Maximum variety: Difference between revisions

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When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between ''unconditionally MV3'' scale patterns or ''abstractly MV3'' ones, patterns that are MV3 regardless of what concrete sizes a, b, and c have, and ''conditionally MV3'' patterns, which have tunings that are not MV3. For example, MMLs is conditionally MV3 because it is only MV3 when L, M and s are chosen such that MM = Ls. When we say that an abstract scale pattern is MV3, the former meaning is usually intended.

=== MV3 Structure Theorem (conjectured) ===

(TODO: Investigate Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023) which seems to have something similar to this)

Consider a(n unconditionally, so independently of tuning) max-variety-3 scale with 3 different step sizes. It is a mathematical fact that, with only one exception, at least two of the three steps must occur '''the same number of times'''. For example, it is possible to have a max-variety-3 scale with 3 small steps, 5 medium steps, and 3 large steps, because there are the same number of small steps as large steps. But a max-variety-3 scale with 3 small steps, 5 medium steps, and 4 large steps is impossible. (The one exception to this rule is "aabacab", along with its repetitions "aabacabaabacab", etc.)

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 When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between ''unconditionally MV3'' scale patterns or ''abstractly MV3'' ones, patterns that are MV3 regardless of what concrete sizes a, b, and c have, and ''conditionally MV3'' patterns, which have tunings that are not MV3. For example, MMLs is conditionally MV3 because it is only MV3 when L, M and s are chosen such that MM = Ls. When we say that an abstract scale pattern is MV3, the former meaning is usually intended.
 === MV3 Structure Theorem (conjectured) ===
+(TODO: Investigate Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023) which seems to have something similar to this)
 Consider a(n unconditionally, so independently of tuning) max-variety-3 scale with 3 different step sizes. It is a mathematical fact that, with only one exception, at least two of the three steps must occur '''the same number of times'''. For example, it is possible to have a max-variety-3 scale with 3 small steps, 5 medium steps, and 3 large steps, because there are the same number of small steps as large steps. But a max-variety-3 scale with 3 small steps, 5 medium steps, and 4 large steps is impossible. (The one exception to this rule is "aabacab", along with its repetitions "aabacabaabacab", etc.)