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.

== Other definitions ==

== Other definitions and assumptions ==

* A strengthening of the generator-offset property, tentatively named the ''swung-generator-alternant property'' (SGA), states that the alternants g1 and g2 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, only xyxz satisfies this property. 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.

* Given a linear or cyclic word ''S'' with a step size X, define ''E''X(''S'') as the scale word resulting from deleting all instances of X from ''S''.

* By a ''subword'', ''substring'', or ''slice~~'' on index ''i~~'' of a word ''S'', denoted ''S''[''i'' : ''j''],

* By a ''subword'', ''substring'', or ''slice'' of a word ''S'', denoted ''S''[''i'' : ''j''] (i ≥ j), we mean ''S''[''i''] ''S''[''i'' + 1] ... ''S''[''j'' − 1] ''S''[''j''].

** ''S''[''i''] ''S''[''i~~'' + 1] ... ''S''[''j''~~ &~~minus~~; ~~1] ''S''[''~~j~~''] if ''j'' > ''i''~~,

** 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''[-].

** ''S''[''i''] ''S''[''i'' + ~~1] ... ''S''[len(''S'')] S[~~1] ...''S''[''j'' − 1] ''S''[''j''] ~~if ''j'' < ''i'',~~

** ~~''S''[''i''] when ''j'' =~~ ''i''.

* The indices above are 1-indexed. If indices are out of bounds we first replace ''i~~'' and ''j~~'' with ''i'' % len(''S'') + 1 ~~and ''j'' % len(~~''S''~~) + 1~~.

* 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.

* An ''n''-''ary'' scale is a scale with ''n'' different step sizes. ''Binary'' and ''ternary'' are used when ''n'' = 2 and 3 respectively.

* Indices for all words are 1-indexed.

== Theorems ==

@@ Line 24: / Line 24: @@
 Note: On this page, non-italicized Latin variables refer to interval sizes, for example step sizes.
-== Other definitions ==
+== Other definitions and assumptions ==
 * 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, only xyxz satisfies this property. 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.
 * Given a linear or cyclic word ''S'' with a step size X, define ''E''<sub>X</sub>(''S'') as the scale word resulting from deleting all instances of X from ''S''.
-* By a ''subword'', ''substring'', or ''slice'' on index ''i'' of a word ''S'', denoted ''S''[''i'' : ''j''],
+* By a ''subword'', ''substring'', or ''slice'' of a word ''S'', denoted ''S''[''i'' : ''j''] (i &ge; j), we mean ''S''[''i''] ''S''[''i'' + 1] ... ''S''[''j''  &minus; 1] ''S''[''j''].
-** ''S''[''i''] ''S''[''i'' + 1] ... ''S''[''j''  &minus; 1] ''S''[''j''] if ''j'' > ''i'',
+** 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''[-].
-** ''S''[''i''] ''S''[''i'' + 1] ... ''S''[len(''S'')] S[1] ...''S''[''j''  &minus; 1] ''S''[''j''] if ''j'' < ''i'',
-** ''S''[''i''] when ''j'' = ''i''.
-* The indices above are 1-indexed. If indices are out of bounds we first replace ''i'' and ''j'' with ''i'' % len(''S'') + 1 and ''j'' % len(''S'') + 1.
 * 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.
 * An ''n''-''ary'' scale is a scale with ''n'' different step sizes. ''Binary'' and ''ternary'' are used when ''n'' = 2 and 3 respectively.
+* Indices for all words are 1-indexed.
 == Theorems ==