Ternary scale theorems: Difference between revisions

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

# A single-period MV3 is either (1) equivalent to XYZYX, (2) equivalent to XYXZXYX, (3) constructed from ~~aX bZ~~ with a even and gcd(a, b) = 1 by replacing every other X with Y, (4) constructed from ~~2aX 2bY~~ with a odd and gcd(a, b) = 1 by replacing every other X with Y, or (5) a "twisted" word constructed as follows:

# A single-period MV3 is either (1) equivalent to XYZYX, (2) equivalent to XYXZXYX, (3) constructed from ''a''X ''b''Z with ''a'' even and gcd(''a'', ''b'') = 1 by replacing every other X with Y, (4) constructed from 2''a''X 2''b''Z with ''a'' odd and gcd(''a'', ''b'') = 1 by replacing every other X with Y, or (5) a "twisted" word constructed as follows:

## Start with a power of a multimos word ''w''(X, Z) = ''ka''X ''kb''Z such that ''a'' is even and each ''a''X ''b''Z subword of ''w'' is of the form X''P''(X, Z)Z where ''P''(X, Z) is a palindrome.

## 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''.

# Single-period 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.

# Single-period 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.

# Single-period 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 ===

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 == Theorem 5 (Classification of MV3 scales) ==
-# A single-period MV3 is either (1) equivalent to XYZYX, (2) equivalent to XYXZXYX, (3) constructed from aX bZ with a even and gcd(a, b) = 1 by replacing every other X with Y, (4) constructed from 2aX 2bY with a odd and gcd(a, b) = 1 by replacing every other X with Y, or (5) a "twisted" word constructed as follows:
+# A single-period MV3 is either (1) equivalent to XYZYX, (2) equivalent to XYXZXYX, (3) constructed from ''a''X ''b''Z with ''a'' even and gcd(''a'', ''b'') = 1 by replacing every other X with Y, (4) constructed from 2''a''X 2''b''Z with ''a'' odd and gcd(''a'', ''b'') = 1 by replacing every other X with Y, or (5) a "twisted" word constructed as follows:
 ## Start with a power of a multimos word ''w''(X, Z) = ''ka''X ''kb''Z such that ''a'' is even and each ''a''X ''b''Z subword of ''w'' is of the form X''P''(X, Z)Z where ''P''(X, Z) is a palindrome.
 ## 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''.
-# Single-period 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.
+# Single-period 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.
 # Single-period 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 ===