Normal forms: Difference between revisions
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== Normal val list == | == Normal val list == | ||
=== | === Canonical form === | ||
If | If A is a list of ''n'' vals, we may write it as an ''n''×''m'' matrix, where the rows of the matrix are the vals, and ''m'' = π (''p''), where ''p'' is the prime limit. To get the '''canonical form''', we do the following: | ||
# Defactor it. | |||
# Hermite reduce it. | |||
# Throw away all rows which consist of nothing but zeros, resulting in a ''k''×''m'' matrix. | |||
=== Positive generator form === | |||
To obtain the '''positive generator form''', we find the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]] of the ''k''×''m'' matrix, and multiply this from the left by the row vector 1200·[1 log<sub>2</sub>3 log<sub>2</sub>5 … log<sub>2</sub>''p'']. If the ''i''-th entry in the result is negative, multiply the corresponding val by -1. Return the result. | |||
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; an example would be [{{val| 1 0 -4 -13 }}, {{val| 0 1 4 10 }}], the mapping for septimal meantone. Using this as input, the generators are 1199.25¢, 1899.45¢ (tempered 2/1, tempered 3/1). | |||
The | === 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. | |||
=== | === 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. | |||
[[Graham Breed]]'s [http://x31eq.com/temper/ temperament finder] uses this form for all rank-2 temperaments. | |||
== Normal interval list == | == Normal interval list == | ||
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The canonical form of the comma list, however, requires that the normal comma sequence be defactored. There is only one such form of any temperament. For example, both [27/25, 36/35] and [27/25, 49/48] are normal, and they both characterize Beep. But the latter is enfactored, so the former is Beep's canonical form. | The canonical form of the comma list, however, requires that the normal comma sequence be defactored. There is only one such form of any temperament. For example, both [27/25, 36/35] and [27/25, 49/48] are normal, and they both characterize Beep. But the latter is enfactored, so the former is Beep's canonical form. | ||
=== | === Minimal ratio form === | ||
In each temperament page on this wiki, however, the "comma lists" are not normal interval lists as defined above. Instead, it shows ratio-wise the simplest comma sequence sufficient to define the temperament, which may be called the Tenney-minimal form. | In each temperament page on this wiki, however, the "comma lists" are not normal interval lists as defined above. Instead, it shows ratio-wise the simplest comma sequence sufficient to define the temperament, which may be called the Tenney-minimal form. | ||