Ternary scale theorems: Difference between revisions
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This proves that the set of ''j''-steps is balanced. When ''m'' is even, take the equave-complement of the set of ''j''-steps to reduce to the above case. {{qed}} | This proves that the set of ''j''-steps is balanced. When ''m'' is even, take the equave-complement of the set of ''j''-steps to reduce to the above case. {{qed}} | ||
== Theorem 6 (Generator-offset structure of diregular | == Theorem 6 (Generator-offset structure of diregular scales) == | ||
=== Definition (Diregular scale) === | === Definition (Diregular scale) === | ||
A primitive ternary scale ''s'' is ''diregular'' if len(''s'') is even and ''s'' is equivalent to a word constructed from taking the MOS 2''a'''''X'''2''c'''''Z''' with ''a'' odd and gcd(''a'', ''c'') = 1, and replacing every other '''X''' with '''Y'''. In particular, ''s'' has [[step signature]] equivalent to ''a'''''X'''''a'''''Y'''''b'''''Z''' with ''a'' odd and ''b'' even. For example, '''LsLsLmsLsLsm''' (achiral [[diachrome]], 5'''L'''2'''m'''5'''s''') is a diregular scale. | A primitive ternary scale ''s'' is ''diregular'' if len(''s'') is even and ''s'' is equivalent to a word constructed from taking the MOS 2''a'''''X'''2''c'''''Z''' with ''a'' odd and gcd(''a'', ''c'') = 1, and replacing every other '''X''' with '''Y'''. In particular, ''s'' has [[step signature]] equivalent to ''a'''''X'''''a'''''Y'''''b'''''Z''' with ''a'' odd and ''b'' even. For example, '''LsLsLmsLsLsm''' (achiral [[diachrome]], 5'''L'''2'''m'''5'''s''') is a diregular scale. | ||