Interleaving: Difference between revisions
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== Ternary interleaved scales == | == Ternary interleaved scales == | ||
Given a [[ternary]] [[step signature]] of the form {{nowrap|''a'''''X'''''b'''''Y'''(''a'' + ''b'')'''Z'''}} where gcd(''a'', ''b'') = 1, there exists a unique single-period (abstractly) 2-interleaved ternary [[word|scale word]] with that step signature. This ternary scale word consists of ''a''-many '''XZ''' subwords and ''b''-many '''YZ''' subwords arranged in a MOS pattern (like the steps of ''a'''''L'''''b'''''s''') and consists of an interleaved pair of two {{nowrap|''a''('''X''' + '''Z''')''b''('''Y''' + '''Z''')}} subsets, offset by '''Z'''. [[Blackdye]] ('''sLmLsLmLsL''', 5'''L'''2'''m'''3'''s''') and [[whitedye]] ('''LsLsLsmsLsLsms''', 5'''L'''2'''m'''7'''s''') are examples of this. | Conjecture: Given a [[ternary]] [[step signature]] of the form {{nowrap|''a'''''X'''''b'''''Y'''(''a'' + ''b'')'''Z'''}} where gcd(''a'', ''b'') = 1, there exists a unique single-period (abstractly) 2-interleaved ternary [[word|scale word]] with that step signature. This ternary scale word consists of ''a''-many '''XZ''' subwords and ''b''-many '''YZ''' subwords arranged in a MOS pattern (like the steps of ''a'''''L'''''b'''''s''') and consists of an interleaved pair of two {{nowrap|''a''('''X''' + '''Z''')''b''('''Y''' + '''Z''')}} subsets, offset by '''Z'''. [[Blackdye]] ('''sLmLsLmLsL''', 5'''L'''2'''m'''3'''s''') and [[whitedye]] ('''LsLsLsmsLsLsms''', 5'''L'''2'''m'''7'''s''') are examples of this. | ||
Proof: Suppose the scale is made of two interleaved subsets offset by the abstract interval '''δ''' | Proof: Suppose the scale is made of two interleaved subsets offset by the abstract interval '''δ'''. | ||
Case 1: ''k'' = 1. | Case 1: gcd(2(''a'' + ''b''), ''k'') = 1. If ''k'' = 1, then any '''Z'''<sup>''q''</sup> a maximal subword of consecutive '''Z'''s has ''q'' odd, and they all must be the same length and separated by one non-'''Z''' letter '''W''': | ||
'''ZZ ZZ ... ZW''' (in strand ''S''<sub>1</sub>) | '''ZZ ZZ ... ZW''' (in strand ''S''<sub>1</sub>) | ||
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Scoot ''w''<sub>1</sub> to the right one step at a time until it loses one '''Z''', or scoot ''w''<sub>2</sub> to the right until it gains one '''Z'''. Because of the offset and because either ''w''<sub>1</sub> or ''w''<sub>2</sub> begins in ''S''<sub>1</sub> (because ''a'' + ''b'' is odd), this proves that a non-'''Z''' letter is equal to '''Z'''. Hence ''q'' = 1, as desired. | Scoot ''w''<sub>1</sub> to the right one step at a time until it loses one '''Z''', or scoot ''w''<sub>2</sub> to the right until it gains one '''Z'''. Because of the offset and because either ''w''<sub>1</sub> or ''w''<sub>2</sub> begins in ''S''<sub>1</sub> (because ''a'' + ''b'' is odd), this proves that a non-'''Z''' letter is equal to '''Z'''. Hence ''q'' = 1, as desired. | ||
If ''k'' > 1, stack the word of ''k''-steps in the scale, yielding a circular word ''T''. 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. (To be continued) | |||
Case 3: ''k'' = ''a'' + ''b''. | Case 3: gcd(2(''a'' + ''b''), ''k'') > 1. ''k'' being even contradicts the interleaving property, hence ''k'' = ''a'' + ''b''. (To be continued) | ||
== Generalizations == | == Generalizations == | ||