Ternary scale theorems: Difference between revisions

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In the following, ''equivalent'' means "is the same circular word after permuting '''X''', '''Y''', and '''Z'''." This means that '''XYXZXYX''' is equivalent to '''YZYXYZY''', or '''XZXYXZX''', and so on.

=== Theorem 7.1 (Classification of ternary balanced scales) ===

# A primitive [[balanced]] MV3 scale ''s'' is one of the following:

# A primitive [[balanced]] MV3 scale ''s'' satisfies one of the following:

## '''sporadic balanced''': ''s'' is equivalent to '''XYXZXYX''', the ternary [[Fraenkel word]], with step signature 4'''X'''2'''Y'''1'''Z'''.

## '''odd-regular''': len(''s'') is odd, and ''s'' is equivalent to a word constructed from taking the brightest mode of the MOS ''c'''''X'''''b'''''Z''' with ''c'' even and gcd(''c'', ''b'') = 1, and replacing every other '''X''' with '''Y'''. We assume '''X''' > '''Z''' when constructing the MOS. In particular, ''s'' has [[step signature]] ''a'''''X'''''a'''''Y'''''b'''''Z''' where ''b'' is odd (with ''a'' = ''c''/2).

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# All primitive balanced ternary scales are MV3.

# A balanced primitive ternary scale is SV3 if and only if it is not even-regular.

# Odd-~~egular~~ balanced primitive ternary scales have a generator sequence of period 2.

# Odd-regular balanced primitive ternary scales have a generator sequence of period 2.

(Condensed: All single-period balanced ternary scales that are not the Fraenkel word are a'''X''' a'''Y''' b'''Z'''. In this case, if b is odd, then the scale is odd-regular. If b is even, then the scale is even-regular.)

==== Proof ====

The following proof is adapted from the proof of Theorem 3.1.1 in Bulgakova, Buzhinsky and Goncharov (2023), "[https://arxiv.org/pdf/2012.15818 On balanced and abelian properties of circular words over a ternary alphabet]".

Note: The xen term "brightest MOS word" is equivalent to "Christoffel word" in the paper, and similarly "brightest multiMOS word" is equivalent to "powers of a Christoffel word". Also see [[Glossary for combinatorics on words]] for more equivalents between xen community terms and standard academic terminology.

=== Theorem 7.2 (Classification of MV3 scales) ===

@@ Line 257: / Line 257: @@
 In the following, ''equivalent'' means "is the same circular word after permuting '''X''', '''Y''', and '''Z'''." This means that '''XYXZXYX''' is equivalent to '''YZYXYZY''', or '''XZXYXZX''', and so on.
 === Theorem 7.1 (Classification of ternary balanced scales) ===
-# A primitive [[balanced]] MV3 scale ''s'' is one of the following:
+# A primitive [[balanced]] MV3 scale ''s'' satisfies one of the following:
 ## '''sporadic balanced''': ''s'' is equivalent to '''XYXZXYX''', the ternary [[Fraenkel word]], with step signature 4'''X'''2'''Y'''1'''Z'''.
 ## '''odd-regular''': len(''s'') is odd, and ''s'' is equivalent to a word constructed from taking the brightest mode of the MOS ''c'''''X'''''b'''''Z''' with ''c'' even and gcd(''c'', ''b'') = 1, and replacing every other '''X''' with '''Y'''. We assume '''X''' > '''Z''' when constructing the MOS. In particular, ''s'' has [[step signature]] ''a'''''X'''''a'''''Y'''''b'''''Z''' where ''b'' is odd (with ''a'' = ''c''/2).
@@ Line 263: / Line 263: @@
 # All primitive balanced ternary scales are MV3.
 # A balanced primitive ternary scale is SV3 if and only if it is not even-regular.
-# Odd-egular balanced primitive ternary scales have a generator sequence of period 2.
+# Odd-regular balanced primitive ternary scales have a generator sequence of period 2.
 (Condensed: All single-period balanced ternary scales that are not the Fraenkel word are a'''X''' a'''Y''' b'''Z'''. In this case, if b is odd, then the scale is odd-regular. If b is even, then the scale is even-regular.)
+==== Proof ====
+The following proof is adapted from the proof of Theorem 3.1.1 in Bulgakova, Buzhinsky and Goncharov (2023), "[https://arxiv.org/pdf/2012.15818 On balanced and abelian properties of circular words over a ternary alphabet]".
+Note: The xen term "brightest MOS word" is equivalent to "Christoffel word" in the paper, and similarly "brightest multiMOS word" is equivalent to "powers of a Christoffel word". Also see [[Glossary for combinatorics on words]] for more equivalents between xen community terms and standard academic terminology.
 === Theorem 7.2 (Classification of MV3 scales) ===