Parallelogram substring scale: Difference between revisions
Created page with "A '''quasi-parallelogram scale''' is a scale whose pitch class lattice forms either a parallelogram or an incomplete traversal of one. == Mathematical definition == 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'''-equ..." |
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Here the scale is thought as traversing a series of rows one step of the row at a time, and | Here the scale is thought as traversing a series of rows one step of the row at a time, and | ||
* <math>\{\mathbf{a} + i\mathbf{v}\}_{i=a}^{n-1}</math> is a tail of the first row | * <math>\{\mathbf{a} + i\mathbf{v}\}_{i=a}^{n-1}</math> is a (nonempty) tail of the first row | ||
* <math>\{\mathbf{a} + i\mathbf{v} + j\mathbf{w}\}_{(i,j) \in [n]_0 \times [m-2]_1}</math> is a (possibly empty) parallelogram where rows are traversed fully | * <math>\{\mathbf{a} + i\mathbf{v} + j\mathbf{w}\}_{(i,j) \in [n]_0 \times [m-2]_1}</math> is a (possibly empty) parallelogram where rows are traversed fully | ||
* <math>\{\mathbf{a} + i\mathbf{v} + (m-1)\mathbf{w}\}_{i=0}^{b}</math> is a prefix of the last row. | * <math>\{\mathbf{a} + i\mathbf{v} + (m-1)\mathbf{w}\}_{i=0}^{b}</math> is a (nonempty) prefix of the last row. | ||