User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions
Add notes about optimizing database lookup. Remove boldface from scalars. |
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== Optimizing database keys == | == Optimizing database keys == | ||
As mentioned before wedgies can be used as unique identifiers of temperaments. The geometric algebra for n-dimensional space is 2<sup>n</sup>-dimensional, but most of the wedgie components are zero and can be dropped. Additionally in the case of rank-2 temperaments there's a neat little [[Wedgies_and_multivals#Constrained_wedgies|trick]] that allows us to drop even more components. In the subgroup ''o''.''p''<sub> | As mentioned before wedgies can be used as unique identifiers of temperaments. The geometric algebra for n-dimensional space is 2<sup>n</sup>-dimensional, but most of the wedgie components are zero and can be dropped. Additionally in the case of rank-2 temperaments there's a neat little [[Wedgies_and_multivals#Constrained_wedgies|trick]] that allows us to drop even more components. In the subgroup ''o''.''p''<sub>2</sub>.''p''<sub>3</sub>.(…).''p''<sub>n</sub> calculate | ||
:<math>\mathbf{T}' = < 1, \log_o(p_2), \log_o(p_3), \ldots, \log_o(p_n) ] \wedge < 0, W_{12}, W_{13}, \ldots, W_{1n} ]</math> | :<math>\mathbf{T}' = < 1, \log_o(p_2), \log_o(p_3), \ldots, \log_o(p_n) ] \wedge < 0, W_{12}, W_{13}, \ldots, W_{1n} ]</math> | ||
If the temperament is reasonably close to just intonation it can be recovered by rounding <math>\mathbf{T}'</math> to the nearest integer. | If the temperament is reasonably close to just intonation it can be recovered by rounding <math>\mathbf{T}'</math> to the nearest integer. | ||
Examples in 2.3.5 include [[Meantone_family#Meantone|Meantone]] ~ [1, 4], [[Augmented_family#Augmented|Augmented]] ~ [3, 0] and [[Limmic_temperaments#5-limit_.28blackwood.29|Blackwood]] ~ [0, 5], [[Slendro_clan#Semaphore|Semaphore]] ~ [2, 1] in 2.3.7 and [[Gamelismic_clan#Miracle|Miracle]] ~ [6, -7, -2] in 2.3.5.7. | Examples in 2.3.5 include [[Meantone_family#Meantone|Meantone]] ~ [1, 4], [[Augmented_family#Augmented|Augmented]] ~ [3, 0] and [[Limmic_temperaments#5-limit_.28blackwood.29|Blackwood]] ~ [0, 5], [[Slendro_clan#Semaphore|Semaphore]] ~ [2, 1] in 2.3.7 and [[Gamelismic_clan#Miracle|Miracle]] ~ [6, -7, -2] in 2.3.5.7. | ||