Aberrismic theory: Difference between revisions

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Mathematically, this difference corresponds to choices of <math>\mathbb{Z}</math>-linear maps

<math>\alpha : \mathbb{Z}^3\langle\mathbf{L}, \mathbf{m}, \mathbf{s}\rangle \to \mathrm{JI}( 2, p_1, ..., p_s )/\mathrm{ker}(T)</math> (here <math>T</math> is a temperament map defined on the 2.p1....ps subgroup), determined by differing choices of <math>\alpha(\mathbf{L}), \alpha(\mathbf{m}), \alpha(\mathbf{s})</math> and subject to the constraint that <math>a\alpha(\mathbf{L}) + b\alpha(\mathbf{m}) + c\alpha(\mathbf{s}) = T(2)</math> for the ternary scale ''a'''''L'''''b'''''m'''''c'''''s'''. Thus there are two choices involved in interpreting a ~~givem~~ ternary scale, namely the choice of temperament and the choice of where to map the scale steps. The assignment of scale steps to tempered intervals is chosen to improve coverage of important LCJI intervals.

<math>\alpha : \mathbb{Z}^3\langle\mathbf{L}, \mathbf{m}, \mathbf{s}\rangle \to \mathrm{JI}( 2, p_1, ..., p_s )/\mathrm{ker}(T)</math> (here <math>T</math> is a temperament map defined on the 2.p1....ps subgroup), determined by differing choices of <math>\alpha(\mathbf{L}), \alpha(\mathbf{m}), \alpha(\mathbf{s})</math> and subject to the constraint that <math>a\alpha(\mathbf{L}) + b\alpha(\mathbf{m}) + c\alpha(\mathbf{s}) = T(2)</math> for the ternary scale ''a'''''L'''''b'''''m'''''c'''''s'''. Thus there are two choices involved in interpreting a given ternary scale, namely the choice of temperament and the choice of where to map the scale steps. The assignment of scale steps to tempered intervals is chosen to improve coverage of important LCJI intervals.

=== Example: blackdye ===

The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.

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 Mathematically, this difference corresponds to choices of <math>\mathbb{Z}</math>-linear maps
-<math>\alpha : \mathbb{Z}^3\langle\mathbf{L}, \mathbf{m}, \mathbf{s}\rangle \to \mathrm{JI}( 2, p_1, ..., p_s )/\mathrm{ker}(T)</math> (here <math>T</math> is a temperament map defined on the 2.p<sub>1</sub>....p<sub>s</sub> subgroup), determined by differing choices of <math>\alpha(\mathbf{L}), \alpha(\mathbf{m}), \alpha(\mathbf{s})</math> and subject to the constraint that <math>a\alpha(\mathbf{L}) + b\alpha(\mathbf{m}) + c\alpha(\mathbf{s}) = T(2)</math> for the ternary scale ''a'''''L'''''b'''''m'''''c'''''s'''. Thus there are two choices involved in interpreting a givem ternary scale, namely the choice of temperament and the choice of where to map the scale steps. The assignment of scale steps to tempered intervals is chosen to improve coverage of important LCJI intervals.
+<math>\alpha : \mathbb{Z}^3\langle\mathbf{L}, \mathbf{m}, \mathbf{s}\rangle \to \mathrm{JI}( 2, p_1, ..., p_s )/\mathrm{ker}(T)</math> (here <math>T</math> is a temperament map defined on the 2.p<sub>1</sub>....p<sub>s</sub> subgroup), determined by differing choices of <math>\alpha(\mathbf{L}), \alpha(\mathbf{m}), \alpha(\mathbf{s})</math> and subject to the constraint that <math>a\alpha(\mathbf{L}) + b\alpha(\mathbf{m}) + c\alpha(\mathbf{s}) = T(2)</math> for the ternary scale ''a'''''L'''''b'''''m'''''c'''''s'''. Thus there are two choices involved in interpreting a given ternary scale, namely the choice of temperament and the choice of where to map the scale steps. The assignment of scale steps to tempered intervals is chosen to improve coverage of important LCJI intervals.
 === Example: blackdye ===
 The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.