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=== Abstractly SV4 scale patterns === | === Abstractly SV4 scale patterns === | ||
Abstractly SV4 scale patterns (patterns that are SV4 for any choice of distinct cent values for the four steps): | Abstractly SV4 scale patterns (patterns that are SV4 for any choice of distinct cent values for the four steps): | ||
* 4 notes: '''1234''' | |||
* 5 notes: '''12134''' | |||
* 6 notes: none | * 6 notes: none | ||
* 7 notes: '''0123210''', '''0102013''', and '''3102010''' (The last two patterns are a chiral pair) | * 7 notes: '''0123210''', '''0102013''', and '''3102010''' (The last two patterns are a chiral pair) | ||
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* 11 notes: none | * 11 notes: none | ||
* 12 notes: none | * 12 notes: none | ||
* 13 notes: none | |||
(Note that abstract SV4-ness implies that a scale pattern is [[primitive]], or single-period.) | (Note that abstract SV4-ness implies that a scale pattern is [[primitive]], or single-period.) | ||
== Open questions == | == Open questions == | ||
* Why are (abstractly) SV4 scale patterns seemingly so rare? | * Why are (abstractly) SV4 scale patterns seemingly so rare? | ||
** Conjecture: There are only finitely many SV4 [[circular word]]s of length | ** Conjecture: There are only finitely many SV4 [[circular word]]s of finite length. | ||
** Related may be the following conjecture: For a sufficiently long [[arity|ternary]] linear word, there exists ''k'' > 1 such that the interval class of ''k''-steps has at least 3 sizes and the interval class of (''k'' − 1)-steps also has at least 3 sizes. | ** Related may be the following conjecture: For a sufficiently long [[arity|ternary]] linear word, there exists ''k'' > 1 such that the interval class of ''k''-steps has at least 3 sizes and the interval class of (''k'' − 1)-steps also has at least 3 sizes. | ||
[[Category:Scale]] | [[Category:Scale]] | ||