Mathematics of MOS: Difference between revisions
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== Proofs == | == Proofs == | ||
=== MOS scales are generated === | |||
See [[Recursive structure of MOS scales#Proofs]]. | |||
=== Binary generated scales with #L coprime to #s within each period are MOS === | === 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 claim that any interval class not ''p''-equivalent to 0 has ''exactly'' 2 sizes in any scale satisfying the antecedent. | 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. | ||