Mathematics of MOS: Difference between revisions

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== Proofs ==

=== Binary generated scales with #L coprime to #s within each period are MOS ===

By ''generatedness'', we mean that every interval in the scale is of the form ''jg'' + ''kp'' where ''g'' is a generator, ''p'' is the period, and ''j, k'' ∈ '''Z''', and that either ''g'' or ''−g'' occurs on every note~~. We have shown that the chunking procedure yields a scale that is generated and binary and that has gcd(#L, #s) = 1 within each period. We need only show that any such scale is a MOS~~. We claim that any interval class not ''p''-equivalent to 0 has ''exactly'' 2 sizes.

By ''generatedness'', we mean that every interval in the scale is of the form ''jg'' + ''kp'' where ''g'' is a generator, ''p'' is the period, and ''j, k'' ∈ '''Z''', and that either ''g'' or ''−g'' occurs on every note. We claim that any interval class not ''p''-equivalent to 0 has ''exactly'' 2 sizes in any scale satisfying the antecedent.

Suppose that such a scale ''S'' (with ''n'' ≥ 2 notes) has ''a''-many L steps and ''b''-many s steps per period ''p'', and has generator ''g''. Since ''S'' is generated, the interval sizes modulo ''p'' that occur in ''S'' are:

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 == Proofs ==
 === Binary generated scales with #L coprime to #s within each period are MOS ===
-By ''generatedness'', we mean that every interval in the scale is of the form ''jg'' + ''kp'' where ''g'' is a generator, ''p'' is the period, and ''j, k'' ∈ '''Z''', and that either ''g'' or ''&minus;g'' occurs on every note. We have shown that the chunking procedure yields a scale that is generated and binary and that has gcd(#L, #s) = 1 within each period. We need only show that any such scale is a MOS. We claim that any interval class not ''p''-equivalent to 0 has ''exactly'' 2 sizes.
+By ''generatedness'', we mean that every interval in the scale is of the form ''jg'' + ''kp'' where ''g'' is a generator, ''p'' is the period, and ''j, k'' ∈ '''Z''', and that either ''g'' or ''&minus;g'' occurs on every note. We claim that any interval class not ''p''-equivalent to 0 has ''exactly'' 2 sizes in any scale satisfying the antecedent.
 Suppose that such a scale ''S'' (with ''n'' ≥ 2 notes) has ''a''-many L steps and ''b''-many s steps per period ''p'', and has generator ''g''. Since ''S'' is generated, the interval sizes modulo ''p'' that occur in ''S'' are: