Ternary scale theorems: Difference between revisions
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## Replace every other X with Y in ''w''. | ## Replace every other X with Y in ''w''. | ||
# Primitive MV3 scales not of type (4) are always SV3, and those of type (4) are SV3 with the exception of the ''n''/2-step (''n'' = scale length) which is variety 2. | # Primitive MV3 scales not of type (4) are always SV3, and those of type (4) are SV3 with the exception of the ''n''/2-step (''n'' = scale length) which is variety 2. | ||
# Primitive MV3 scales not of type (5) and not of the form XYZYX are ''balanced'': for any ''k'', any pair of k-steps has a difference that contains +1, -1, or 0 of each step size. | # Primitive MV3 scales not of type (5) and not of the form XYZYX are ''[[balanced]]'': for any ''k'', any pair of k-steps has a difference that contains +1, -1, or 0 of each step size. | ||
=== Proof === | === Proof === | ||
Proven by Bulgakova, Buzhinsky and Goncharov (2023), "[https://www.sciencedirect.com/science/article/pii/S0304397522006417 On balanced and abelian properties of circular words over a ternary alphabet]" (and Theorem 4). Note that odd GO scales are type (3) in this classification. | Proven by Bulgakova, Buzhinsky and Goncharov (2023), "[https://www.sciencedirect.com/science/article/pii/S0304397522006417 On balanced and abelian properties of circular words over a ternary alphabet]" (and Theorem 4). Note that odd GO scales are type (3) in this classification. | ||