Generator-offset property: Difference between revisions
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=== Theorem 4 (PWF implies SV3 and either GO or abacaba) === | === Theorem 4 (PWF implies SV3 and either GO or abacaba) === | ||
Let ''S''(a, b, c) be a scale word in three '''Z'''-linearly independent step sizes a, b, c. Suppose S is PWF. Then S is SV3 and has an odd number of notes. Moreover, ''S'' is either GO or equivalent to the scale word ''abacaba''. | Let ''S''(a, b, c) be a scale word in three '''Z'''-linearly independent step sizes a, b, c. Suppose ''S'' is PWF. 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 PWF ===== | ===== If the generator of a projection of ''S'' is a ''k''-step, the word of stacked ''k''-steps in ''S'' is PWF ===== | ||
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. | ||
index: 1 2 3 4 ... n | index: 1 2 3 4 ... ''n'' | ||
Σ<sub>1</sub>: P P P P ... P I | Σ<sub>1</sub>: P P P P ... P I | ||
Σ<sub>2</sub>: [some mos] | Σ<sub>2</sub>: [some mos] | ||
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Either β′′ = β or β′′ = β′. Assume β′′ = β′. Then γ′′ = γ, and Λ<sub>3</sub> is (λ + 1)γ (μ − 1)γ′. Also assume that the first ''k''-step in Σ is Q. Then we have: | Either β′′ = β or β′′ = β′. Assume β′′ = β′. Then γ′′ = γ, and Λ<sub>3</sub> is (λ + 1)γ (μ − 1)γ′. Also assume that the first ''k''-step in Σ is Q. Then we have: | ||
1 … n | 1 … ''n'' | ||
Σ = Q ''W''(Q, R) T | Σ = Q ''W''(Q, R) T | ||
Λ<sub>1</sub> = α … α α′ | Λ<sub>1</sub> = α … α α′ | ||
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For Λ<sub>2</sub> to be a mos, the first, and only, occurrence of R must be at either ''f'' = floor(''n''/2) or ceil(''n''/2). We may assume that it is at ''f''; otherwise flip the chain and reindex the words to start at 2''f''. | For Λ<sub>2</sub> to be a mos, the first, and only, occurrence of R must be at either ''f'' = floor(''n''/2) or ceil(''n''/2). We may assume that it is at ''f''; otherwise flip the chain and reindex the words to start at 2''f''. | ||
1 … f … | 1 … ''f'' … 2''f'' ''n'' | ||
Σ = Q … Q R Q … Q T | Σ = Q … Q R Q … Q T | ||
Λ<sub>1</sub> = α … α α α … α α′ | Λ<sub>1</sub> = α … α α α … α α′ | ||