Normal forms: Difference between revisions

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=== Minimal generator form ===

The '''minimal generator form''' (or '''mingen form''') is a form specific to rank-2 temperaments, where the matrix is normalized such that the generator is positive and no greater than half the period.

The '''minimal generator form''' (or '''mingen form''') is a form specific to rank-2 temperaments, where the matrix is normalized such that the generator is positive and no greater than half the period.<ref>This is somewhat like octave reduction combined with octave inversion, because 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>

[[Graham Breed]]'s [http://x31eq.com/temper/ temperament finder] uses this form for all rank-2 temperaments. Septimal meantone in minimal generator form is [{{val| 1 2 4 7 }}, {{val| 0 -1 -4 -10 }}], corresponding to generators of ~2/1 and ~4/3.

~~For more information~~, ~~see~~ [[~~mingen~~]].

Beyond rank-2, the mingen form of a temperament is no longer unique. You can always get smaller and smaller generators. This is why on Graham Breed's temperament finding tool, beyond rank-2 he simply uses the Hermite Normal Form.

Consider the example in the diagram given here: [[Generator size manipulation#beyond rank-2]]. 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.

== Normal interval lists ==

@@ Line 162: / Line 162: @@
 === Minimal generator form ===
-The '''minimal generator form''' (or '''mingen form''') is a form specific to rank-2 temperaments, where the matrix is normalized such that the generator is positive and no greater than half the period.
+The '''minimal generator form''' (or '''mingen form''') is a form specific to rank-2 temperaments, where the matrix is normalized such that the generator is positive and no greater than half the period.<ref>This is somewhat like octave reduction combined with octave inversion, because 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>
 [[Graham Breed]]'s [http://x31eq.com/temper/ temperament finder] uses this form for all rank-2 temperaments. Septimal meantone in minimal generator form is [{{val| 1 2 4 7 }}, {{val| 0 -1 -4 -10 }}], corresponding to generators of ~2/1 and ~4/3.
-For more information, see [[mingen]].
+Beyond rank-2, the mingen form of a temperament is no longer unique. You can always get smaller and smaller generators. This is why on Graham Breed's temperament finding tool, beyond rank-2 he simply uses the Hermite Normal Form.
+Consider the example in the diagram given here: [[Generator size manipulation#beyond rank-2]]. 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.
 == Normal interval lists ==