Interleaving: Difference between revisions
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If ''k'' > 1, stack the word of ''k''-steps in the scale, yielding a circular word ''T'', which traverses all notes of ''S'' since gcd(''k'', 2(''a'' + ''b'')) = 1. Since ''k'' is odd, the letters of this word alternate between beginning in ''S''<sub>1</sub> and beginning in ''S''<sub>2</sub>. By a reasoning similar to the above, ''T'' has a letter '''δ''' between its two mutually interleaved strands. By the lemma this scale has step variety ''r'' > 3, say with letters '''W'''<sub>1</sub>, ..., '''W'''<sub>''r''</sub>. Let ''k' '' be the inverse of ''k'' mod 2(''a'' + ''b''). By stacking ''k' ''-step subwords of ''T'', we end up with at least 4 different linear equations with 3 unknowns '''X''', '''Y''', and '''Z''', implying a linear relation between '''X''', '''Y''', and '''Z''' (?). This is a contradiction as '''X''', '''Y''', and '''Z''' are assumed to not have a linear relation. | If ''k'' > 1, stack the word of ''k''-steps in the scale, yielding a circular word ''T'', which traverses all notes of ''S'' since gcd(''k'', 2(''a'' + ''b'')) = 1. Since ''k'' is odd, the letters of this word alternate between beginning in ''S''<sub>1</sub> and beginning in ''S''<sub>2</sub>. By a reasoning similar to the above, ''T'' has a letter '''δ''' between its two mutually interleaved strands. By the lemma this scale has step variety ''r'' > 3, say with letters '''W'''<sub>1</sub>, ..., '''W'''<sub>''r''</sub>. Let ''k' '' be the inverse of ''k'' mod 2(''a'' + ''b''). By stacking ''k' ''-step subwords of ''T'', we end up with at least 4 different linear equations with 3 unknowns '''X''', '''Y''', and '''Z''', implying a linear relation between '''X''', '''Y''', and '''Z''' (?). This is a contradiction as '''X''', '''Y''', and '''Z''' are assumed to not have a linear relation. | ||
Case 3: gcd(2(''a'' + ''b''), ''k'') > 1. ''k'' being even contradicts the interleaving property, hence ''k'' = ''a'' + ''b''. (To be continued) | Case 3: gcd(2(''a'' + ''b''), ''k'') > 1. ''k'' being even contradicts the interleaving property, hence ''k'' = ''a'' + ''b'' which must be odd. (To be continued) | ||
== Generalizations == | == Generalizations == | ||