Self-Similar scales

I might say a few words about self-similar scales myself.
Brian is talking about several different kinds of self-similarity.
The type exhibited by the Harrison/Lucy system is true of any
cyclic tuning, whether equally-tempered, linearly tempered (meantone,
Helmholtzian, Sabat-Garibaldi's, etc.), or just (Pythagorean, cycles
of 5/4's, etc.). Each new tone has a constant relation to the one
preceding: either the cyclic interval or its complement re the
interval of equivalence (octave inversion if the IE is 2/1).

The Golden Ratio or Section tuning (Kornerup, Dudon) has another type
of self-similarity. Many of the small intervals of the scale are in
the GS relation to each other. In this case, the chromatic semitone/
diatonic semitone, the diatonic semitone/whole tone, the whole tone/
minor third, and the minor third/perfect fourth are in the GS relation. From
these relations and the derivation of these intervals by a cycle of
fifths, the "golden" fifth may be found. From this fifth (and/or its
complement) all the intervals may be generated.

A third type is described in Brian's Xenharmoniko^n papers. In this case,
a given relation is iteratively applied within the the IE. For example,
iterating the Golden Ratio relation (.618034..) in the octave yields
1200, 742, 459, 283....cents. A possible problem with this kind of
scale is that the terms quickly converge to inaudible intervals.

A variation on this last type may be obtained by limiting the iterations
before the intervals become unusably small and applying the technique to
the larger intervals already generated. This procedure yields scales
with more evenly distributed intervals while preserving the sense of
proportion.

Hahn's question about the perceptual character of these scales is
still relevant. I can only say that some successful pieces have
been written using unusual relations, e.g., John Chowning's STRIA,
which uses the 9th root of PHI (aka the GS). I believe Brian answered my
own question by making a distinction between the acoustic and compositional
levels of organization. I think the only way to address these concerns
is to compose, record, and distribute more music and let the listeners
decide.

--John


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