Normal forms: Difference between revisions
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=== Positive generator form === | === Positive generator form === | ||
The '''positive generator form''' is based on the canonical form, but the vals on the list correspond to a list of generators which are all positive (written additively) or equivalently greater than 1 (written multiplicatively). | |||
To obtain this form, we find the [[Frobenius generator]]s of the temperament – to break this down, we find the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]] of the ''k''×''m'' matrix, A<sup>+</sup>, and multiply this from the left by the row vector of [[JIP]], J<sub>0</sub> = [1 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'']. | |||
<math>G_\text{F} = J_0 A^+</math> | |||
If the ''i''-th entry in the result is negative, multiply the corresponding val by -1. Return the result. | |||
The "mapping" (though not the "Map to lattice") listed on temperament pages of this wiki are in this form. The generators in canonical form of septimal meantone is positive already, so its positive generator form is the same as its canonical form, [{{val| 1 0 -4 -13 }}, {{val| 0 1 4 10 }}], corresponding to generators of ~2/1 and ~3/1. | The "mapping" (though not the "Map to lattice") listed on temperament pages of this wiki are in this form. The generators in canonical form of septimal meantone is positive already, so its positive generator form is the same as its canonical form, [{{val| 1 0 -4 -13 }}, {{val| 0 1 4 10 }}], corresponding to generators of ~2/1 and ~3/1. | ||
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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. | ||
[[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. | [[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. | ||
== Normal interval list == | == Normal interval list == | ||