Normal forms: Difference between revisions

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In the case of interval lists, the most common format they are presented in is as ratios, not vectors, e.g. [81/80, 64/63] rather than [{{vector|-4 4 1 0}} {{vector|-6 2 0 1}}]. So you may need to convert ratios to vectors and back when working with these forms.

=== Minimal form ===

The '''minimal form''', or '''reduced form''' shows the simplest comma list sufficient to define the temperament, where the commas are rated by a complexity measure such as the [[Tenney height]], in which case it is called the ''Tenney-minimal form''. It is obvious why this form is useful, but unfortunately, it is not easy to find since it is equivalent to the ''shortest basis problem'', an NP-hard problem. The [[LLL reduction]] algorithm gives a good approximation. Otherwise, it is typically solved heuristically by hand.

'''Note''': This is the form shown in the "comma lists" of each temperament on this wiki.

=== Antitransposed defactored Hermite form ===

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The positive ratio form of septimal meantone is [81/80, 59049/57344].

~~=== Minimal form ===~~

The '''minimal form''', or '''reduced form''' shows the simplest comma list sufficient to define the temperament, where the commas are rated by a complexity measure such as the [[Tenney height]], in which case it is called the ''Tenney-minimal form''. It is obvious why this form is useful, but unfortunately, it is not easy to find since it is equivalent to the ''shortest basis problem'', an NP-hard problem. The [[LLL reduction]] algorithm gives a good approximation. Otherwise, it is typically solved heuristically by hand.

~~'''Note''': This is the form shown in the "comma lists" of each temperament on this wiki.~~

=== Integer reduced row echelon form (IRREF) ===

@@ Line 190: / Line 190: @@
 In the case of interval lists, the most common format they are presented in is as ratios, not vectors, e.g. [81/80, 64/63] rather than [{{vector|-4 4 1 0}} {{vector|-6 2 0 1}}]. So you may need to convert ratios to vectors and back when working with these forms.
+=== Minimal form ===
+The '''minimal form''', or '''reduced form''' shows the simplest comma list sufficient to define the temperament, where the commas are rated by a complexity measure such as the [[Tenney height]], in which case it is called the ''Tenney-minimal form''. It is obvious why this form is useful, but unfortunately, it is not easy to find since it is equivalent to the ''shortest basis problem'', an NP-hard problem. The [[LLL reduction]] algorithm gives a good approximation. Otherwise, it is typically solved heuristically by hand.
+'''Note''': This is the form shown in the "comma lists" of each temperament on this wiki.
 === Antitransposed defactored Hermite form ===
@@ Line 278: / Line 283: @@
 The positive ratio form of septimal meantone is [81/80, 59049/57344].
-=== Minimal form ===
-The '''minimal form''', or '''reduced form''' shows the simplest comma list sufficient to define the temperament, where the commas are rated by a complexity measure such as the [[Tenney height]], in which case it is called the ''Tenney-minimal form''. It is obvious why this form is useful, but unfortunately, it is not easy to find since it is equivalent to the ''shortest basis problem'', an NP-hard problem. The [[LLL reduction]] algorithm gives a good approximation. Otherwise, it is typically solved heuristically by hand.
-'''Note''': This is the form shown in the "comma lists" of each temperament on this wiki.
 === Integer reduced row echelon form (IRREF) ===