply tonality--
because the root motions between three-
pitch "facets" (i.e., triads) in ratio space
are ratios like 3/2, 4/3, &c. Yet, paradoxically,
the CPS structure itself in ratio space
has no 1/1 and thus no tonal center--until
it is collapsed down into acoustical space
by dividing the entire set of ratio space
points by the ratio of a single ratio space
point. (I.e., by choosing some ratio as a
1/1 and dividing all the other ratios by
that ratio.)
This process has the effect both of centering
the acoustic representation around one
particular point in ratio space--in effect,
making some arbitrary point in the n-space
lattice the origin of the ratio space
coordinates--and it also has the effect
of making the individual acoustic intervals
more complex than they were in the original
abstract ratio-space representation because
they're now all multiplied by the inverse of
the ratio chosen as 1/1 (that is, divided by
the ratio chosen as 1/1).
Erv Wilson's CPS tunings are not really
atonal when low integer generators are used.
They become increasingly atonal-sounding
as higher and ever higher integers are used
as members of the generating set.
However, because any other point of the
ratio space lattice can be chosen at any
time as a different 1/1, the Wilson CPS
tunings allow maximum modulatory
flexibility with a minimum set of pitches.
Partch's diamond is less efficient in modulatory
terms than Wilson's CPS structures in
ratio space because moving from one
section of the two-dimensional Partch
ratio-space diamond to the next does not
produce a maximum change in triadic notes
with a minimum length of ratio-space
movement.
You could think of this in projective geometry
terms: the length of the geodesic in n-space
between two points separated by more than
2 dimensions will always be shorter than
the length of the geodesic on a plane between
the same two points projected on a plane.
This is nothing more complicated that the
very familiar problem of projecting 3-D
objects onto a 2-D plane. Map-makers have
for millenia known that projecting coordinates
on a sphere down onto a 2-D plane creates
large distortions in some of the distances
between some of the coordinates. Some
distances which are short in the 3-D original
are unnaturally and deceptively long on
the 2-D plane; and so it is with ratio space
projected onto a plane--as with Partch's
ratio diamond, which can be (as has been,
by Erv Wilson) visualized as a much more
compact higher-dimensional ratio-space
structure.
Another way of thinking of the Wilson CPS
method is that it provides a ratio-space
coordinate system within which acoustic
structures can be rotated with maximum
symmetry. To put it another way, as
the size of the Partch diamond increases,
the amount of symmetry-breaking increases
as the square as the number of prime
generators. By "amount of symmetry-breaking"
I mean here the number of anisometric points
in the grid, each of which has the potential
to break symmetry upon rotation of a musical
structure across the grid.
Example: In the 4 5 6 7 1/4 1/5 1/6 1/7
Partch diamond with 4 generators there
are 4 1/1's--1/1, 3/3, 5/5 and 7/7.
This anisometric diagonal of 1/1s acts
in effect like an axis. It has the effect of
breaking the symmetry of 2-D ratio
space structures when they're rotated
through the 2-D ratio space of the Partch
diamond. But in a Wilson CPS *there is
no 1/1*, and thus there's *never* any
example of symmetry-breaking when
substructures are rotated through
ratio space. (Wilson CPS structures
are inherently symmetrical: thus their
symmetry cannot collapse in ratio space
but only when they are projected
into some anisometric lower-dimensional
coordinate system.)
Physicists know that many physical
processes can be described in terms
of broken symmetry: the nucleation of
snowflakes in a supersaturated
air, the spontaneous transformation
of isotherms into tornadic weather
patterns, the formation of new crystals through
screw dislocations, and the fracture of the
single unified force after the intial few
femtoseconds of the Big Bang into the 4
fundamental universal forces of gravitation,
electromagnetism, strong and weak nuclear
forces, are all classic examples of symmetry-
breaking which leads to emergent order.
In the same way, symmetry breaking is
required to generate an emergent order
out of the abstract ratio space structures
of the Wilson CPS or the Partch diamond.
This leads us to a concise explanation of
why I sometimes add an "extraneous"
1/1 to the Wilson CPS.
Seen from the perspective of symmetry
breaking as a means of generating scalar
order, it's not obvious why choosing
one of the CPS lattice points as a 1/1
is significantly different from adding
a 1/1 point at the lowest-dimensional point
of the structure in ratio space. Both
procedures break the symmetry of the
abstract ratio-space CPS; both procedures
collapse the higher-dimensional structure
down into a lower-dimensional acoustic
space.
The advantages of adding a 1/1 at the lowest-
dimensional point of the ratio space Wilson
CPS are: [A] the acoustic ratios are maximally
simplified and so it's especially easy to hear
the underlying harmonic progressions. [B] the
acoustic distance from the 1/1 to any given
scale-step is minimized. The disadvantages
of adding such an "extraneous" 1/1 in ratio
space are: [C] modulation is no longer
maximally efficient--that is, modulations
no longer produce maximum change in
number of chord components for minimum
change in root position; [D] it is no longer
simple to modulate to another substructure
embedded within the ratio space CPS.
These advantages and disadvantages must
be weighed carefully. In fact, Erv Wilson
often adds "extraneous" pitches to his
CPS tunings--there is no reason whatever
why I can't add "extraneous" pitches too.
The point is that the process is not black-&-
white, good-or-bad. Simply because Erv
Wilson doesn't do something (and even Erv
sometimes refers to the "nonexistent"
1/1 at the lowest-dimensional point in
ratio space in the process of theorizing;
he regards as the null operation in which
all axes of the ratio space are raised to
the zeroth power) doesn't mean it can't
or oughtn't to be done.
--mclaren




Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Tue, 13 Aug 1996 21:16 +0200
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA02160; Tue, 13 Aug 1996 21:17:01 +0200
Received: from eartha.mills.edu by ns (smtpxd); id XA02166
Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
for id MAA17431; Tue, 13 Aug 1996 12:16:58 -0700
Date: Tue, 13 Aug 1996 12:16:58 -0700
Message-Id: <13960813191331/0005695065PK5EM@MCIMAIL.COM>
Errors-To: madole@ella.mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu