CPSs with no factor of 1

Actually, CPSs with no factor of 1 arise naturally in the construction
of Euler genera from CPS building blocks. For example, the Euler genus
3.5.7 can be constructed from the CPSs 1)4[1,3,5,7], 2)4[1,3,5,7], and
3)4[1,3,5,7]. The intersection of the first two is the CPS 1)3[3,5,7],
and the intersection of the last two is 2)3[3,5,7]. So CPSs without a
factor of 1 are "where you put the glue" when constructing Euler genera.
In the triangular lattice, the CPSs are symmetrical structures, while
the Euler genera are skewed parallelopipeds. In fact, according to the
triangular lattice philosophy, there are many other structures just as
simple as Euler genera. For example, if you use 1)3 [1,3,5] and 2)3
[1,3,5] as the connecting faces for the first three CPS mentioned above,
you get such a structure. (The individual CPSs will, in general, have to
be transposed with respect to the raw multiplication results; the fact
that you don't have to transpose them in the construction of the Euler
genus is a mere mathematical curiosity.) More interesting is what
happens when you use all possible connecting faces of a given CPS; at
least in some cases, this results in what Erv Wilson called a
"stellation." Or using all possible 1)n or (n-1))n CPSs connected to a
given point; that results in a Partch diamond if the n factors are
consecutive odd numbers starting with 1. . .

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David Finnamore wrote:

''But I think it's a different kind of dimensionality, if you will, from
the
prime type. I can't fully explain this concept yet, but to me, prime, odd,
and scale-interval-generator are three distinctly different and equally
valid
kinds of dimensionality''

I believe that definitions like these have a lot to do with what exactly
you want
your music theory to do. If you are making a pre-compositional theory, then
you
will probably want to locate and organize aspects of the materials that you
intend
to project structurally, so as to make these aspects them audible throughthe way
that you articulate them. If you are making a theory for interpreting (asa
listener
or a performer) an existing work, then the theory should explain how the
surface
articulations reflect physical characteristics of the materials. Both of
these kinds
of theories are based upon the assumption that the way musical materials
are
articulated is essential to their interpretation.

On the other hand, if your intention is to make a claim for the
psychophysical status
of your materials, you are initially freed from accounting for such
personal or cultural
constructions as described above, but the completion of a psychophysical
approach
depends upon eventually being able to address these concerns. In my
experience, existing psychophysical theories are either weak or culturally local.
(There are some minor innate behavior mechanisms associated with sounds - my favorite
example is the
taste of Carlsberg Beer while listening to particular sine wave frequencies
- but the
culturally aquired patterns seem to be overriding; see, for example,
Gilbert Rouget _Music and Trance_).

Let's look at some real examples: (1) I compose a piece consisting only of chromatically planed (parallel)
Major ninth
chords without fifths in just intonation. If the orchestration is such that
a listener can hear the
individual voices of these chords, then 9 is being articulated
independently of 3. Since
9 is here relatively prime, how could 9 be here distinguished a 'real'
prime?

(2) I write a chorale in triads that goes I - IV - V - I. There is a 9:8
relationship
between IV and V chords, but this is usually weakly articulated and the Vof V (3^2)
relationship is heard as more important. (3) I compose a sound installation consisting of standing sine waves using
only prime
multiples of a given fundamental and 9 times that fundamental. Depending
upon where
I am in the room, I will hear the 9 as either an independent entity or inconnection with 3.

(4) Using a TX81Z (with a resolution of 1.56 cent deviations from 12tet),I
perform
a melody based upon random walks over a lattice with two dimensions: 3s and
9s. The
best approximation of 9 does not coincide with the best approximation of
3^2. For all practical purposes, 3 and 9, in this temperament, are
relatively prime.

(5) A composition is played one day on a gamelan tuned to a 7 pitches outof 9TET, on
the next to a Lou Harrison-style just intonation pelog, on the next to a
Wilson
style linear series in just intonation, and finally on a real
unsystematically-tuned Javanese instrument. To what degree is the composition intonation-transparent? Itis interesting to hear how certain structures survive radically differentmappings intact as well as to hear what features are projected differently
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