MOS substitution: Difference between revisions

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== Mathematical facts ==

=== A ternary scale whose L = m and s = 0 temperings are MOS comes from MOS substitution ===

If a ternary scale with step signature ''a'''''L'''''b'''''m'''''c'''''s''' satisfies:

# the result of identifying '''L''' steps with '''m''' steps is a MOS;

# the result of deleting all '''s''' steps is a MOS,

then it is a MOS substitution scale, namely subst((''a''+''b'')'''X'''''c'''''s'''(''i''), '''X''', ''a'''''L'''''b'''''m'''(''j'')) for some brightnesses ''i'' and ''j''.

In particular, all monotone-MOS{{idiosyncratic}} scales (i.e. such that the results of '''L''' = '''m''', '''m''' = '''s''', and '''s''' = '''0''' temperings are MOSes) arise from MOS substitution in this way.

=== If the template MOS is primitive, MOS substitution yields generator sequences ===

The following holds for <math>S = \mathsf{MOS\_subst}(a, b, c; \mathbf{L}, \mathbf{s}; k)</math> (and after switching <math>\mathbf{L}</math> with <math>\mathbf{m}</math> and <math>a</math> with <math>b,</math> for <math>\mathsf{MOS\_subst}(a, b, c; \mathbf{m}, \mathbf{s}; k)</math> as well):

@@ Line 132: / Line 132: @@
 == Mathematical facts ==
+=== A ternary scale whose L = m and s = 0 temperings are MOS comes from MOS substitution ===
+If a ternary scale with step signature ''a'''''L'''''b'''''m'''''c'''''s''' satisfies:
+# the result of identifying '''L''' steps with '''m''' steps is a MOS;
+# the result of deleting all '''s''' steps is a MOS,
+then it is a MOS substitution scale, namely subst((''a''+''b'')'''X'''''c'''''s'''(''i''), '''X''', ''a'''''L'''''b'''''m'''(''j'')) for some brightnesses ''i'' and ''j''.
+In particular, all monotone-MOS{{idiosyncratic}} scales (i.e. such that the results of '''L''' = '''m''', '''m''' = '''s''', and '''s''' = '''0''' temperings are MOSes) arise from MOS substitution in this way.
 === If the template MOS is primitive, MOS substitution yields generator sequences ===
 The following holds for <math>S = \mathsf{MOS\_subst}(a, b, c; \mathbf{L}, \mathbf{s}; k)</math> (and after switching <math>\mathbf{L}</math> with <math>\mathbf{m}</math> and <math>a</math> with <math>b,</math> for <math>\mathsf{MOS\_subst}(a, b, c; \mathbf{m}, \mathbf{s}; k)</math> as well):