Ternary scale theorems: Difference between revisions

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*# The scale is ''well-formed'' with respect to g, i.e. all occurrences of the generator g are ''k''-steps for a fixed ''k''.

* A ''scale'' or ''scale word'' is a circular word with a chosen size for its equave. As we're not working with scales with distinct equaves simultaneously, all three terms are effectively synonymous for our purposes.

* A scale is ''primitive''~~, or ''single-period'',~~ 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.

* 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.

* 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 generator-offset scales are SGA, and aside from odd generator-offset scales, the only ternary scales to satisfy SGA are (xy)''r''xz, ''r'' ≥ 1. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is generator-offset but not SGA.

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 *# The scale is ''well-formed'' with respect to g, i.e. all occurrences of the generator g are ''k''-steps for a fixed ''k''.
 * A ''scale'' or ''scale word'' is a circular word with a chosen size for its equave. As we're not working with scales with distinct equaves simultaneously, all three terms are effectively synonymous for our purposes.
-* A scale is ''primitive'', or ''single-period'', 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.
+* 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.
 * 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 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'' &ge; 1. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is generator-offset but not SGA.