MOS substitution: Difference between revisions
| Line 200: | Line 200: | ||
Consider the mode of the template MOS <math>T = T(\mathbf{m},\mathbf{X}) = (a+c)\mathbf{X}b\mathbf{m}(0).</math> This is the mode of <math>T</math> that has the most <math>\mathbf{X}</math> steps near the end. If <math>T</math> is [[primitive]], let <math>r</math> be the count of <math>\mathbf{X}</math> steps in a chosen (reduced) generator of <math>T.</math> Since <math>r</math> must be coprime to <math>a+c,</math> <math>r</math>-steps in the filling MOS <math>F = a\mathbf{L}c\mathbf{s}(k)</math> come in exactly 2 sizes, <math>i\mathbf{L}+j\mathbf{s}</math> and <math>(i-1)\mathbf{L}+(j+1)\mathbf{s}.</math> Since the detempering of the imperfect generator of <math>T</math> occurs only once in <math>S</math>, <math>S</math> admits a particularly elegant well-formed binary (using two distinct generators) [[generator sequence]] of length <math>q = \frac{a+c}{\gcd(a,c)},</math> the period of the filling MOS. The generator sequence corresponds to the circle of <math>r</math>-steps in the filling MOS. Letting <math>\mathsf{GS}(g_1, ..., g_{q})</math> be this generator sequence, <math>g_j</math> is either <math>p\mathbf{m} + i\mathbf{L} + j\mathbf{s}</math> or <math>p\mathbf{m} + (i-1)\mathbf{L} + (j+1)\mathbf{s},</math> according as the <math>j</math>-th <math>r</math>-step in the sequence of stacked <math>r</math>-steps on the chosen mode of <math>F</math> is <math>i\mathbf{L} + j\mathbf{s}</math> or <math>(i-1)\mathbf{L} + (j+1)\mathbf{s}.</math> (We could have chosen to use the mode of <math>T</math> on the other extreme of its generator arc instead, which corresponds to taking the circle of <math>(a+c - r)</math>-steps in <math>F</math> and is thus also valid.) The generator of the template MOS serves as the "guide generator" for this generator sequence. | Consider the mode of the template MOS <math>T = T(\mathbf{m},\mathbf{X}) = (a+c)\mathbf{X}b\mathbf{m}(0).</math> This is the mode of <math>T</math> that has the most <math>\mathbf{X}</math> steps near the end. If <math>T</math> is [[primitive]], let <math>r</math> be the count of <math>\mathbf{X}</math> steps in a chosen (reduced) generator of <math>T.</math> Since <math>r</math> must be coprime to <math>a+c,</math> <math>r</math>-steps in the filling MOS <math>F = a\mathbf{L}c\mathbf{s}(k)</math> come in exactly 2 sizes, <math>i\mathbf{L}+j\mathbf{s}</math> and <math>(i-1)\mathbf{L}+(j+1)\mathbf{s}.</math> Since the detempering of the imperfect generator of <math>T</math> occurs only once in <math>S</math>, <math>S</math> admits a particularly elegant well-formed binary (using two distinct generators) [[generator sequence]] of length <math>q = \frac{a+c}{\gcd(a,c)},</math> the period of the filling MOS. The generator sequence corresponds to the circle of <math>r</math>-steps in the filling MOS. Letting <math>\mathsf{GS}(g_1, ..., g_{q})</math> be this generator sequence, <math>g_j</math> is either <math>p\mathbf{m} + i\mathbf{L} + j\mathbf{s}</math> or <math>p\mathbf{m} + (i-1)\mathbf{L} + (j+1)\mathbf{s},</math> according as the <math>j</math>-th <math>r</math>-step in the sequence of stacked <math>r</math>-steps on the chosen mode of <math>F</math> is <math>i\mathbf{L} + j\mathbf{s}</math> or <math>(i-1)\mathbf{L} + (j+1)\mathbf{s}.</math> (We could have chosen to use the mode of <math>T</math> on the other extreme of its generator arc instead, which corresponds to taking the circle of <math>(a+c - r)</math>-steps in <math>F</math> and is thus also valid.) The generator of the template MOS serves as the "guide generator" for this generator sequence. | ||
