Generator form manipulation: Difference between revisions
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{{Interwiki | |||
| en = Generator form manipulation | |||
| ja = ジェネレーター読み替え操作 | |||
}} | |||
A [[canonical_form|canonical mapping form]] is an important standard to have as a community for uniquely identifying [[temperaments]], but it is not the only mapping form one should ever need, because one may wish to use differently-sized [[generators]] (to ultimately generate the same tempered intervals). Several such forms with different generator sizes have been presented, such as [[Normal_lists#Positive_generator_form|positive generator form]], [[Normal_lists#Equave-reduced_generator_form|equave-reduced generator form]], and [[Normal_lists#Minimal_generator_form|minimal-generator form]]. | A [[canonical_form|canonical mapping form]] is an important standard to have as a community for uniquely identifying [[temperaments]], but it is not the only mapping form one should ever need, because one may wish to use differently-sized [[generators]] (to ultimately generate the same tempered intervals). Several such forms with different generator sizes have been presented, such as [[Normal_lists#Positive_generator_form|positive generator form]], [[Normal_lists#Equave-reduced_generator_form|equave-reduced generator form]], and [[Normal_lists#Minimal_generator_form|minimal-generator form]]. | ||
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So at this point, the generators are an octave and a negative perfect fifth. It's time for our second step. | So at this point, the generators are an octave and a negative perfect fifth. It's time for our second step. | ||
The second step of achieving a generator with the size of a perfect fourth should be familiar: we need to increase our negative perfect fifth to a perfect fourth, and so we need to add one octave, and therefore we need to change <math>𝒎_1</math> to be <math>𝒎_1 - | The second step of achieving a generator with the size of a perfect fourth should be familiar: we need to increase our negative perfect fifth to a perfect fourth, and so we need to add one octave, and therefore we need to change <math>𝒎_1</math> to be <math>𝒎_1 - 𝒎_2</math>, so we end up with: | ||
<math> | <math> | ||
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</math> | </math> | ||
And we're done, having found the mapping {{rket|{{map|1 2 4}} {{map|0 -1 -4}}}}. | And we're done, having found the mapping {{rket|{{map|1 2 4}} {{map|0 -1 -4}}}}. | ||
=== Beyond rank-2 === | === Beyond rank-2 === | ||
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Using these two tricks, you do not have to worry about enfactoring the mapping, i.e. introducing a common factor in one of the maps. This is because neither of these tricks ever involve replacing a map with a multiple of that map; we always replace a map with a combination of at least one each of two different maps, as in trick 1, or with the map negated, as in trick 2. | Using these two tricks, you do not have to worry about enfactoring the mapping, i.e. introducing a common factor in one of the maps. This is because neither of these tricks ever involve replacing a map with a multiple of that map; we always replace a map with a combination of at least one each of two different maps, as in trick 1, or with the map negated, as in trick 2. | ||
== Tuning | == Tuning == | ||
In order to | In order to determine the relative size of the generators, you need to have chosen a tuning for each them. The tuning we're using here comes from the [[minimax-ES]] tuning scheme, because this scheme is decent enough and easy to compute.<ref>Note from Douglas Blumeyer: though if I had written this article today having done a lot of tuning theory and built a library for optimizing tunings, I would have gone with TILT minimax-U instead.</ref> | ||
== Easy instructions table to achieve mingen form for rank-2 mapping == | == Easy instructions table to achieve mingen form for rank-2 mapping == | ||
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|1201.4 | |1201.4 | ||
|−1898.4 | |−1898.4 | ||
|g < −p | |<math>g < −p</math> | ||
|g + p | |<math>g + p</math> | ||
|𝒎₁ − 2𝒎₂ | |<math>𝒎₁ − 2𝒎₂</math> | ||
| | | | ||
|yes | |yes | ||
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|1201.4 | |1201.4 | ||
|−697.049 | |−697.049 | ||
|−p <= g < −p/2 | |<math>−p <= g < −p/2</math> | ||
|p + g | |<math>p + g</math> | ||
|𝒎₁ − 𝒎₂ | |<math>𝒎₁ − 𝒎₂</math> | ||
| | | | ||
|no, you're done | |no, you're done | ||
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|1201.4 | |1201.4 | ||
|−504.348 | |−504.348 | ||
|−p/2 <= g < 0 | |<math>−p/2 <= g < 0</math> | ||
|−g | |<math>−g</math> | ||
| | | | ||
|−𝒎₂ | |<math>−𝒎₂</math> | ||
|no, you're done | |no, you're done | ||
|- | |- | ||
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|1201.4 | |1201.4 | ||
|504.4 | |504.4 | ||
|0 <= g <= p/2 | |<math>0 <= g <= p/2</math> | ||
|g | |<math>g</math> | ||
| | | | ||
| | | | ||
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|1201.4 | |1201.4 | ||
|697.049 | |697.049 | ||
|p/2 < g <= p | |<math>p/2 < g <= p</math> | ||
|p - g | |<math>p - g</math> | ||
|𝒎₁ + 𝒎₂ | |<math>𝒎₁ + 𝒎₂</math> | ||
|−𝒎₂ | |<math>−𝒎₂</math> | ||
|no, you're done | |no, you're done | ||
|- | |- | ||
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|1201.4 | |1201.4 | ||
|1898.4 | |1898.4 | ||
|p < g | |<math>p < g</math> | ||
|g - p | |<math>g - p</math> | ||
|𝒎₁ + 2𝒎₂ | |<math>𝒎₁ + 2𝒎₂</math> | ||
|−𝒎₂ | |<math>−𝒎₂</math> | ||
|yes | |yes | ||
|} | |} | ||