MOS substitution: Difference between revisions
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==Examples== | ==Examples== | ||
In the following tables, the generators stacked in the generating sequence is such that the perfect generator has fewer <math>\mathbf{X}</math> steps than the imperfect counterpart. | |||
=== 5L2m4s === | === 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: | ||
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|colspan=2| GS('''L'''+'''s''', '''L'''+'''s''', '''L'''+'''m''') || yes | |colspan=2| GS('''L'''+'''s''', '''L'''+'''s''', '''L'''+'''m''') || yes | ||
|} | |} | ||
( | === 5L2m6s === | ||
{| class="wikitable" | |||
|+ <math>5\mathbf{L}2\mathbf{m}6\mathbf{s}</math> as <math>\mathsf{MOS\_subst}(5, 2, 6; \mathbf{m}, \mathbf{s}; k)</math> | |||
|- | |||
!rowspan=2| <math>k</math> | |||
!rowspan=2| filling MOS (1 period) | |||
!rowspan=2| [[UDP]] for filling MOS | |||
!colspan=2| step pattern | |||
!colspan=2| generator sequence | |||
!rowspan=2| MOS for <math>\mathbf{s} = \mathbf{0}</math> | |||
|- | |||
!| template MOS: | |||
|| <code>LXLXXLXLXXLXX</code> | |||
!| intvl. class of gen.:(*) | |||
|| 5-steps | |||
|- | |||
| 3 || <code>msss</code> || 6|0(2) | |||
|colspan=2 style="text-align:right;"| <code>LmLssLsLmsLss</code> | |||
|colspan=2| GS((2'''L'''+'''m'''+2'''s''')<sup>3</sup>, 2'''L'''+3'''s''') || yes | |||
|- | |||
| 2 || <code>smss</code> || 4|2(2) | |||
|colspan=2 style="text-align:right;"| <code>LsLmsLsLsmLss</code> | |||
|colspan=2| GS((2'''L'''+'''m'''+2'''s''')<sup>2</sup>, 2'''L'''+3'''s''', 2'''L'''+'''m'''+2'''s''') || yes | |||
|- | |||
| 1 || <code>ssms</code> || 2|4(2) | |||
|colspan=2 style="text-align:right;"| <code>LsLsmLsLssLms</code> | |||
|colspan=2| GS(2'''L'''+'''m'''+2'''s''', 2'''L'''+3'''s''', (2'''L'''+'''m'''+2'''s''')<sup>2</sup>) || yes | |||
|- | |||
| 0 || <code>sssm</code> || 0|6(2) | |||
|colspan=2 style="text-align:right;"| <code>LsLssLmLssLsm</code> | |||
|colspan=2| GS(2'''L'''+3'''s''', (2'''L'''+'''m'''+2'''s''')<sup>3</sup>) || yes | |||
|} | |||
=== 6L7m9s === | === 6L7m9s === | ||