Normal forms: Difference between revisions

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Even though by using the HNF the defactored Hermite form ensures that the pivot (first nonzero entry) of each mapping row is a positive ''number'', this does not necessarily mean that the corresponding generators are all positive ''in pitch''. For example, the defactored Hermite form of porcupine is the matrix {{rket| {{map| 1 2 3 }} {{map| 0 3 5 }} }}. The second column of this matrix tells us it takes 2 of the first generator and 3 of the second generator to reach its approximation of 3/1. But as we can tell from the first column of this matrix, it takes only 1 of the first generator and nothing else to reach its approximation of 2/1. Therefore, if we move by 2 of the first generator, we are already at this temperament's approximation of 4/1, and so if we still need to move by 3 of the second generator to reach its approximation of 3/1, then the second generator must be negative. Indeed, it is about 163 cents ''downward'' in pitch. Negative generators like this can be surprising and confusing, and so the '''positive generator form''' was developed to address this concern.

To obtain this form, we first need to know whether each generator is positive or negative in pitch. Many methods are available for finding this information, but the one which is the easiest (and therefore the one this form is defined as using) is to find the [[Frobenius generator]]s of the temperament. To break this down, we find the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]] of the mapping V, which we notate V+, and multiply this from the left by the row vector of [[JIP]], {{nowrap|J {{=}} ~~{{!(}}~~1 log23 log25 … log2''p''~~{{)!}}~~}}.

To obtain this form, we first need to know whether each generator is positive or negative in pitch. Many methods are available for finding this information, but the one which is the easiest (and therefore the one this form is defined as using) is to find the [[Frobenius generator]]s of the temperament. To break this down, we find the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]] of the mapping V, which we notate V+, and multiply this from the left by the row vector of [[JIP]], {{nowrap|J {{=}} [1 log23 log25 … log2''p'']}}.

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 Even though by using the HNF the defactored Hermite form ensures that the pivot (first nonzero entry) of each mapping row is a positive ''number'', this does not necessarily mean that the corresponding generators are all positive ''in pitch''. For example, the defactored Hermite form of porcupine is the matrix {{rket| {{map| 1 2 3 }} {{map| 0 3 5 }} }}. The second column of this matrix tells us it takes 2 of the first generator and 3 of the second generator to reach its approximation of 3/1. But as we can tell from the first column of this matrix, it takes only 1 of the first generator and nothing else to reach its approximation of 2/1. Therefore, if we move by 2 of the first generator, we are already at this temperament's approximation of 4/1, and so if we still need to move by 3 of the second generator to reach its approximation of 3/1, then the second generator must be negative. Indeed, it is about 163 cents ''downward'' in pitch. Negative generators like this can be surprising and confusing, and so the '''positive generator form''' was developed to address this concern.
-To obtain this form, we first need to know whether each generator is positive or negative in pitch. Many methods are available for finding this information, but the one which is the easiest (and therefore the one this form is defined as using) is to find the [[Frobenius generator]]s of the temperament. To break this down, we find the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]] of the mapping V, which we notate V<sup>+</sup>, and multiply this from the left by the row vector of [[JIP]], {{nowrap|J {{=}} {{!(}}1 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p''{{)!}}}}.
+To obtain this form, we first need to know whether each generator is positive or negative in pitch. Many methods are available for finding this information, but the one which is the easiest (and therefore the one this form is defined as using) is to find the [[Frobenius generator]]s of the temperament. To break this down, we find the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]] of the mapping V, which we notate V<sup>+</sup>, and multiply this from the left by the row vector of [[JIP]], {{nowrap|J {{=}} [1 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'']}}.
 <math>G = JV^+</math>