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 aLbs is primitive iff gcd(a, b) = 1. This corresponds to the term ''single-period'' in common xen parlance. | * 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 aLbs is primitive iff gcd(a, b) = 1. This corresponds to the term ''single-period'' in common xen parlance. | ||
* An ''n''-''ary'' scale is a scale with ''n'' different step sizes. ''Binary'' and ''ternary'' are used when ''n'' = 2 and 3 respectively. | * An ''n''-''ary'' scale is a scale with ''n'' different step sizes. ''Binary'' and ''ternary'' are used when ''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 binary generating 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''', ''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'''<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 binary generating 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''', ''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. | * An ''odd-step'' is a ''k''-step where ''k'' is odd; an ''even-step'' is defined similarly. | ||