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.

=== definition ===

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. Mingen form is when each successive generator is less than half the size of the previous generator.<ref>Like octave reduction combined with octave inversion, since you can't just add or subtract half octaves until it's between 0 and 600 cents. You have to add or subtract octaves until it's between -600 and +600 cents, then multiply by -1 if it's negative.</ref><ref>You could always find a smaller and smaller generator by going negative, so this assumes positive generators.</ref>

Mingen form is when each successive generator is less than half the size of the previous generator.<ref>Like octave reduction combined with octave inversion, since you can't just add or subtract half octaves until it's between 0 and 600 cents. You have to add or subtract octaves until it's between -600 and +600 cents, then multiply by -1 if it's negative.</ref><ref>You could always find a smaller and smaller generator by going negative, so this assumes positive generators.</ref>

=== tuning strategy ===

In order to define the size of the generators, you need to specify a tuning strategy. Though if you have two generators that are close enough that their size ranking depends on the tuning, then you probably have other problems. In any case, the tuning strategy that we'll be using here is T2, because it's decent enough and easy to compute.

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Consider the example in the diagram above. We begin with {{vector|{{map|1 2 0 -1}} {{map|0 -1 6 10}} {{map|0 0 -1 -2}}}} with generators of 1200.6¢, 499.841¢, and 214.024¢, which therefore already satisfies the condition that each generator is less than half the previous generator. But we can transform it into {{vector|{{map|1 2 2 3}} {{map|0 -1 1 0}} {{map|0 0 -1 -2}}}} which has a third generator of 116.013¢ instead. This is accomplished by adding row 3 to row 2 five times, which decreases generator 3 by the size of five times row 2, from 214.024¢ by 5 × 499.841 = 2499.205¢ to -2285.18¢; and then subtracting row 3 from row 1 twice, which increases generator 3 by the size of two times row 1, from -2285.18¢ by 2 × 1200.6¢ = 2401.2¢ to 116.013¢. And we can get that generator even smaller if we had instead moved up by 499.841 twice to 1213.71¢ and then down by 1200.6¢ once to 13.109¢ (that's a final mapping of {{vector|{{map|1 2 -1 -3}} {{map|0 -1 8 14}} {{map|0 0 -1 -2}}}}.

You could find smaller and smaller generators if you wanted, by essentially finding increasingly small "commas" between the other generators' sizes (e.g. 5 × 1200.6¢ versus 12 × 499.841¢ is a difference of only 4.908¢) and then shifting generators by those commas.

This problem also precludes the possibility of a definitive maximum generator which is still less than half of the previous generator.

== References ==

@@ Line 106: / Line 106: @@
 == 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.
 === definition ===
-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. Mingen form is when each successive generator is less than half the size of the previous generator.<ref>Like octave reduction combined with octave inversion, since you can't just add or subtract half octaves until it's between 0 and 600 cents. You have to add or subtract octaves until it's between -600 and +600 cents, then multiply by -1 if it's negative.</ref><ref>You could always find a smaller and smaller generator by going negative, so this assumes positive generators.</ref>
+Mingen form is when each successive generator is less than half the size of the previous generator.<ref>Like octave reduction combined with octave inversion, since you can't just add or subtract half octaves until it's between 0 and 600 cents. You have to add or subtract octaves until it's between -600 and +600 cents, then multiply by -1 if it's negative.</ref><ref>You could always find a smaller and smaller generator by going negative, so this assumes positive generators.</ref>
+=== tuning strategy ===
 In order to define the size of the generators, you need to specify a tuning strategy. Though if you have two generators that are close enough that their size ranking depends on the tuning, then you probably have other problems. In any case, the tuning strategy that we'll be using here is T2, because it's decent enough and easy to compute.
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 Consider the example in the diagram above. We begin with {{vector|{{map|1 2 0 -1}} {{map|0 -1 6 10}} {{map|0 0 -1 -2}}}} with generators of 1200.6¢, 499.841¢, and 214.024¢, which therefore already satisfies the condition that each generator is less than half the previous generator. But we can transform it into {{vector|{{map|1 2 2 3}} {{map|0 -1 1 0}} {{map|0 0 -1 -2}}}} which has a third generator of 116.013¢ instead. This is accomplished by adding row 3 to row 2 five times, which decreases generator 3 by the size of five times row 2, from 214.024¢ by 5 × 499.841 = 2499.205¢ to -2285.18¢; and then subtracting row 3 from row 1 twice, which increases generator 3 by the size of two times row 1, from -2285.18¢ by 2 × 1200.6¢ = 2401.2¢ to 116.013¢. And we can get that generator even smaller if we had instead moved up by 499.841 twice to 1213.71¢ and then down by 1200.6¢ once to 13.109¢ (that's a final mapping of {{vector|{{map|1 2 -1 -3}} {{map|0 -1 8 14}} {{map|0 0 -1 -2}}}}.
 You could find smaller and smaller generators if you wanted, by essentially finding increasingly small "commas" between the other generators' sizes (e.g. 5 × 1200.6¢ versus 12 × 499.841¢ is a difference of only 4.908¢) and then shifting generators by those commas.
+This problem also precludes the possibility of a definitive maximum generator which is still less than half of the previous generator.
 == References ==