Generator-offset property: Difference between revisions
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Note: On this page, non-italicized Latin variables refer to interval sizes, for example step sizes. | Note: On this page, non-italicized Latin variables refer to interval sizes, for example step sizes. | ||
== Other definitions | == Other definitions == | ||
* A strengthening of the generator-offset property, tentatively named the ''swung-generator-alternant property'' (SGA), states that the alternants g<sub>1</sub> and g<sub>2</sub> can be taken to always subtend the same number of scale steps, thus both representing "detemperings" of a generator of a single-period [[mos]] scale (otherwise known as a well-formed scale). All odd GO scales are SGA, and aside from odd GO scales, the only ternary scales to satisfy SGA are (xy)<sup>''r''</sup>xz, ''r'' ≥ 1. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is GO but not SGA. | * A strengthening of the generator-offset property, tentatively named the ''swung-generator-alternant property'' (SGA), states that the alternants g<sub>1</sub> and g<sub>2</sub> can be taken to always subtend the same number of scale steps, thus both representing "detemperings" of a generator of a single-period [[mos]] scale (otherwise known as a well-formed scale). All odd GO scales are SGA, and aside from odd GO scales, the only ternary scales to satisfy SGA are (xy)<sup>''r''</sup>xz, ''r'' ≥ 1. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is GO but not SGA. | ||
* An ''odd-step'' is a ''k''-step where ''k'' is odd; an ''even-step'' is defined similarly. | * An ''odd-step'' is a ''k''-step where ''k'' is odd; an ''even-step'' is defined similarly. | ||
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** If the index ''i'' is out of bounds we first replace ''i'' with ''i'' % len(''S'') + 1 before using it as an argument in ''S''[-]. | ** If the index ''i'' is out of bounds we first replace ''i'' with ''i'' % len(''S'') + 1 before using it as an argument in ''S''[-]. | ||
* Given a mos aX bY, a ''chunk'' of X's is a maximal substring made of X's, bounded by Y's, possibly empty. We do not include the boundary Y's. | * Given a mos aX bY, a ''chunk'' of X's is a maximal substring made of X's, bounded by Y's, possibly empty. We do not include the boundary Y's. | ||
== Theorems == | == Theorems == | ||