Ternary scale theorems: Difference between revisions

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# A [[balanced]] primitive MV3 scale is either

## equivalent to XYXZXYX

## constructed from ''a~~'''''~~X~~''' ''b'''''~~Z~~'''~~ with ''a'' even and gcd(''a'', ''b'') = 1 by replacing every other '''X''' with '''Y'''

## equivalent to a word constructed from <math>\mathsf{Christoffel}_{a,b}(\mathbf{X}, \mathbf{Z})</math> MOS with ''a'' even and gcd(''a'', ''b'') = 1 by replacing every other '''X''' with '''Y'''. Here <math>\mathsf{Christoffel}_{a,b}(\mathbf{X}, \mathbf{Z})</math> is the brightest mode of ''a'''''X''' ''b'''''Z''' taking '''X''' > '''Z'''.

~~## constructed from 2~~''a'''''X''' 2''b'''''Z''' with ''a'' odd and gcd(''a'', ''b'') = 1 by replacing every other '''X''' with '''Y'''

## equivalent to a word constructed from <math>\mathsf{Christoffel}_{a,b}(\mathbf{X}, \mathbf{Z})^2</math> with ''a'' odd and gcd(''a'', ''b'') = 1 by replacing every other '''X''' with '''Y'''

# Balanced primitive ternary scales not of type (3) are always SV3, and those of type (3) are SV3 with the exception of the ''n''/2-step (''n'' = scale length) which is variety 2.

# Balanced primitive ternary scales of type (2) have a generator sequence of period 2.

=== Classification of MV3 scales ===

# A primitive MV3 scale is either (1) balanced, (2) equivalent to XYZYX, or (3) a "twisted" word constructed as follows:

@@ Line 223: / Line 223: @@
 # A [[balanced]] primitive MV3 scale is either
 ## equivalent to XYXZXYX
-## constructed from ''a'''''X''' ''b'''''Z''' with ''a'' even and gcd(''a'', ''b'') = 1 by replacing every other '''X''' with '''Y'''
+## equivalent to a word constructed from <math>\mathsf{Christoffel}_{a,b}(\mathbf{X}, \mathbf{Z})</math> MOS with ''a'' even and gcd(''a'', ''b'') = 1 by replacing every other '''X''' with '''Y'''. Here <math>\mathsf{Christoffel}_{a,b}(\mathbf{X}, \mathbf{Z})</math> is the brightest mode of ''a'''''X''' ''b'''''Z''' taking '''X''' > '''Z'''.
-## constructed from 2''a'''''X''' 2''b'''''Z''' with ''a'' odd and gcd(''a'', ''b'') = 1 by replacing every other '''X''' with '''Y'''
+## equivalent to a word constructed from <math>\mathsf{Christoffel}_{a,b}(\mathbf{X}, \mathbf{Z})^2</math> with ''a'' odd and gcd(''a'', ''b'') = 1 by replacing every other '''X''' with '''Y'''
 # Balanced primitive ternary scales not of type (3) are always SV3, and those of type (3) are SV3 with the exception of the ''n''/2-step (''n'' = scale length) which is variety 2.
 # Balanced primitive ternary scales of type (2) have a generator sequence of period 2.
 === Classification of MV3 scales ===
 # A primitive MV3 scale is either (1) balanced, (2) equivalent to XYZYX, or (3) a "twisted" word constructed as follows: