Normal forms: Difference between revisions

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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 form manipulation #Beyond rank-2]]. We begin with {{rket| {{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 {{rket| {{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 {{rket| {{map| 1 2 -1 -3 }} {{map| 0 -1 8 14 }} {{map| 0 0 -1 -2 }} }}.

Consider the example in the diagram given here: [[Generator form manipulation #Beyond rank-2]]. We begin with {{rket| {{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 {{rket| {{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 {{nowrap|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 {{rket| {{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. {{nowrap|5 × 1200.6¢}} versus {{nowrap|12 × 499.841¢}} is a difference of only 4.908¢) and then shifting generators by those commas.

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 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 form manipulation #Beyond rank-2]]. We begin with {{rket| {{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 {{rket| {{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 {{rket| {{map| 1 2 -1 -3 }} {{map| 0 -1 8 14 }} {{map| 0 0 -1 -2 }} }}.
+Consider the example in the diagram given here: [[Generator form manipulation #Beyond rank-2]]. We begin with {{rket| {{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 {{rket| {{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 &minus;2285.18¢ by {{nowrap|2 &times; 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 {{rket| {{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. {{nowrap|5 &times; 1200.6¢}} versus {{nowrap|12 &times; 499.841¢}} is a difference of only 4.908¢) and then shifting generators by those commas.