Optimization: Difference between revisions
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In [[regular temperament theory]], '''optimization''' is the theory and practice to find low-error tunings of regular temperaments. | In [[regular temperament theory]], '''optimization''' is the theory and practice to find low-error tunings of regular temperaments. | ||
A regular temperament is defined by a [[mapping]] or a [[comma basis]]. It does not contain specific tuning information. To tune a temperament, one must define a [[tuning map]] by specifying the size of each [[Periods and generators|generator]]. The question is what it should be. In general, a temperament is an approximation to [[just intonation]] (JI). Any tuning will unavoidably introduce errors on some intervals for sure. The art of tempering seems to be about compromises – to find a sweet spot where the concerning intervals have the least overall error, so that the harmonic qualities of JI are best preserved. | |||
== Taxonomy == | == Taxonomy == | ||
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<math>\displaystyle W = \operatorname {diag} (1/\log_2 (Q)) </math> | <math>\displaystyle W = \operatorname {diag} (1/\log_2 (Q)) </math> | ||
indicates that the prime harmonic ''q'' in Q = {{val| 2 3 5 … }} has the importance of 1/log<sub>2</sub>(''q''). Since the tuning space and the interval space are [[dual]] to each other, such a rating of importance in the tuning space has the dual effect in the interval space: the prime harmonic ''q'' has the complexity log<sub>2</sub>(''q''). The more complex it is, the more error will be allowed for it. | indicates that the prime harmonic ''q'' in Q = {{val| 2 3 5 … }} has the importance of 1/log<sub>2</sub>(''q''). Since the tuning space and the interval space are [[Wikipedia:Dual (mathematics)|dual]] to each other, such a rating of importance in the tuning space has the dual effect in the interval space: the prime harmonic ''q'' has the complexity log<sub>2</sub>(''q''). The more complex it is, the more error will be allowed for it. | ||
=== Skew === | === Skew === | ||
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== General formulation == | == General formulation == | ||
In general, the temperament optimization problem (except for the destretch) can be defined as follows. Given a temperament mapping A and the JIP J<sub>0</sub>, we specify a weight W, a skew X, and a ''p''-norm. | In general, the temperament optimization problem (except for the destretch) can be defined as follows. Given a temperament mapping ''A'' and the [[just intonation point]] (JIP) ''J''<sub>0</sub>, we specify a weight ''W'', a skew ''X'', and a ''p''-norm. An optional eigenmonzo list ''B''<sub>C</sub> can be added. The goal is to find the generator list ''G'' by | ||
Minimize | Minimize | ||
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<math>\displaystyle (GA - J_0)B_{\rm C} = O </math> | <math>\displaystyle (GA - J_0)B_{\rm C} = O </math> | ||
where V is the weight-skewed mapping and J the weight-skewed JIP, found by | where ''V'' is the weight-skewed mapping and ''J'' the weight-skewed JIP, found by | ||
<math>\displaystyle | <math>\displaystyle | ||
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== See also == | == See also == | ||
* [[:File:MiddlePath2015.pdf|''A Middle Path between Just Intonation and the Equal Temperaments – Part 1'']] ("middle path") by [[Paul Erlich]] | * [[:File:MiddlePath2015.pdf|''A Middle Path between Just Intonation and the Equal Temperaments – Part 1'']] ("middle path") by [[Paul Erlich]] | ||
== External links == | |||
* [http://x31eq.com/temper/primerr.pdf|''Prime Based Error and Complexity Measures''] ("primerr.pdf") by [[Graham Breed]] | * [http://x31eq.com/temper/primerr.pdf|''Prime Based Error and Complexity Measures''] ("primerr.pdf") by [[Graham Breed]] | ||