MOS substitution: Difference between revisions

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* ''a'''''X'''''b'''''Y'''(''k'') denotes the mode of ''a'''''X'''''b'''''Y''' which would have [[UDP]] notation <math>dk|d(a/d+b/d-1-k)\ (d), \ d = \gcd(a,b)</math> under the assumption '''X''' > '''Y''' > '''0'''.

== Formal definition ==

Suppose ''a'', ''b'', and ''c'' are positive integers. Suppose the noncircular word ''t'' is the brightest mode of the MOS scale ''a'''''x'''(''b'' + ''c'')'''X''' when treating '''x''' as larger than '''X''' and the noncircular word ''f'' is one rotation of the MOS scale ''b'''''y'''''c'''''z'''. Then the result of replacing the ''i''th occurrence of '''X''' in ''t'' replaced with the ''i''th letter of ''f'', treated as a circular word, is said to be a ''MOS substitution scale word'' with ''template MOS'' ''a'''''x'''(''b'' + ''c'')'''X''' and ''filling MOS'' ''b'''''y'''''c'''''z'''.

== Original derivation ==

MOS substitution was developed by [[Inthar]] for the purpose of adding [[aberrisma]] steps in an orderly manner to a MOS pattern <math>a\mathbf{L}b\mathbf{m}</math> (which we write in place of <math>a\mathbf{L}b\mathbf{s}</math> for convenience's sake, since <math>\mathbf{s}</math> denotes the new aberrisma steps added to the MOS) in the context of groundfault's [[aberrismic theory]]. MOS substitution is intended to take advantage of extra potential symmetry when <math>a, c</math> or <math>b, c</math> is not a coprime pair and mildly generalize the congruence substitution procedure for building [[balanced]] words to obtain non-balanced but still more "even" scales with simple [[generator sequence]] expressions (in the sense of being binary, i.e. using only two distinct generators).

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 * ''a'''''X'''''b'''''Y'''(''k'') denotes the mode of ''a'''''X'''''b'''''Y''' which would have [[UDP]] notation <math>dk|d(a/d+b/d-1-k)\ (d), \ d = \gcd(a,b)</math> under the assumption '''X''' > '''Y''' > '''0'''.
+== Formal definition ==
+Suppose ''a'', ''b'', and ''c'' are positive integers. Suppose the noncircular word ''t'' is the brightest mode of the MOS scale ''a'''''x'''(''b'' + ''c'')'''X''' when treating '''x''' as larger than '''X''' and the noncircular word ''f'' is one rotation of the MOS scale ''b'''''y'''''c'''''z'''. Then the result of replacing the ''i''th occurrence of '''X''' in ''t'' replaced with the ''i''th letter of ''f'', treated as a circular word, is said to be a ''MOS substitution scale word'' with ''template MOS'' ''a'''''x'''(''b'' + ''c'')'''X''' and ''filling MOS'' ''b'''''y'''''c'''''z'''.
 == Original derivation ==
 MOS substitution was developed by [[Inthar]] for the purpose of adding [[aberrisma]] steps in an orderly manner to a MOS pattern <math>a\mathbf{L}b\mathbf{m}</math> (which we write in place of <math>a\mathbf{L}b\mathbf{s}</math> for convenience's sake, since <math>\mathbf{s}</math> denotes the new aberrisma steps added to the MOS) in the context of groundfault's [[aberrismic theory]]. MOS substitution is intended to take advantage of extra potential symmetry when <math>a, c</math> or <math>b, c</math> is not a coprime pair and mildly generalize the congruence substitution procedure for building [[balanced]] words to obtain non-balanced but still more "even" scales with simple [[generator sequence]] expressions (in the sense of being binary, i.e. using only two distinct generators).