Normal forms: Difference between revisions
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# Hermite reduce it. | # Hermite reduce it. | ||
# Throw away all rows which consist of nothing but zeros, resulting in a ''k''×''m'' matrix. | # Throw away all rows which consist of nothing but zeros, resulting in a ''k''×''m'' matrix. | ||
For example, septimal meantone has the canonical form of [{{val| 1 0 -4 -13 }}, {{val| 0 1 4 10 }}], corresponding to generators of ~2/1 and ~3/1. | |||
=== Positive generator form === | === Positive generator form === | ||
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Now the vals on the list correspond to a list of generators which are all positive (written additively) or equivalently greater than 1 (written multiplicatively). | Now the vals on the list correspond to a list of generators which are all positive (written additively) or equivalently greater than 1 (written multiplicatively). | ||
The "mapping" (though not the "Map to lattice") listed on temperament pages of this wiki are in this form | 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. | ||
=== Equave-reduced generator form === | === Equave-reduced generator form === | ||
The '''equave-reduced generator form''' is similar to the positive generator form, but the matrix is further normalized such that each generator is reduced by the formal prime represented by the first column of the matrix. | The '''equave-reduced generator form''' is similar to the positive generator form, but the matrix is further normalized such that each generator is reduced by the formal prime represented by the first column of the matrix. | ||
Septimal meantone in equave-reduced generator form is [{{val| 1 1 0 -3 }}, {{val| 0 1 4 10 }}], corresponding to generators of ~2/1 and ~3/2. | |||
=== Minimal generator form === | === 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. | ||
[[Graham Breed]]'s [http://x31eq.com/temper/ temperament finder] uses this form for all rank-2 temperaments. | [[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 == | ||