Ternary scale theorems: Difference between revisions

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

# A primitive [[balanced]] ~~MV3~~ scale ''s'' is pairwise-MOS and satisfies one of the following:

# A primitive [[balanced]] ternary scale ''s'' is MV3 and pairwise-MOS, and 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 {{nowrap|gcd(''c'', ''b'') {{=}} 1}}, and replacing every other '''X''' with '''Y'''. We assume {{nowrap|'''X''' > '''Z'''}} when constructing the MOS. In particular, ''s'' has [[step signature]] ''a'''''X'''''a'''''Y'''''b'''''Z''' where ''b'' is odd (with {{nowrap|''a'' {{=}} ''c''/2}}).

@@ Line 269: / Line 269: @@
 === Theorem 7.1 (Classification of ternary balanced scales) ===
-# A primitive [[balanced]] MV3 scale ''s'' is pairwise-MOS and satisfies one of the following:
+# A primitive [[balanced]] ternary scale ''s'' is MV3 and pairwise-MOS, and 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 {{nowrap|gcd(''c'', ''b'') {{=}} 1}}, and replacing every other '''X''' with '''Y'''. We assume {{nowrap|'''X''' &gt; '''Z'''}} when constructing the MOS. In particular, ''s'' has [[step signature]] ''a'''''X'''''a'''''Y'''''b'''''Z''' where ''b'' is odd (with {{nowrap|''a'' {{=}} ''c''/2}}).