Recursive structure of MOS scales: Difference between revisions

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=== Binary generated scales 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'''. We have shown that the result of chunking and reduction is generated and binary ~~and the result of expanding from a reduced word is generated and binary. We~~ need only show that binary generated scales are MOS. Specifically, we claim that any such scale has ''Myhill's property with respect to p'', which we define to mean 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'''. We have shown that the result of chunking and reduction is generated and binary, so we need only show that binary generated scales are MOS. Specifically, we claim that any such scale has ''Myhill's property with respect to p'', which we define to mean that any interval class not ''p''-equivalent to 0 has ''exactly'' 2 sizes.

Suppose that such a scale ''S'' (with ''n'' ≥ 2 notes) has ''a''-many L steps and ''b''-many s steps per period ''p'', where gcd(''a'', ''b'') = 1, and has generator ''g''. First assume that ''g'' and ''p'' are linearly independent, so the step ratio of ''S'' is irrational. Since ''S'' is generated, the interval sizes modulo ''p'' that occur in ''S'' are:

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 === Binary generated scales 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'''. We have shown that the result of chunking and reduction is generated and binary and the result of expanding from a reduced word is generated and binary. We need only show that binary generated scales are MOS. Specifically, we claim that any such scale has ''Myhill's property with respect to p'', which we define to mean 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'''. We have shown that the result of chunking and reduction is generated and binary, so we need only show that binary generated scales are MOS. Specifically, we claim that any such scale has ''Myhill's property with respect to p'', which we define to mean that any interval class not ''p''-equivalent to 0 has ''exactly'' 2 sizes.
 Suppose that such a scale ''S'' (with ''n'' ≥ 2 notes) has ''a''-many L steps and ''b''-many s steps per period ''p'', where gcd(''a'', ''b'') = 1, and has generator ''g''. First assume that ''g'' and ''p'' are linearly independent, so the step ratio of ''S'' is irrational. Since ''S'' is generated, the interval sizes modulo ''p'' that occur in ''S'' are: