|
|
| Line 23: |
Line 23: |
| * the perfect generator of the filling MOS corresponds to advancing from ''C''<sub>''i''</sub> to ''C''<sub>''i''+1</sub> | | * the perfect generator of the filling MOS corresponds to advancing from ''C''<sub>''i''</sub> to ''C''<sub>''i''+1</sub> |
| * 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'' − 1 notes thereafter are advanced by 1 note in the chain. | | * 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'' − 1 notes thereafter are advanced by 1 note in the chain. |
|
| |
| ==Examples==
| |
| 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:
| |
| {| 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>
| |
| |-
| |
| !rowspan=2| <math>k</math>
| |
| !rowspan=2| filling MOS
| |
| !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>LXLXLXLXLXX</code>
| |
| !| intvl. class of gen.:
| |
| || 2-steps
| |
| |-
| |
| | 2 || <code>mssmss</code> || 4|0(2)
| |
| |colspan=2 style="text-align:right;"| <code>LmLsLsLmLss</code>
| |
| |colspan=2| GS('''L'''+'''m''', '''L'''+'''s''', '''L'''+'''s''') || yes
| |
| |-
| |
| | 1 || <code>smssms</code> || 2|2(2)
| |
| |colspan=2 style="text-align:right;"| <code>LsLmLsLsLms</code>
| |
| |colspan=2| GS('''L'''+'''s''', '''L'''+'''m''', '''L'''+'''s''') || yes
| |
| |-
| |
| | 0 || <code>ssmssm</code> || 0|4(2)
| |
| |colspan=2 style="text-align:right;"| <code>LsLsLmLsLsm</code>
| |
| |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
| |
| |}
| |
| Here the notation ''G''<sup>''k''</sup> denotes repeating the generator ''G'' ''k'' times in the generator sequence.
| |
|
| |
| These are four of the 8 billiard words that have pattern 5L2m6s. The other four billiard words have length-3 subwords of non-X letters, unlike the MOS substitution scales.
| |
|
| |
| This scale pattern is available in [[37edo]] with step ratio 5:3:1; the generator sequence in the tuning has 2'''L'''+'''m'''+2'''s''' = 486.5 (~4/3) and 2'''L'''+3'''s''' = 421.6 (~14/11), and notably this tuning represents all primes from 3 to 13 with only 3 being inaccurate. 65edo's 9:7:1 is another optimum for 2.3.5.11.13, and is given by a GS using three 4/3's and one 5/4.
| |
|
| |
| === 6L7m9s ===
| |
| {| class="wikitable"
| |
| |+ <math>6\mathbf{L}7\mathbf{m}9\mathbf{s}</math> as <math>\mathsf{MOS\_subst}(6, 7, 9; \mathbf{L}, \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>mXXmXXmXXmXXmXXmXXmXXX</code>
| |
| !| intvl. class of gen.:
| |
| || 3-steps
| |
| |-
| |
| | 4 || <code>LsLss</code> || 12|0(3)
| |
| |colspan=2 style="text-align:right;"| <code>mLsmLsmsLmsLmssmLsmLss</code>
| |
| |colspan=2| GS('''L'''+'''m'''+'''s''', '''L'''+'''m'''+'''s''', '''L'''+'''m'''+'''s''', '''L'''+'''m'''+'''s''', '''m'''+2'''s''') || yes
| |
| |-
| |
| | 3 || <code>LssLs</code> || 9|3(3)
| |
| |colspan=2 style="text-align:right;"| <code>mLsmsLmsLmssmLsmLsmsLs</code>
| |
| |colspan=2| GS('''L'''+'''m'''+'''s''', '''L'''+'''m'''+'''s''', '''L'''+'''m'''+'''s''', '''m'''+2'''s''', '''L'''+'''m'''+'''s''') || yes
| |
| |-
| |
| | 2 || <code>sLsLs</code> || 6|6(3)
| |
| |colspan=2 style="text-align:right;"| <code>msLmsLmssmLsmLsmsLmsLs</code>
| |
| |colspan=2| GS('''L'''+'''m'''+'''s''', '''L'''+'''m'''+'''s''', '''m'''+2'''s''', '''L'''+'''m'''+'''s''', '''L'''+'''m'''+'''s''') || yes
| |
| |-
| |
| | 1 || <code>sLssL</code> || 3|9(3)
| |
| |colspan=2 style="text-align:right;"| <code>msLmssmLsmLsmsLmsLmssL</code>
| |
| |colspan=2| GS('''L'''+'''m'''+'''s''', '''m'''+2'''s''', '''L'''+'''m'''+'''s''', '''L'''+'''m'''+'''s''', '''L'''+'''m'''+'''s''') || yes
| |
| |-
| |
| | 0 || <code>ssLsL</code> || 0|12(3)
| |
| |colspan=2 style="text-align:right;"| <code>mssmLsmLsmsLmsLmssmLsL</code>
| |
| |colspan=2| GS('''m'''+2'''s''', '''L'''+'''m'''+'''s''', '''L'''+'''m'''+'''s''', '''L'''+'''m'''+'''s''', '''L'''+'''m'''+'''s''') || no
| |
| |}
| |
|
| |
|
| == Open questions == | | == Open questions == |