Recursive structure of MOS scales: Difference between revisions

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By ''generatedness'', we mean that every interval in the scale is of the form ''jg'' + ''kp'' where ''g'' is the 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 (i.e. satisfy Myhill's property).

~~First suppose that ''g'' and ''p'' are linearly independent, so the step ratio of ''S'' is irrational.~~

Suppose that such a scale ''S'' (with ''n'' ≥ 2 notes) has ''a''-many L steps and ''b''-many s steps and has generator ''g'' and period ''p''. 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:

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

{(−''n'' + 1)''g'', ..., −''g'', 0, ''g'', ..., (''n'' − 1)''g''}.

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 By ''generatedness'', we mean that every interval in the scale is of the form ''jg'' + ''kp'' where ''g'' is the 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 (i.e. satisfy Myhill's property).
-First suppose that ''g'' and ''p'' are linearly independent, so the step ratio of ''S'' is irrational.
+Suppose that such a scale ''S'' (with ''n'' ≥ 2 notes) has ''a''-many L steps and ''b''-many s steps and has generator ''g'' and period ''p''. 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:
-Suppose that such a scale ''S'' (with ''n'' ≥ 2 notes) has ''a''-many L steps and ''b''-many s steps and has generator ''g'' and period ''p''. Since ''S'' is generated, the interval sizes modulo ''p'' that occur in ''S'' are:
 {(&minus;''n'' + 1)''g'', ..., &minus;''g'', 0, ''g'', ..., (''n'' &minus; 1)''g''}.