Generator-offset property: Difference between revisions
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=== Theorem 4 (Classification of pairwise well-formed scales) === | === 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 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. | 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 (equivalently, all its projections are single-period mosses). 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 pairwise well-formed ===== | ===== If the generator of a projection of ''S'' is a ''k''-step, the word of stacked ''k''-steps in ''S'' is pairwise well-formed ===== | ||