Two papers by Kazuo Kondo

John Chalmers very kindly sent me two papers by Kondo, a professor of something or other in Japan. The basic thesis is that scales "must" be defined in terms of the non-zero elements of finite fields. His reasons for claiming this are gibberish; he seems to be saying that since sine waves are defined in terms of complex exponentials and complex exponentials live in an infinte field, for a scale to exist the infinite field (presumably, the complex numbers) must map to a finite field. This makes no sense, and in any case field homomorphims of this type don't exist; he would need to shift gears to a ring of algebraic integers instead, and even so it still would make no sense.

However, once we drop the "undoubtable lemma" he claims to have "theoretically discovered" and simply ask what he is doing, it comes down to this:

(1) Scales must have a number of degrees equal to p^n - 1 to the octave, where p is a prime. p=2 is particularly favored, so 7 and 15 note scales are his babies. You'd think 31 would interest him also, but he seems to have no notion that small integer ratios are in any way interesting.

(2) The chords of the scale are defined by its structure as a projective space.

In particular, the diatonic scales "must" have the structure of a projective plane of order 7, so that the seven triads of it in one form are FCD, CGA, GDE, DAB, AEF, EBC, and BFG if I am understanding him correcly. I don't guarantee I am, since he's none too clear. The idea of trying to jam musical meaning into finite combinatorial structures with nice automorphism groups such as finite projective spaces or designs is one I've had also, but it is rather artificial.