MOS substitution: Difference between revisions
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Hence these particular MOS substitution scales satisfy a property that we call ''almost parallelogram''{{User:Inthar/Template:adhoc}}. An '''e'''-equivalent scale is ''almost a 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 | Hence these particular MOS substitution scales satisfy a property that we call ''almost parallelogram''{{User:Inthar/Template:adhoc}}. An '''e'''-equivalent scale is ''almost a 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>\{\mathbf{a} + i\mathbf{v}\}_{i=a}^{n-1} \cup \{\mathbf{a} | <math> | ||
\{\mathbf{a} + i\mathbf{v}\}_{i=a}^{n-1} <!--tail of first row--> | |||
\cup \{\mathbf{a} + i\mathbf{v} + j\mathbf{w}\}_{(i,j) \in [n]_0 \times [m-2]_1} <!-- parallelogram, maybe empty --> | |||
\cup \{\mathbf{a} + i\mathbf{v} + (m-1)\mathbf{w}\}_{i=0}^{b}. <!--first part of last row --> | |||
</math> | |||
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>''r''</sup>, '''X''', ''F''<sup>''r''</sup>)}}. | 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>''r''</sup>, '''X''', ''F''<sup>''r''</sup>)}}. | ||