=== If the template is a primitive MOS, and for some perfect generators <math>p_T, p_F, \ r := \left|p_T\right|_\mathbf{X} = \left|p_F\right|,</math> then MOS substitution yields | === If the template is a primitive MOS, and for some perfect generators <math>p_T, p_F, \ r := \left|p_T\right|_\mathbf{X} = \left|p_F\right|,</math> then MOS substitution yields a quasi-parallelogram in the lattice === | ||
With the additional assumption that the number of '''X''' letters in a perfect generator ''p''<sub>''T''</sub> of the template MOS be a generator class of the filling MOS, the generator sequence yields ''q'' parallel chains ''C''<sub>1</sub>, | With the additional assumption that the number of '''X''' letters in a perfect generator ''p''<sub>''T''</sub> of the template MOS be a generator class of the filling MOS, the generator sequence yields ''q'' parallel chains ''C''<sub>1</sub>, | ||
..., ''C''<sub>''q''</sub> of the aggregate generator, the sum of the generators in the GS. The offset between ''C''<sub>''i''</sub> and ''C''<sub>''i''+1</sub> is equal to subst(''p''<sub>''T''</sub>, '''X''', ''p''<sub>''F''</sub>), where ''p''<sub>''T''</sub> and ''p''<sub>''F''</sub> are perfect generators (of appropriate lengths) of the template and filling MOSes, respectively. The aggregate generator is subst((''p''<sub>''T''</sub>)<sup>''q''</sup>, '''X''', ''G''<sup>''r''</sup>), where ''G'' is the period of the filling MOS. | ..., ''C''<sub>''q''</sub> of the aggregate generator, the sum of the generators in the GS. The offset between ''C''<sub>''i''</sub> and ''C''<sub>''i''+1</sub> is equal to subst(''p''<sub>''T''</sub>, '''X''', ''p''<sub>''F''</sub>), where ''p''<sub>''T''</sub> and ''p''<sub>''F''</sub> are perfect generators (of appropriate lengths) of the template and filling MOSes, respectively. The aggregate generator is subst((''p''<sub>''T''</sub>)<sup>''q''</sup>, '''X''', ''G''<sup>''r''</sup>), where ''G'' is the period of the filling MOS. | ||
| Line 208: | Line 208: | ||
* 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 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'' − 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 '' | Hence these particular MOS substitution scales satisfy a property that we call ''quasi-parallelogram''{{User:Inthar/Template:adhoc}}. An '''e'''-equivalent scale is a ''quasi-parallelogram'' if there exist non-negative integers ''m'', ''n'', 0 < ''a'' < ''n'', 0 < ''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> | ||
| Line 223: | Line 223: | ||
In the above case, {{nowrap| ''n'' {{=}} ''q'' | '''v''' {{=}} subst(''p''<sub>''T''</sub>, '''X''', ''p''<sub>''F''</sub>) | and '''w''' {{=}} subst((''p''<sub>''T''</sub>)<sup>''q''</sup>, '''X''', ''G''<sup>''r''</sup>) (the aggregate generator)}}. | In the above case, {{nowrap| ''n'' {{=}} ''q'' | '''v''' {{=}} subst(''p''<sub>''T''</sub>, '''X''', ''p''<sub>''F''</sub>) | and '''w''' {{=}} subst((''p''<sub>''T''</sub>)<sup>''q''</sup>, '''X''', ''G''<sup>''r''</sup>) (the aggregate generator)}}. | ||
The converse is false, as the scale in 5 letters [9/8 28/27 9/8 64/63 9/8 28/27 243/224 28/27 64/63 567/512 64/63] is | The converse is false, as the scale in 5 letters [9/8 28/27 9/8 64/63 9/8 28/27 243/224 28/27 64/63 567/512 64/63] is a quasi-parallelogram. | ||
=== MOS substitution scales have block balance at most 2 === | === MOS substitution scales have block balance at most 2 === | ||