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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.
[[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 ==
== Formal definition ==
Suppose ''a'', ''b'', and ''c'' are positive integers. Suppose that the noncircular binary word ''t''('''x''', '''X''') is the brightest mode of the MOS scale word with step signature ''a'''''x'''(''b'' + ''c'')'''X''' when treating '''x''' as larger than '''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 ==
== Original derivation ==
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In particular, all [[monotone-MOS scale]]s (i.e. such that the results of {{nowrap|'''L''' {{=}} '''m''' | '''m''' {{=}} '''s'''}}, and {{nowrap|'''s''' {{=}} '''0'''}} temperings are MOSes) arise from MOS substitution in this way.
In particular, all [[monotone-MOS scale]]s (i.e. such that the results of {{nowrap|'''L''' {{=}} '''m''' | '''m''' {{=}} '''s'''}}, and {{nowrap|'''s''' {{=}} '''0'''}} temperings are MOSes) arise from MOS substitution in this way.
=== If a ternary scale satisfies all three possible MOS-substitution types, then it is pairwise-MOS and deletion-MOS ===
This fact is immediate. (See [[pairwise-MOS]] and [[deletion-MOS]].)
Corollary (by [[Ternary scale theorems]]): Such a scale is [[Fraenkel word|Fraenkel]], [[odd-regular]], or [[even-regular]].


=== If the template MOS is primitive, MOS substitution yields binary well-formed generator sequences ===
=== If the template MOS is primitive, MOS substitution yields binary well-formed generator sequences ===
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* the imperfect generator of the filling MOS corresponds to looping back to ''C''<sub>1</sub> but on the next note of ''C''<sub>1</sub>, so it and the ''q'' &minus; 1 notes thereafter are advanced by 1 note from any predecessor notes in the chains.
* the imperfect generator of the filling MOS corresponds to looping back to ''C''<sub>1</sub> but on the next note of ''C''<sub>1</sub>, so it and the ''q'' &minus; 1 notes thereafter are advanced by 1 note from any predecessor notes in the chains.


Hence these particular MOS substitution scales satisfy a property that we call ''parallelogram substring''. An '''e'''-equivalent scale is a ''parallelogram substring'' if there exist integers ''m'' > 0, ''n'' > 0, 0 &le; ''a'' < ''n'', 0 &le; ''b'' < ''n'', a vector '''a''', and two linearly independent vectors '''v''' and '''w''' such that the set of notes in the scale as a subset of the lattice of '''e'''-equivalent pitches is  
Hence these particular MOS substitution scales satisfy a property that we call ''[[parallelogram substring scale|parallelogram substring]]''. An '''e'''-equivalent scale is a ''parallelogram substring'' if there exist integers ''m'' > 0, ''n'' > 0, 0 &le; ''a'' < ''n'', 0 &le; ''b'' < ''n'', a vector '''a''', and two linearly independent vectors '''v''' and '''w''' such that the set of notes in the scale as a subset of the lattice of '''e'''-equivalent pitches is  


<math>
<math>
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Case 2: Neither of ''k'' and {{nowrap|''k'' + 1}} equals ({{nowrap|''b'' + ''c''}}). Here, if '''Y''' occurs ''j'' or {{nowrap|''j'' + 1}} times in a window of size ''k'', then '''Y''' occurs {{nowrap|''j'' + 1}} or {{nowrap|''j'' + 2}} times in a window of size {{nowrap|''k'' + 2}}.
Case 2: Neither of ''k'' and {{nowrap|''k'' + 1}} equals ({{nowrap|''b'' + ''c''}}). Here, if '''Y''' occurs ''j'' or {{nowrap|''j'' + 1}} times in a window of size ''k'', then '''Y''' occurs {{nowrap|''j'' + 1}} or {{nowrap|''j'' + 2}} times in a window of size {{nowrap|''k'' + 2}}.
=== Ternary parallelogram scales are MOS substitution scales ===
:''Main article: [[Ternary parallelogram scales are MOS substitution]]''


== MOS substitution scales and RTT ==
== MOS substitution scales and RTT ==
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