Interleaving: Difference between revisions

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== Ternary interleaved scales ==

~~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.

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'''.

=== Lemma ===

If ''S'' consists of the subwords '''XZ''' and '''YZ''' arranged in the pattern of a single-period binary circular word ''w'', and ''k'' is an odd number greater than 1 and less than |''S''|, then the class of ''k''-steps has more than 3 abstract intervals.

Proof: Let ''A'' be the set of all (''k'' - 1)/2-step intervals of ''w''. |''A''| = 1 implies that ''k'' = 1 mod |''w''|, so |''A''| ≥ 2 and contains at least two intervals '''w'''1 and '''w'''2.

Say |''A''| = 2. (If |''A''| ≥ 3 then the proof is easy.) Suppose the (''k'' - 1)-step subword σ of ''S'' has size '''w'''1, and assume WOLOG that σ ends in '''Z'''. Then '''Z'''σ and σ'''W''' are subwords, where '''W''' is a non-'''Z''' letter. Assume WOLOG '''W''' = '''X'''. Then we have that the word σ'''X''' = '''V'''τ for some non-'''Z''' letter '''V''' and subword τ.

=== Proof of Theorem ===

Proof of Conjecture: Suppose the scale is made of two interleaved subsets offset by the abstract interval '''δ'''.

Case 1: gcd(2(''a'' + ''b''), ''k'') = 1. If ''k'' = 1, then any '''Z'''''q'' 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''':

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 == Ternary interleaved scales ==
-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.
+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'''.
+=== Lemma ===
+If ''S'' consists of the subwords '''XZ''' and '''YZ''' arranged in the pattern of a single-period binary circular word ''w'', and ''k'' is an odd number greater than 1 and less than |''S''|, then the class of ''k''-steps has more than 3 abstract intervals.
+Proof: Let ''A'' be the set of all (''k'' - 1)/2-step intervals of ''w''. |''A''| = 1 implies that ''k'' = 1 mod |''w''|, so |''A''| &ge; 2 and contains at least two intervals '''w'''<sub>1</sub> and '''w'''<sub>2</sub>.
+Say |''A''| = 2. (If |''A''| &ge; 3 then the proof is easy.) Suppose the (''k'' - 1)-step subword σ of ''S'' has size '''w'''<sub>1</sub>, and assume WOLOG that σ ends in '''Z'''. Then '''Z'''σ and σ'''W''' are subwords, where '''W''' is a non-'''Z''' letter. Assume WOLOG '''W''' = '''X'''. Then we have that the word σ'''X''' = '''V'''τ for some non-'''Z''' letter '''V''' and subword τ.
+=== Proof of Theorem ===
+Proof of Conjecture: Suppose the scale is made of two interleaved subsets offset by the abstract interval '''δ'''.
 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''':