Generator-offset property: Difference between revisions
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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 ==== | ||
===== The word of stacked k-steps | ===== 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, km is ''not'' divisible by n and km-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, km is ''not'' divisible by n and km-steps come in ''exactly'' two sizes; hence both Σ<sub>2</sub> and Σ<sub>3</sub> are single-period mosses. | ||