Maximum variety: Difference between revisions
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## Start with a power of the mos word ''w''(X, Z) that begins with a X and ending with a Z and has an even number of X's. | ## Start with a power of the mos word ''w''(X, Z) that begins with a X and ending with a Z and has an even number of X's. | ||
## Interchange some of the Z's and X's at some of the borders of these copies of the mos word ''w''. | ## Interchange some of the Z's and X's at some of the borders of these copies of the mos word ''w''. | ||
## Replace every other X with Y in ''w''. The PWF scales are exactly the single-period rank-3 [[billiard scale]]s. | ## Replace every other X with Y in ''w''. | ||
# The PWF scales are exactly the single-period rank-3 [[billiard scale]]s. | |||
# Non-twisted single-period MV3 scales are always SV3. | # Non-twisted single-period MV3 scales are always SV3. | ||
# If the scale is PWF, with one exception abacaba, there always exists some "generator" interval such that the scale can be expressed as '''two parallel chains''' of this generator which are almost equal in length (the lengths are either equal, or differ by 1). This property is called the [[generator-offset property]] (GO). | # If the scale is PWF, with one exception abacaba, there always exists some "generator" interval such that the scale can be expressed as '''two parallel chains''' of this generator which are almost equal in length (the lengths are either equal, or differ by 1). This property is called the [[generator-offset property]] (GO). | ||
Statements 1 | Statements 1 through 3 are proven in Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023). Statement 4 is proven in [[generator-offset property]]. | ||
=== Generating MV3 scales === | === Generating MV3 scales === | ||