Generator-offset property: Difference between revisions

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* 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.

== Theorems ==

== Theorems ==<!--

=== Proposition 1 (Properties of SGA scales) ===

Let ''S'' be a 3-step-size scale word in L, M, and s of length ''n'', and suppose ''S'' is SGA. Then:

# The length of ''S'' is odd, or ''S'' is equivalent to xyxz.

# The length of ''S'' is odd, or ''S'' is equivalent to xxx...xyxz.

# ''S'' is of the form ''a''x ''b''y ''b''z for some permutation (x, y, z) of (L, M, s).

# If ''n'' is odd, ''S'' is abstractly SV3 (i.e. SV3 for almost all tunings).

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For (6), consider the mos ''a''X 2''b''W as chunks of X separated by W (tempering Y and Z together into W). Eliminating every other W turns it into a mos, because the sum of sizes of consecutive chunks of X (1st chunk with 2nd chunk, 3rd with 4th, ...) must form a mos. This is because the chunk sizes of X form a mos, and taking every ''k''th note of an ''n''-note mos where ''k'' divides ''n'' yields a mos. Since the result of setting X = 0 is the mos ''b''Y ''b''Z, ''S'' is elimination-mos.

-->

=== Proposition 2 (Odd GO scales are SGA) ===

Suppose that a periodic scale satisfies the following:

@@ Line 27: / Line 27: @@
 * 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.
-== Theorems ==
+== Theorems ==<!--
 === Proposition 1 (Properties of SGA scales) ===
 Let ''S'' be a 3-step-size scale word in L, M, and s of length ''n'', and suppose ''S'' is SGA. Then:
-# The length of ''S'' is odd, or ''S'' is equivalent to xyxz.
+# The length of ''S'' is odd, or ''S'' is equivalent to xxx...xyxz.
 # ''S'' is of the form ''a''x ''b''y ''b''z for some permutation (x, y, z) of (L, M, s).
 # If ''n'' is odd, ''S'' is abstractly SV3 (i.e. SV3 for almost all tunings).
@@ Line 99: / Line 99: @@
 For (6), consider the mos ''a''X 2''b''W as chunks of X separated by W (tempering Y and Z together into W). Eliminating every other W turns it into a mos, because the sum of sizes of consecutive chunks of X (1st chunk with 2nd chunk, 3rd with 4th, ...) must form a mos. This is because the chunk sizes of X form a mos, and taking every ''k''th note of an ''n''-note mos where ''k'' divides ''n'' yields a mos. Since the result of setting X = 0 is the mos ''b''Y ''b''Z, ''S'' is elimination-mos.
+-->
 === Proposition 2 (Odd GO scales are SGA) ===
 Suppose that a periodic scale satisfies the following: