Generator-offset property: Difference between revisions

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* '''Case 3.9''': (''x'', ''y'') = (floor(''n''/μ), 2*floor(''n''/μ) + 2)

* '''Case 3.10''': (''x'', ''y'') = (floor(''n''/μ), 2*floor(''n''/μ) + 3)

=== Theorem 5 (Classification of MV3 scales) ===

# A single-period MV3 is PWF or equivalent to abcba. The PWF scales are exactly the single-period rank-3 [[billiard scale]]s. These scales are all SV3.

# A multiperiod MV3 is either a repetition of a single-period MV3 pattern or a "twisted" word constructed as follows:

## Start with a power of the mos word ''w'' 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''.

## Replace every other X with Y in ''w''.

==== Proof ====

Proven in Bulgakova, Buzhinsky and Goncharov (2023), "On balanced and abelian properties of circular words over a ternary alphabet".

== Open conjectures ==

~~=== Conjecture ("SV3 Structure Theorem") ===~~

~~(TODO: Investigate Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023) which seems to have a proof of this)~~

~~==== Conjecture 1 ====~~

~~If a 3-step size scale word that is ''a''x ''b''y ''c''z is~~

~~# abstractly SV3 (strict variety 3, i.e. every ''k''-step except multiples of the equave comes in ''exactly'' 3 sizes, for almost all tunings),~~

~~# has ''odd'' length,~~

~~# has gcd(''a'', ''b'', ''c'') = 1,~~

~~# is not of the form xyzyx or xyxzxyx,~~

~~then it is GO (and therefore SGA). (a converse to Proposition 1)~~

~~By Theorem 4, this conjecture would follow from proving that such scales are PWF.~~

~~==== Conjecture 2 ====~~

~~If a 3-step size scale word that is ''a''x ''b''y ''c''z is abstractly SV3, then two of ''a'', ''b'', ''c'' are equal.~~

=== Conjecture ("MV3 Sequences") ===

Given any two generators, we can iterate them to any number of notes and see what the maximum-variety of the resulting scale is. In particular, we can look at those scale sizes which are MV3, and thus compute the '''MV3 sequence''' for the pair of generators (similar to the "MOS sequence" one can compute for one generator). Thus, for any pair of generators, we can form the associated sequence of increasingly large MV3 scales.

@@ Line 222: / Line 222: @@
 * '''Case 3.9''': (''x'', ''y'') = (floor(''n''/μ), 2*floor(''n''/μ) + 2)
 * '''Case 3.10''': (''x'', ''y'') = (floor(''n''/μ), 2*floor(''n''/μ) + 3)
+=== Theorem 5 (Classification of MV3 scales) ===
+# A single-period MV3 is PWF or equivalent to abcba. The PWF scales are exactly the single-period rank-3 [[billiard scale]]s. These scales are all SV3.
+# A multiperiod MV3 is either a repetition of a single-period MV3 pattern or a "twisted" word constructed as follows:
+## Start with a power of the mos word ''w''  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''.
+## Replace every other X with Y in ''w''.
+==== Proof ====
+Proven in Bulgakova, Buzhinsky and Goncharov (2023), "On balanced and abelian properties of circular words over a ternary alphabet".
 == Open conjectures ==
-=== Conjecture ("SV3 Structure Theorem") ===
-(TODO: Investigate Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023) which seems to have a proof of this)
-==== Conjecture 1 ====
-If a 3-step size scale word that is ''a''x ''b''y ''c''z is
-# abstractly SV3 (strict variety 3, i.e. every ''k''-step except multiples of the equave comes in ''exactly'' 3 sizes, for almost all tunings),
-# has ''odd'' length,
-# has gcd(''a'', ''b'', ''c'') = 1,
-# is not of the form xyzyx or xyxzxyx,
-then it is GO (and therefore SGA). (a converse to Proposition 1)
-By Theorem 4, this conjecture would follow from proving that such scales are PWF.
-==== Conjecture 2 ====
-If a 3-step size scale word that is ''a''x ''b''y ''c''z is abstractly SV3, then two of ''a'', ''b'', ''c'' are equal.
 === Conjecture ("MV3 Sequences") ===
 Given any two generators, we can iterate them to any number of notes and see what the maximum-variety of the resulting scale is. In particular, we can look at those scale sizes which are MV3, and thus compute the '''MV3 sequence''' for the pair of generators (similar to the "MOS sequence" one can compute for one generator). Thus, for any pair of generators, we can form the associated sequence of increasingly large MV3 scales.

Generator-offset property: Difference between revisions

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