Ternary scale theorems: Difference between revisions

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* The notation ''s''('''X'''1, ..., '''X'''''r'') is used for an ''r''-ary scale word with variables '''X'''1, ..., '''X'''''r'' possibly standing in for any sizes. If ''s''('''X''', '''Y''') = '''XXY''' then ''s''('''A''', '''B''') = '''AAB'''.

* We leave the distinction between linear words (words in the ordinary sense) and circular words up to context. We usually also elide the distinction between subwords and the dyad sizes that subtend them.

* For a word ''w'' and letter '''x''', |''w''|'''x''' denotes the number of occurrences of the letter '''x''' in ''w''. For ~~an abstract dyad~~ size '''v''', |'''v'''|'''x''' is similar.

* For a word ''w'' and letter '''x''', |''w''|'''x''' denotes the number of occurrences of the letter '''x''' in ''w''. For a step vector size '''v''', |'''v'''|'''x''' is similar.

== Definitions ==

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All pairwise-well-formed scales are [[balanced]].

=== Proof ===

Let ''s'' be a PWF (thus primitive) scale. The case where ''s'' is equivalent to '''XYXZXYX''' can be manually verified, so by Theorem 4, the only remaining case is when ''s'' can be constructed by stacking two alternating sizes, '''g'''1 and '''g'''2, of ''k''-steps. We assume that ''s'' has [[step signature]] ''a'''''X''' ''b'''''Y''' ''b'''''Z''' where ''a'' is odd. This ''k'' corresponds to a class of generators of the primitive MOS ''a'''''X''' 2''b'''''W'''. This MOS is obtained from ''s'' by applying the letterwise substitution function π such that π('''X''') = '''X''' and π('''Y''') = π('''Z''') = '''W'''. Naturally, π applies to linear words, circular words, and ~~abstract dyad~~ sizes. Additionally, we can choose ''k'' so that the two sizes of ''k''-steps in π(''s'') are:

Let ''s'' be a PWF (thus primitive) scale. The case where ''s'' is equivalent to '''XYXZXYX''' can be manually verified, so by Theorem 4, the only remaining case is when ''s'' can be constructed by stacking two alternating sizes, '''g'''1 and '''g'''2, of ''k''-steps. We assume that ''s'' has [[step signature]] ''a'''''X''' ''b'''''Y''' ''b'''''Z''' where ''a'' is odd. This ''k'' corresponds to a class of generators of the primitive MOS ''a'''''X''' 2''b'''''W'''. This MOS is obtained from ''s'' by applying the letterwise substitution function π such that π('''X''') = '''X''' and π('''Y''') = π('''Z''') = '''W'''. Naturally, π applies to linear words, circular words, and step vector sizes. Additionally, we can choose ''k'' so that the two sizes of ''k''-steps in π(''s'') are:

* the perfect generator '''g''' = ''t'''''X''' + (''k'' − ''t'')'''W''' (note (''k'' − ''t'') is odd by a previous proof), and

* the imperfect generator '''i''' = (''t'' + 1)'''X''' + (''k'' − ''t'' − 1)'''W'''.

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 * The notation ''s''('''X'''<sub>1</sub>, ..., '''X'''<sub>''r''</sub>) is used for an ''r''-ary scale word with variables '''X'''<sub>1</sub>, ..., '''X'''<sub>''r''</sub> possibly standing in for any sizes. If ''s''('''X''', '''Y''') = '''XXY''' then ''s''('''A''', '''B''') = '''AAB'''.
 * We leave the distinction between linear words (words in the ordinary sense) and circular words up to context. We usually also elide the distinction between subwords and the dyad sizes that subtend them.
-* For a word ''w'' and letter '''x''', |''w''|<sub>'''x'''</sub> denotes the number of occurrences of the letter '''x''' in ''w''. For an abstract dyad size '''v''', |'''v'''|<sub>'''x'''</sub> is similar.
+* For a word ''w'' and letter '''x''', |''w''|<sub>'''x'''</sub> denotes the number of occurrences of the letter '''x''' in ''w''. For a step vector size '''v''', |'''v'''|<sub>'''x'''</sub> is similar.
 == Definitions ==
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 All pairwise-well-formed scales are [[balanced]].
 === Proof ===
-Let ''s'' be a PWF (thus primitive) scale. The case where ''s'' is equivalent to '''XYXZXYX''' can be manually verified, so by Theorem 4, the only remaining case is when ''s'' can be constructed by stacking two alternating sizes, '''g'''<sub>1</sub> and '''g'''<sub>2</sub>, of ''k''-steps. We assume that ''s'' has [[step signature]] ''a'''''X''' ''b'''''Y''' ''b'''''Z''' where ''a'' is odd. This ''k'' corresponds to a class of generators of the primitive MOS ''a'''''X''' 2''b'''''W'''. This MOS is obtained from ''s'' by applying the letterwise substitution function π such that π('''X''') = '''X''' and π('''Y''') = π('''Z''') = '''W'''. Naturally, π applies to linear words, circular words, and abstract dyad sizes. Additionally, we can choose ''k'' so that the two sizes of ''k''-steps in π(''s'') are:
+Let ''s'' be a PWF (thus primitive) scale. The case where ''s'' is equivalent to '''XYXZXYX''' can be manually verified, so by Theorem 4, the only remaining case is when ''s'' can be constructed by stacking two alternating sizes, '''g'''<sub>1</sub> and '''g'''<sub>2</sub>, of ''k''-steps. We assume that ''s'' has [[step signature]] ''a'''''X''' ''b'''''Y''' ''b'''''Z''' where ''a'' is odd. This ''k'' corresponds to a class of generators of the primitive MOS ''a'''''X''' 2''b'''''W'''. This MOS is obtained from ''s'' by applying the letterwise substitution function π such that π('''X''') = '''X''' and π('''Y''') = π('''Z''') = '''W'''. Naturally, π applies to linear words, circular words, and step vector sizes. Additionally, we can choose ''k'' so that the two sizes of ''k''-steps in π(''s'') are:
 * the perfect generator '''g''' = ''t'''''X''' + (''k'' &minus; ''t'')'''W''' (note (''k'' &minus; ''t'') is odd by a previous proof), and
 * the imperfect generator '''i''' = (''t'' + 1)'''X''' + (''k'' &minus; ''t'' &minus; 1)'''W'''.