Generator-offset property: Difference between revisions

Line 25:

Note: On this page, non-italicized Latin variables refer to interval sizes, for example step sizes.

== Other definitions and assumptions ==

* 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 GO scales are SGA, and aside from odd GO scales, only ~~xyxz satisfies this property~~. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is GO but not SGA.

* 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 GO scales are SGA, and aside from odd GO 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 GO but not SGA.

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

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

@@ Line 25: / Line 25: @@
 Note: On this page, non-italicized Latin variables refer to interval sizes, for example step sizes.
 == Other definitions and assumptions ==
-* 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 GO scales are SGA, and aside from odd GO scales, only xyxz satisfies this property. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is GO but not SGA.
+* 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 GO scales are SGA, and aside from odd GO 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 GO but not SGA.
 * An ''odd-step'' is a ''k''-step where ''k'' is odd; an ''even-step'' is defined similarly.
 * Given a linear or cyclic 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''.