MOS substitution: Difference between revisions

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In the following tables, the interval class of the generators stacked in the generator sequence is such that the perfect generator has fewer <math>\mathbf{X}</math> steps than the imperfect counterpart.

=== 5L2m4s ===

To derive groundfault's diamech scale which has step pattern <math>5\mathbf{L}2\mathbf{m}4\mathbf{s}</math> as <math>\mathsf{MOS\_subst}(5, 2, 4; \mathbf{m}, \mathbf{s}; k)</math>, we exploit <math>(b, c) = 2</math> and substitute <math>2\mathbf{m}4\mathbf{s}</math> into the template MOS <math>5\mathbf{L}6\mathbf{X}</math> (<math>\mathbf{LXLXLXLXLXX}</math>). Since <math>2\mathbf{m}4\mathbf{s}</math> has three distinct modes (<math>\mathbf{ssmssm}, \mathbf{smssms}, \mathbf{mssmss}</math>) and <math>5\mathbf{L}6\mathbf{X}</math> is primitive, we obtain three distinct scales, all of which admit length-3 generator sequences of 2-steps, representing all 3 possible rotations of <math>(\mathbf{L}+\mathbf{m}, \mathbf{L}+\mathbf{s}, \mathbf{L}+\mathbf{s})</math> as displayed in the following table:

To derive groundfault's [[diamech]] scale which has step pattern <math>5\mathbf{L}2\mathbf{m}4\mathbf{s}</math> as <math>\mathsf{MOS\_subst}(5, 2, 4; \mathbf{m}, \mathbf{s}; k)</math>, we exploit <math>(b, c) = 2</math> and substitute <math>2\mathbf{m}4\mathbf{s}</math> into the template MOS <math>5\mathbf{L}6\mathbf{X}</math> (<math>\mathbf{LXLXLXLXLXX}</math>). Since <math>2\mathbf{m}4\mathbf{s}</math> has three distinct modes (<math>\mathbf{ssmssm}, \mathbf{smssms}, \mathbf{mssmss}</math>) and <math>5\mathbf{L}6\mathbf{X}</math> is primitive, we obtain three distinct scales, all of which admit length-3 generator sequences of 2-steps, representing all 3 possible rotations of <math>(\mathbf{L}+\mathbf{m}, \mathbf{L}+\mathbf{s}, \mathbf{L}+\mathbf{s})</math> as displayed in the following table:

{| class="wikitable"

|+ <math>5\mathbf{L}2\mathbf{m}4\mathbf{s}</math> as <math>\mathsf{MOS\_subst}(5, 2, 4; \mathbf{m}, \mathbf{s}; k)</math>

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|colspan=2| GS('''L'''+'''s''', '''L'''+'''s''', '''L'''+'''m''') || yes

|}

=== 5L2m6s ===

{| class="wikitable"

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 In the following tables, the interval class of the generators stacked in the generator sequence is such that the perfect generator has fewer <math>\mathbf{X}</math> steps than the imperfect counterpart.
 === 5L2m4s ===
-To derive groundfault's diamech scale which has step pattern <math>5\mathbf{L}2\mathbf{m}4\mathbf{s}</math> as <math>\mathsf{MOS\_subst}(5, 2, 4; \mathbf{m}, \mathbf{s}; k)</math>, we exploit <math>(b, c) = 2</math> and substitute <math>2\mathbf{m}4\mathbf{s}</math> into the template MOS <math>5\mathbf{L}6\mathbf{X}</math> (<math>\mathbf{LXLXLXLXLXX}</math>). Since <math>2\mathbf{m}4\mathbf{s}</math> has three distinct modes (<math>\mathbf{ssmssm}, \mathbf{smssms}, \mathbf{mssmss}</math>) and <math>5\mathbf{L}6\mathbf{X}</math> is primitive, we obtain three distinct scales, all of which admit length-3 generator sequences of 2-steps, representing all 3 possible rotations of <math>(\mathbf{L}+\mathbf{m}, \mathbf{L}+\mathbf{s}, \mathbf{L}+\mathbf{s})</math> as displayed in the following table:
+To derive groundfault's [[diamech]] scale which has step pattern <math>5\mathbf{L}2\mathbf{m}4\mathbf{s}</math> as <math>\mathsf{MOS\_subst}(5, 2, 4; \mathbf{m}, \mathbf{s}; k)</math>, we exploit <math>(b, c) = 2</math> and substitute <math>2\mathbf{m}4\mathbf{s}</math> into the template MOS <math>5\mathbf{L}6\mathbf{X}</math> (<math>\mathbf{LXLXLXLXLXX}</math>). Since <math>2\mathbf{m}4\mathbf{s}</math> has three distinct modes (<math>\mathbf{ssmssm}, \mathbf{smssms}, \mathbf{mssmss}</math>) and <math>5\mathbf{L}6\mathbf{X}</math> is primitive, we obtain three distinct scales, all of which admit length-3 generator sequences of 2-steps, representing all 3 possible rotations of <math>(\mathbf{L}+\mathbf{m}, \mathbf{L}+\mathbf{s}, \mathbf{L}+\mathbf{s})</math> as displayed in the following table:
 {| class="wikitable"
 |+ <math>5\mathbf{L}2\mathbf{m}4\mathbf{s}</math>  as <math>\mathsf{MOS\_subst}(5, 2, 4; \mathbf{m}, \mathbf{s}; k)</math>
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 |colspan=2| GS('''L'''+'''s''', '''L'''+'''s''', '''L'''+'''m''') || yes
 |}
 === 5L2m6s ===
 {| class="wikitable"