Generator-offset property: Difference between revisions
→Theorem 4 (PWF implies SV3 and either GO or abacaba): Uploading proof. Will format later... |
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Let S(a,b,c) be a scale word in three step sizes a, b, c. Suppose S is PWF. Then S is SV3. Moreover, S is either GO or equivalent to the scale word abacaba. | Let S(a,b,c) be a scale word in three step sizes a, b, c. Suppose S is PWF. Then S is SV3. Moreover, S is either GO or equivalent to the scale word abacaba. | ||
==== Proof ==== | ==== Proof ==== | ||
Suppose S has n notes (by excluding small cases, n >= 7) and S projects to single-period mosses S1 (a ~ b), S2 (b ~ c) and S3 (c ~ a). Suppose S1'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 S2 and S3, respectively. These words, which we call Σ2 and Σ3, must be mosses. Both Σ2 and Σ3 are single period, since (k, n) = 1. | Suppose S has n notes (by excluding small cases, n >= 7) and S projects to single-period mosses S1 (via identifying a ~ b), S2 (via identifying b ~ c) and S3 (via identifying c ~ a). Suppose S1'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 S2 and S3, respectively. These words, which we call Σ2 and Σ3, must be mosses. Both Σ2 and Σ3 are single period, since (k, n) = 1. | ||
index: 1 2 3 4 ... n | index: 1 2 3 4 ... n | ||