MOS substitution: Difference between revisions

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'''MOS substitution''' is a procedure for obtaining a ternary (3 step sizes) scale from two [[MOS]] patterns. It consists of substituting the step pattern of one MOS pattern (called the filling MOS), scale step for scale step, for all occurrences of a chosen step size of another MOS pattern (called the template MOS). Unlike MV3 scales, a MOS substitution scale may have any combination of step sizes.

'''MOS substitution''' is a procedure for obtaining a ternary (3 step sizes) scale from two [[MOS]] patterns. It consists of substituting the step pattern of one MOS pattern (called the ''filling MOS''), scale step for scale step, for all occurrences of a chosen step size of another MOS pattern (called the ''template MOS''). Unlike MV3 scales, a MOS substitution scale may have any combination of step sizes.

[[Aberrismic theory]] uses MOS substitution. In fact, groundfault reports having come up with a similar concept but not following up on it.

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== Formal definition ==

~~Suppose~~ ''a'', ''b''~~, and~~ ''c'' ~~are positive integers. Suppose that the noncircular binary word~~ ''t''('''x''', ''~~'X'~~''~~) is the lexicographically first mode of the MOS~~ scale word ~~with step signature ''~~a'''''x'''(~~''b'' + ''c''~~)~~'''X''' under the ordering~~ '''x''' < ~~'''X''', and that the noncircular binary word ''f''~~(~~'''y''', '''z'''~~) ~~is one rotation of the MOS scale word with step signature ''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'' for 0 ≤ ''i'' < |''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''', or~~ in ~~shorthand,~~ a ~~"subst ''a'''''x'''~~(''~~b'''''y'''''c'''''z'~~'')".

A ternary scale word ''w''('''x'''1, '''x'''2, '''x'''3) is a ''MOS substitution'' scale word if there exists a permutation <math>\pi \in S_3</math> such that the following holds:

* identifying '''x'''π(1) and '''x'''π(2) results in a MOS (called the ''template MOS'') and

* deleting all instances of '''x'''π(3) (called the ''slot letter'') results in a MOS (called the ''filling MOS'').

== Original derivation ==

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-'''MOS substitution''' is a procedure for obtaining a ternary (3 step sizes) scale from two [[MOS]] patterns. It consists of substituting the step pattern of one MOS pattern (called the filling MOS), scale step for scale step, for all occurrences of a chosen step size of another MOS pattern (called the template MOS). Unlike MV3 scales, a MOS substitution scale may have any combination of step sizes.
+'''MOS substitution''' is a procedure for obtaining a ternary (3 step sizes) scale from two [[MOS]] patterns. It consists of substituting the step pattern of one MOS pattern (called the ''filling MOS''), scale step for scale step, for all occurrences of a chosen step size of another MOS pattern (called the ''template MOS''). Unlike MV3 scales, a MOS substitution scale may have any combination of step sizes.
 [[Aberrismic theory]] uses MOS substitution. In fact, groundfault reports having come up with a similar concept but not following up on it.
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 == Formal definition ==
-Suppose ''a'', ''b'', and ''c'' are positive integers. Suppose that the noncircular binary word ''t''('''x''', '''X''') is the lexicographically first mode of the MOS scale word with step signature ''a'''''x'''(''b'' + ''c'')'''X''' under the ordering '''x''' < '''X''', and that the noncircular binary word ''f''('''y''', '''z''') is one rotation of the MOS scale word with step signature ''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'' for 0 &le; ''i'' < |''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''', or in shorthand, a "subst&nbsp;''a'''''x'''(''b'''''y'''''c'''''z''')".
+A ternary scale word ''w''('''x'''<sub>1</sub>, '''x'''<sub>2</sub>, '''x'''<sub>3</sub>) is a ''MOS substitution'' scale word if there exists a permutation <math>\pi \in S_3</math> such that the following holds:
+* identifying '''x'''<sub>π(1)</sub> and '''x'''<sub>π(2)</sub> results in a MOS (called the ''template MOS'') and
+* deleting all instances of '''x'''<sub>π(3)</sub> (called the ''slot letter'') results in a MOS (called the ''filling MOS'').
 == Original derivation ==