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.

* For the [[generator-offset property]], see the article.

* A strengthening of the generator-offset property, here the ''swung-generator-alternant property'' (SGA), states that the alternants '''g'''1 and '''g'''2 can be taken to always subtend the same number of scale steps, thus both representing "detemperings" of a generator of a primitive [[MOS]] scale (otherwise known as a well-formed scale). This is simply the property of having a [[generator sequence]] of period 2. All odd generator-offset scales are SGA, and aside from odd generator-offset scales, the only ternary scales to satisfy SGA are ('''XY''')''r'''''XZ''', {{nowrap|''r'' ≥ 1}}. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is generator-offset but not SGA.

* A strengthening of the generator-offset property, here the ''swung-generator-alternant property'' (SGA), states that the alternants '''g'''1 and '''g'''2 can be taken to always subtend the same number of scale steps, thus both representing "detemperings" of a generator of a primitive [[MOS]] scale (otherwise known as a well-formed scale). This is simply the property of having a well-formed [[generator sequence]] of period 2. All odd generator-offset scales are SGA, and aside from odd generator-offset scales, the only ternary scales to satisfy SGA are ('''XY''')''r'''''XZ''', {{nowrap|''r'' ≥ 1}}. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is generator-offset but not SGA.

* An ''odd-step'' is a ''k''-step where ''k'' is odd; an ''even-step'' is defined similarly.

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

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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.
 * For the [[generator-offset property]], see the article.
-* A strengthening of the generator-offset property, here 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 primitive [[MOS]] scale (otherwise known as a well-formed scale). This is simply the property of having a [[generator sequence]] of period 2. All odd generator-offset scales are SGA, and aside from odd generator-offset scales, the only ternary scales to satisfy SGA are ('''XY''')<sup>''r''</sup>'''XZ''', {{nowrap|''r'' &ge; 1}}. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is generator-offset but not SGA.
+* A strengthening of the generator-offset property, here 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 primitive [[MOS]] scale (otherwise known as a well-formed scale). This is simply the property of having a well-formed [[generator sequence]] of period 2. All odd generator-offset scales are SGA, and aside from odd generator-offset scales, the only ternary scales to satisfy SGA are ('''XY''')<sup>''r''</sup>'''XZ''', {{nowrap|''r'' &ge; 1}}. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is generator-offset but not SGA.
 * An ''odd-step'' is a ''k''-step where ''k'' is odd; an ''even-step'' is defined similarly.
 * Given a linear or circular 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''.