Interleaving: Difference between revisions

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Moreover, the offset '''δ''' = '''Z''', i.e. ''S''2 is separated by the interval '''Z''' to the right of ''S''1.

If any maximal subword of consecutive '''Z'''s has ''q'' > 1, then the scale can be split into two subwords of length ''a'' + ''b'', ''w''1 with the maximal number of consecutive '''Z''''s and ''w''2 with the minimal number of '''Z'''s. ~~We can also choose ''w''1 to begin in ''S''1 (because?)~~

If any maximal subword of consecutive '''Z'''s has ''q'' > 1, then the scale can be split into two subwords of length ''a'' + ''b'', ''w''1 with the maximal number of consecutive '''Z''''s and ''w''2 with the minimal number of '''Z'''s. Scoot ''w''1 to the right one step at a time until it loses one '''Z'''. This proves either that a non-'''Z''' letter is equal to '''Z''' or that the offset goes in the wrong direction. Hence ''q'' = 1, as desired.

Scoot ''w''1 to the right one step at a time until it loses one '''Z'''. This proves either that a non-'''Z''' letter is equal to '''Z''' or that the offset goes in the wrong direction. Hence ''q'' = 1, as desired.

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''1 and beginning in ''S''2. 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'''1, ..., '''W'''''r''. 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 nontrivial linear relation between '''X''', '''Y''', and '''Z''' (?). This is a contradiction as '''X''', '''Y''', and '''Z''' are assumed to not have a linear relation.

@@ Line 93: / Line 93: @@
 Moreover, the offset '''δ''' = '''Z''', i.e. ''S''<sub>2</sub> is separated by the interval '''Z''' to the right of ''S''<sub>1</sub>.
-If any maximal subword of consecutive '''Z'''s has ''q'' > 1, then the scale can be split into two subwords of length ''a'' + ''b'', ''w''<sub>1</sub> with the maximal number of consecutive '''Z''''s and ''w''<sub>2</sub> with the minimal number of '''Z'''s. We can also choose ''w''<sub>1</sub> to begin in ''S''<sub>1</sub> (because?)
+If any maximal subword of consecutive '''Z'''s has ''q'' > 1, then the scale can be split into two subwords of length ''a'' + ''b'', ''w''<sub>1</sub> with the maximal number of consecutive '''Z''''s and ''w''<sub>2</sub> with the minimal number of '''Z'''s. Scoot ''w''<sub>1</sub> to the right one step at a time until it loses one '''Z'''. This proves either that a non-'''Z''' letter is equal to '''Z''' or that the offset goes in the wrong direction. Hence ''q'' = 1, as desired.
-Scoot ''w''<sub>1</sub> to the right one step at a time until it loses one '''Z'''. This proves either that a non-'''Z''' letter is equal to '''Z''' or that the offset goes in the wrong direction. Hence ''q'' = 1, as desired.
 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 nontrivial linear relation between '''X''', '''Y''', and '''Z''' (?). This is a contradiction as '''X''', '''Y''', and '''Z''' are assumed to not have a linear relation.