Interleaving: Difference between revisions

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== 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 always exists a single-period 2-interleaved ternary [[word|scale word]] with that step signature imposing no nontrivial linear relations between step sizes. 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 always exists a single-period 2-interleaved ternary [[word|scale word]] with that step signature imposing no nontrivial linear relations between step sizes.(*) 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: This ternary scale word is the ''unique'' one with this property.

(*) Conjecture: This ternary scale word is the ''unique'' one with this property.

== Generalizations ==

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 == 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 always exists a single-period 2-interleaved ternary [[word|scale word]] with that step signature imposing no nontrivial linear relations between step sizes. 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 always exists a single-period 2-interleaved ternary [[word|scale word]] with that step signature imposing no nontrivial linear relations between step sizes.(*) 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: This ternary scale word is the ''unique'' one with this property.
+(*) Conjecture: This ternary scale word is the ''unique'' one with this property.
 == Generalizations ==