Generator form manipulation: Difference between revisions

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== mingen form ==

A number of generator size targets have been described, such as [[UDP#Generalizing_to_Arbitrary_MOS.27s:_Bright_and_Dark_Generators_.28Chroma-Positive_and_Chroma-Negative.29|chroma-positive generators]], "musician's form", and probably more will be described in the future<ref>Such as forms using [https://mathworld.wolfram.com/LLLAlgorithm.html LLL] to get "a simple set of generators", per a post by Graham Breed on Facebook [https://bit.ly/3AhDYP8 here].</ref>. The generator size target we'll be demonstrating here may not have been rigorously described elsewhere, or named<ref>It is the form used by [[Graham Breed]]'s regular temperament finding scripts hosted here: http://x31eq.com/temper/net.html, at least for rank-2 temperaments.</ref>, but these purposes, we'll be calling it minimum-generator, or mingen, for short.

A number of generator size targets have been described, such as [[UDP#Generalizing_to_Arbitrary_MOS.27s:_Bright_and_Dark_Generators_.28Chroma-Positive_and_Chroma-Negative.29|chroma-positive generators]], "musician's form"<ref>probably: the generator with the simplest JI preimage while still being smaller than the previous generator (not smaller than half of the previous generator, as with mingen). For example, for meantone, the musician's form is 3/2. That's because 3/1 is simpler but greater than 2/1, and while 4/3 also works, it's slightly more complex than 3/2.</ref>, and probably more will be described in the future<ref>Such as forms using [https://mathworld.wolfram.com/LLLAlgorithm.html LLL] to get "a simple set of generators", per a post by Graham Breed on Facebook [https://bit.ly/3AhDYP8 here].</ref>. The generator size target we'll be demonstrating here may not have been rigorously described elsewhere, or named<ref>It is the form used by [[Graham Breed]]'s regular temperament finding scripts hosted here: http://x31eq.com/temper/net.html, at least for rank-2 temperaments.</ref>, but these purposes, we'll be calling it minimum-generator, or mingen, for short.

=== definition ===

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 == mingen form ==
-A number of generator size targets have been described, such as [[UDP#Generalizing_to_Arbitrary_MOS.27s:_Bright_and_Dark_Generators_.28Chroma-Positive_and_Chroma-Negative.29|chroma-positive generators]], "musician's form", and probably more will be described in the future<ref>Such as forms using [https://mathworld.wolfram.com/LLLAlgorithm.html LLL] to get "a simple set of generators", per a post by Graham Breed on Facebook [https://bit.ly/3AhDYP8 here].</ref>. The generator size target we'll be demonstrating here may not have been rigorously described elsewhere, or named<ref>It is the form used by [[Graham Breed]]'s regular temperament finding scripts hosted here: http://x31eq.com/temper/net.html, at least for rank-2 temperaments.</ref>, but these purposes, we'll be calling it minimum-generator, or mingen, for short.
+A number of generator size targets have been described, such as [[UDP#Generalizing_to_Arbitrary_MOS.27s:_Bright_and_Dark_Generators_.28Chroma-Positive_and_Chroma-Negative.29|chroma-positive generators]], "musician's form"<ref>probably: the generator with the simplest JI preimage while still being smaller than the previous generator (not smaller than half of the previous generator, as with mingen). For example, for meantone, the musician's form is 3/2. That's because 3/1 is simpler but greater than 2/1, and while 4/3 also works, it's slightly more complex than 3/2.</ref>, and probably more will be described in the future<ref>Such as forms using [https://mathworld.wolfram.com/LLLAlgorithm.html LLL] to get "a simple set of generators", per a post by Graham Breed on Facebook [https://bit.ly/3AhDYP8 here].</ref>. The generator size target we'll be demonstrating here may not have been rigorously described elsewhere, or named<ref>It is the form used by [[Graham Breed]]'s regular temperament finding scripts hosted here: http://x31eq.com/temper/net.html, at least for rank-2 temperaments.</ref>, but these purposes, we'll be calling it minimum-generator, or mingen, for short.
 === definition ===