Generator-offset property: Difference between revisions
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(1) and (2) were proved in the proof of Proposition 1 (the part that we appeal to, from "all multiples of the generator g must be even-steps ..." to "These are all distinct by Z-linear independence", does not rely on ''S'' having the SGA property). (3) is easy to check using (1). | (1) and (2) were proved in the proof of Proposition 1 (the part that we appeal to, from "all multiples of the generator g must be even-steps ..." to "These are all distinct by Z-linear independence", does not rely on ''S'' having the SGA property). (3) is easy to check using (1). | ||
=== Theorem 4 (Classification of | === Theorem 4 (Classification of pairwise well-formed scales) === | ||
Let ''S''(a, b, c) be a scale word in three '''Z'''-linearly independent step sizes a, b, c. Suppose ''S'' is | Let ''S''(a, b, c) be a scale word in three '''Z'''-linearly independent step sizes a, b, c. Suppose ''S'' is pairwise well-formed. Then ''S'' is SV3 and has an odd number of notes. Moreover, ''S'' is either GO or equivalent to the scale word abacaba. | ||
==== Proof ==== | ==== Proof ==== | ||
===== If the generator of a projection of ''S'' is a ''k''-step, the word of stacked ''k''-steps in ''S'' is | ===== If the generator of a projection of ''S'' is a ''k''-step, the word of stacked ''k''-steps in ''S'' is pairwise well-formed ===== | ||
Suppose ''S'' has ''n'' notes (after dealing with small cases, we may assume ''n'' ≥ 7) and ''S'' projects to single-period mosses ''S''<sub>1</sub> (via identifying b with c), ''S''<sub>2</sub> (via identifying a with c) and ''S''<sub>3</sub> (via identifying a with b). Suppose ''S''<sub>1</sub>'s generator is a ''k''-step, which comes in two sizes: P, the perfect ''k''-step, and I, the imperfect ''k''-step. By stacking ''k''-steps, we get two words of length ''n'' of ''k''-steps of ''S''<sub>2</sub> and ''S''<sub>3</sub>, respectively. These two-"step-size" words, which we call Σ<sub>2</sub> and Σ<sub>3</sub>, must be mosses, since ''m''-steps in the new words correspond to ''mk''-steps in the mos words ''S''<sub>1</sub> and ''S''<sub>2</sub>, which come in at most two sizes. Since ''S''<sub>1</sub> is a single-period mos, gcd(''k'', ''n'') = 1. Hence when 0 < ''m'' < ''n'', ''mk'' is ''not'' divisible by ''n'' and ''mk''-steps come in ''exactly'' two sizes; hence both Σ<sub>2</sub> and Σ<sub>3</sub> are single-period mosses. | Suppose ''S'' has ''n'' notes (after dealing with small cases, we may assume ''n'' ≥ 7) and ''S'' projects to single-period mosses ''S''<sub>1</sub> (via identifying b with c), ''S''<sub>2</sub> (via identifying a with c) and ''S''<sub>3</sub> (via identifying a with b). Suppose ''S''<sub>1</sub>'s generator is a ''k''-step, which comes in two sizes: P, the perfect ''k''-step, and I, the imperfect ''k''-step. By stacking ''k''-steps, we get two words of length ''n'' of ''k''-steps of ''S''<sub>2</sub> and ''S''<sub>3</sub>, respectively. These two-"step-size" words, which we call Σ<sub>2</sub> and Σ<sub>3</sub>, must be mosses, since ''m''-steps in the new words correspond to ''mk''-steps in the mos words ''S''<sub>1</sub> and ''S''<sub>2</sub>, which come in at most two sizes. Since ''S''<sub>1</sub> is a single-period mos, gcd(''k'', ''n'') = 1. Hence when 0 < ''m'' < ''n'', ''mk'' is ''not'' divisible by ''n'' and ''mk''-steps come in ''exactly'' two sizes; hence both Σ<sub>2</sub> and Σ<sub>3</sub> are single-period mosses. | ||
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=== Theorem 5 (Classification of MV3 scales) === | === Theorem 5 (Classification of MV3 scales) === | ||
# A single-period MV3 is either (1) | # A single-period MV3 is either (1) pairwise well-formed, (2) equivalent to XYZYX, or (3) a "twisted" word constructed as follows: | ||
## Start with a power of a multimos word ''w''(X, Z) = ''ka''X ''kb''Z such that ''a'' is even and each ''a''X ''b''Z subword of ''w'' is of the form X''P''(X, Z)Z where ''P''(X, Z) is a palindrome. | ## Start with a power of a multimos word ''w''(X, Z) = ''ka''X ''kb''Z such that ''a'' is even and each ''a''X ''b''Z subword of ''w'' is of the form X''P''(X, Z)Z where ''P''(X, Z) is a palindrome. | ||
## Interchange some of the Z's and X's at some (possibly none) of the borders of these copies of the mos word ''w''. | ## Interchange some of the Z's and X's at some (possibly none) of the borders of these copies of the mos word ''w''. | ||