Normal forms: Difference between revisions
some more canonical -> defactored hermite form stuff I missed; added notes that "enfactoring" = torsion/contorsion |
Cmloegcmluin (talk | contribs) various revisions post-Mike's adjustment of name, Tom Price's critique, typos, etc. |
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=== Defactored Hermite Form === | === Defactored Hermite Form === | ||
This is | 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. | ||
We may write a list of <span><math>k</math></span> vals as an <span><math>k×d</math></span> matrix, where the rows of the matrix are the vals, and <span><math>d</math></span> is the ''dimensionality'' of the system<ref>Calling the [[wikipedia: Prime-counting function|prime-counting function]], written π(x), on the prime limit will give us this number. For examples, π(2) = 1, π(3) = 2, π(5) = 3, π(7) = 4, π(11) = 5, etc.</ref>. To get the '''Defactored Hermite form''', we do the following: | We may write a list of <span><math>k</math></span> vals as an <span><math>k×d</math></span> matrix, where the rows of the matrix are the vals, and <span><math>d</math></span> is the ''dimensionality'' of the system<ref>Calling the [[wikipedia: Prime-counting function|prime-counting function]], written π(x), on the prime limit will give us this number. For examples, π(2) = 1, π(3) = 2, π(5) = 3, π(7) = 4, π(11) = 5, etc.</ref>. To get the '''Defactored Hermite form''', we do the following: | ||
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# Then, put this result into HNF. | # Then, put this result into HNF. | ||
For example, septimal meantone has the defactored Hermite | For example, septimal meantone has the defactored Hermite form of {{ket|{{map| 1 0 -4 -13 }} {{map| 0 1 4 10 }}}}, corresponding to generators of ~2/1 and ~3/1. | ||
The HNF step is not necessary to defactor the mapping, but it is important for canonicalizing into a convenient form. | |||
The key advantage of the defactored Hermite form is its purity and simplicity, while sidestepping many of the issues with [[contorsion]]/[[enfactoring]] matrices. If your primary need is uniquely identifying temperaments, this is the ideal choice. The remaining normal forms each introduce further constraints on the sizes of the generators, which can be nice and convenient if that matters for your use case, but otherwise is unnecessary. | The key advantage of the defactored Hermite form is its purity and simplicity, while sidestepping many of the issues with [[contorsion]]/[[enfactoring]] matrices. If your primary need is uniquely identifying temperaments, this is the ideal choice. The remaining normal forms each introduce further constraints on the sizes of the generators, which can be nice and convenient if that matters for your use case, but otherwise is unnecessary. | ||
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=== Defactored Hermite form === | === 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. | 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]] | ||
The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any ''p''-limit group it lives inside. The normalized list contains a minimal set of ratios, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we normalize [81/80, 126/125] we obtain [80/81, 57344/59049]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit. Because [[regular temperament]]s, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. The above normal interval list, for example, characterizes septimal meantone, defining the normal comma list of septimal meantone. | The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any ''p''-limit group it lives inside. The normalized list contains a minimal set of ratios, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we normalize [81/80, 126/125] we obtain [80/81, 57344/59049]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit. Because [[regular temperament]]s, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. The above normal interval list, for example, characterizes septimal meantone, defining the normal comma list of septimal meantone. | ||