Normal forms: Difference between revisions

== Hermite normal form ==

[[Wikipedia: Hermite normal form|Hermite normal form]], or HNF for short, is an important normal form that was defined for integer matrices in the mathematical field of linear algebra. An integer matrix is simply a 2D array of integers, and so we can easily think of lists of vals or commas as integer matrices and therefore leverage HNF in regular temperament theory.  

== Normal val lists ==

First we will review the set of normal forms that are defined for val lists.

This is the "canonical form" for a temperament that was developed by [[Dave Keenan]] and [[Douglas Blumeyer]], formed from [[defactoring]] the matrix (aka removing [[contorsion]]) prior to putting it into Hermite form.

=== Positive generator form ===

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 {{ket|{{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.  

=== 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.<ref>This is somewhat like octave reduction combined with octave inversion, because you can't just add or subtract half octaves until it's between 0 and 600 cents; you have to add or subtract octaves until it's between -600 and +600 cents, then multiply by -1 if it's negative.</ref><ref>You could always find a smaller and smaller generator by going negative, so this assumes positive generators.</ref>

=== Defactored Hermite form ===

Given a matrix of ''p''-limit intervals, we can find its defactored Hermite form in much the same way as a mapping. The only difference is that the matrix must be antitransposed once at the beginning of the operation and once again at the end. For an example, see: [[defactoring#canonical_comma-bases]]

=== Positive ratio form ===

Similar to [[Normal lists #Positive generator form|the situation with mappings]], even though by using the HNF the defactored Hermite form ensures that the first nonzero entry of each comma's prime count vector is a positive ''number'', this does not necessarily mean that the corresponding commas are all positive ''in pitch''.  

=== Minimal ratio form ===

The '''minimal ratio form''' shows ratio-wise the simplest comma list sufficient to define the temperament, where the ratios are rated by the product of the numerator and the denominator, i.e. the [[Benedetti height]].