Ternary scale theorems: Difference between revisions

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

* A ''well-formed generator sequence'' (WFGS) is a [[generator sequence]] GS(''x''1, ..., ''x''''r'') with the following properties:

** There exists a positive integer ''k'' such that for every generator ''x''''i'' in the GS recipe GS(''x''1, ..., ''x''''r''), every occurrence of ''x''''i'' in the scale [[subtend]]s ''k'' steps. This ~~automatically~~ implies that the gap between the next higher equave and the result of stacking len(scale) − 1 of the generators in the recipe, called the ''closing generator'', or the ''imperfect generator'' since it is analogous to the imperfect generator in [[MOS]] scales, also subtends this number of steps.

** There exists a positive integer ''k'' such that for every generator ''x''''i'' in the GS recipe GS(''x''1, ..., ''x''''r''), every occurrence of ''x''''i'' in the scale [[subtend]]s ''k'' steps. This implies that the gap between the next higher equave and the result of stacking len(scale) − 1 of the generators in the recipe, called the ''closing generator'', or the ''imperfect generator'' since it is analogous to the imperfect generator in [[MOS]] scales, also subtends this number of steps.

** The closing generator must be distinct from all of the generators used in the generator sequence and occur only once in the scale.

* The property of having a WFGS of period 2, denoted WFGS-2 in this article, is important as it is equivalent to being an odd-regular MV3 scale; see below. It used to be called the "WFGS-2 property" in past versions of this article.

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 * An ''n''-''ary'' scale is a scale with ''n'' different step sizes. ''Binary'' and ''ternary'' are used when {{nowrap|''n'' {{=}} 2 and 3}}, respectively.
 * A ''well-formed generator sequence'' (WFGS) is a [[generator sequence]] GS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>) with the following properties:
-** There exists a positive integer ''k'' such that for every generator ''x''<sub>''i''</sub> in the GS recipe GS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>), every occurrence of ''x''<sub>''i''</sub> in the scale [[subtend]]s ''k'' steps. This automatically implies that the gap between the next higher equave and the result of stacking len(scale) &minus; 1 of the generators in the recipe, called the ''closing generator'', or the ''imperfect generator'' since it is analogous to the imperfect generator in [[MOS]] scales, also subtends this number of steps.
+** There exists a positive integer ''k'' such that for every generator ''x''<sub>''i''</sub> in the GS recipe GS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>), every occurrence of ''x''<sub>''i''</sub> in the scale [[subtend]]s ''k'' steps. This implies that the gap between the next higher equave and the result of stacking len(scale) &minus; 1 of the generators in the recipe, called the ''closing generator'', or the ''imperfect generator'' since it is analogous to the imperfect generator in [[MOS]] scales, also subtends this number of steps.
 ** The closing generator must be distinct from all of the generators used in the generator sequence and occur only once in the scale.
 * The property of having a WFGS of period 2, denoted WFGS-2 in this article, is important as it is equivalent to being an odd-regular MV3 scale; see below. It used to be called the "WFGS-2 property" in past versions of this article.