Ternary scale theorems: Difference between revisions
m →Theorem 6.1 (Classification of ternary balanced scales): Corrected variable from S to s |
Added a section describing the generator-offset structure of diregular scales |
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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 (Classification of MV3 scales) == | == Theorem 6 (Generator-offset structure of diregular balanced scales) == | ||
=== 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. | |||
=== Theorem === | |||
If ''s'' = ''s''('''X''', '''Y''', '''Z''') is diregular, then: | |||
# ''s'' consists of two generator chains, each with len(''s'')/2 notes; | |||
# the generator has the same interval class as some generator of the MOS 2''a'''''W'''2''c'''''Z'''; | |||
# the two generator chains are offset by a len(''s'')/2-step interval. | |||
=== Proof === | |||
The result of sending '''Y''' to '''X''' (let us call this map ''p'') is the MOS ''M'' = 2''a'''''X'''2''c'''''Z''', which has exactly 2 periods since gcd(''a'', ''c'') = 1. ''M'' thus consists of two generator chains separated by the period of ''M'', which has ''a'' + ''c'' = len(''s'') steps. It thus suffices for there to exist ''k'', 0 < ''k'' < ''a'' + ''b'', such that every perfect ''k''-step generator has the same preimage in ''s'', which will be our desired generator. | |||
There are only two values of ''k'' to check. Suppose that the perfect ''k''-step of ''M'' is ''i'''''W''' + ''j'''''Z''' where 0 < ''i'' < ''a''. Since ''a'' is odd, possibly after taking the period-complement we may assume that ''i'' is even. Hence each subword ''w'' of ''s'' such that its projection ''p''(''w'') spans a perfect ''k''-step satisfies |''w''|<sub>'''X'''</sub> = |''w''|<sub>'''Y'''</sub> (= ''i''/2). It plainly follows that every such ''w'' satisfies |''w''|<sub>'''X'''</sub> = |''w''|<sub>'''Y'''</sub> = ''i''/2 and |''w''|<sub>'''Z'''</sub> = ''j''. {{qed}} | |||
== Theorem 7 (Classification of MV3 scales) == | |||
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. | 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 | === 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'' is one of the following: | ||
## '''sporadic balanced''': ''s'' is equivalent to '''XYXZXYX''', the ternary [[Fraenkel word]], with step signature 4'''X'''2'''Y'''1'''Z'''. | ## '''sporadic balanced''': ''s'' is equivalent to '''XYXZXYX''', the ternary [[Fraenkel word]], with step signature 4'''X'''2'''Y'''1'''Z'''. | ||
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# Regular balanced primitive ternary scales have a generator sequence of period 2. | # Regular balanced primitive ternary scales have a generator sequence of period 2. | ||
=== Theorem | === Theorem 7.2 (Classification of MV3 scales) === | ||
A primitive MV3 scale is either | A primitive MV3 scale is either | ||
# '''balanced''' (classified by the previous theorem), | # '''balanced''' (classified by the previous theorem), | ||
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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. | 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. | ||
For | For 7.1.3: The ternary Fraenkel word may be verified as SV3 by inspection, and we have already shown in Theorem 1 that regular balanced scales are SV3. To show that diregular balanced scales are ''not'' SV3, observe that (''a''+''c'')-steps come in only 2 sizes in such a scale ''s'': floor(''a''/2)'''X''' + ceil(''a''/2)'''Y''' + ''c'''''Z''' and ceil(''a''/2)'''X''' + floor(''a''/2)'''Y''' + ''c'''''Z''', since the underlying MOS 2''a'''''X'''2''c'''''Y''' only has the (''a''+''c'')-step ''a'''''X''' + ''c'''''Z'''. The construction replaces the '''X'''s in these subwords with alternating '''X'''s and '''Y'''s; either of '''X''' or '''Y''' may occur first, corresponding to the two possible sizes, since ''a'' is odd and thus the (''a''+''c'')-step subword ''s''[''k'' : ''k''+''a''+''c''] becomes the subword ''s''[''k''+''a''+''c'' : ''k''+2''a''+2''c''] via interchanging '''X''' and '''Y'''. | ||
Claim | Claim 7.1.4 can be verified by noting that such scales are PWF and using Theorem 4. {{Qed}} | ||
== Open problems == | == Open problems == | ||