Ternary scale theorems: Difference between revisions

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

=== Proof ===

Let ''S'' be a PWF 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 scale 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 function π such that π('''X''') = '''X''' and π('''Y''') = π('''Z''') = '''W'''. Naturally, π applies to abstract dyad sizes. Additionally, we can choose ''k'' so that the two sizes of ''k''-steps in π(''S'') are:

Let ''S'' be a primitive PWF 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 scale 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 function π such that π('''X''') = '''X''' and π('''Y''') = π('''Z''') = '''W'''. Naturally, π applies to abstract dyad 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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 All pairwise-well-formed scales are [[balanced]].
 === Proof ===
-Let ''S'' be a PWF 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 scale 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 function π such that π('''X''') = '''X''' and π('''Y''') = π('''Z''') = '''W'''. Naturally, π applies to abstract dyad sizes. Additionally, we can choose ''k'' so that the two sizes of ''k''-steps in π(''S'') are:
+Let ''S'' be a primitive PWF 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 scale 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 function π such that π('''X''') = '''X''' and π('''Y''') = π('''Z''') = '''W'''. Naturally, π applies to abstract dyad 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'''.