Ternary scale theorems: Difference between revisions

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* A scale is ''primitive'' if its period is the same as its equave. A ''multiMOS'' or ''multiperiod MOS'' is a non-primitive MOS. A MOS ''a'''''L''' ''b'''''s''' is primitive iff {{nowrap|gcd(''a'', ''b'') {{=}} 1}}. This corresponds to the term ''single-period'' in common xen parlance. Any multiMOS can be constructed from a primitive MOS by repeating the MOS pattern multiple times, e.g. if 3'''L''' 2'''s''' is '''LLsLs''', then 9'''L''' 6'''s''' is '''LLsLsLLsLsLLsLs'''.

* 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.~~

* For the ''well-formed generator sequence'' property, see the [[generator sequence]] 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 ~~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'' &ge~~; ~~1}}~~. ~~The Zarlino and diasem scales above are both SGA. [[Blackdye]] is generator-offset but not~~ SGA.

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

* 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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 * A scale is ''primitive'' if its period is the same as its equave. A ''multiMOS'' or ''multiperiod MOS'' is a non-primitive MOS. A MOS ''a'''''L''' ''b'''''s''' is primitive iff {{nowrap|gcd(''a'', ''b'') {{=}} 1}}. This corresponds to the term ''single-period'' in common xen parlance. Any multiMOS can be constructed from a primitive MOS by repeating the MOS pattern multiple times, e.g. if 3'''L''' 2'''s''' is '''LLsLs''', then 9'''L''' 6'''s''' is '''LLsLsLLsLsLLsLs'''.
 * 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.
+* For the ''well-formed generator sequence'' property, see the [[generator sequence]] 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 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.
+* The property of having a WFGS of period 2 is important as it is equivalent to being an odd-regular MV3 scale; see below. It used to be called the "SGA property" in past versions of this article.
 * 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''.