Reply to Doren Garcia

>I am unfamiliar with the aesthetic of the school of 'New Complexity'. Can
>someone offer a synopsis of their ideas?

I don't know much about this, but I don't think it is so much a "school" as
a style defined from without. Perhaps the music journalists concieved of it
a a dialectical antithesis to minimalism, i.e., the "new simplicity." Nor is
it all that new, as a Frank Zappa book from the early 80's said that Steve
Vai's painstaking transcriptions of Zappa's improvisations "look like
something out of the new complexity." One defining feature is a metric and
rhythmic complexity that would scare away most MIDI programmers, let alone
live performers. Some writers have claimed complexity as a coherent movement
in recent thought, lumping the musical style in with choas theory in
physics, etc. Sounds like hogwash to me.

>Can anyone recommend
>a book that will explain concepts such as, Wilson CPS scales, subharmonics
>and the associated mathematical principles involved?

Subharmonics are just upside-down harmonics. The literature on subharmonics
is vast; the opinions on their applicability to music theory range from not
at all to "a useful construct" to an equivalent status with harmonics. I
take the middle view -- the only two ways to arrange consonant intervals
into chords are according to the harmonic or subharmonic series, but the
consonance of the intervals in both cases is due only to harmonics. The
harmonics coincide more often in subharmonic chords, so there is not really
an equivalence here. As soon as roots and combination tones enter the
picture, harmonic configurations become acoustically simpler than
subharmonic ones.

As for Wilson CPS scales, Brian McLaren has recently explained them
adequately, although he confuses the issue by sometimes adding an extraneous
1/1 and calculating ratios from there. In their simplest and (I believe)
originally intended form, an m)n CPS scale takes all possible products of m
out of n specified factors, and interprets the products as frequencies.
(Transposing by octaves is done freely.) The resulting frequency ratios can
often be simplified considerably by renaming one of the notes as 1/1, which
Brian does not do. The concept is more powerful than appears at first
glance. For example, a 1)3 [1,3,5] scale is a just major triad, and a 2)3
[1,3,5] is a just minor triad. Represented as triangles in (3,5) space,
these two triads tile the plane. I recently discovered, in communication
with John Chalmers, that this concept extends to n dimensions, where n
different CPS scales (represented as hyperpolyhedra with triangular faces)
join together to tile n-space. I don't think there's a book on CPS scales,
though I believe Erv Wilson has written articles on them for Xenharmonikon.


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