Normal forms: Difference between revisions

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| en = Normal forms

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| ja = 標準形

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Since a [[regular temperament]] can be represented by multiple equivalent [[vals and tuning space|val]] lists (a.k.a. [[mapping]]s) or [[monzos and interval space|monzo]] lists (a.k.a. [[comma basis|comma bases]]), it can be helpful – e.g. when comparing or cataloguing temperaments – to choose a single one of these equivalent lists to use as its unique identifier. A set of rules that are consistently able to narrow the full set of equivalent lists down to a single list for each temperament may be called a '''normal form''', and accordingly, a list which uniquely identifies a temperament in this way may be called a '''normal list'''.

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=== 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.

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 norm]], 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.

{{Note| This is the form shown in the "comma lists" of each temperament on this wiki. }}

=== Antitransposed defactored Hermite form ===

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+{{Interwiki
+| de =
+| en = Normal forms
+| es =
+| ja = 標準形
+}}
 Since a [[regular temperament]] can be represented by multiple equivalent [[vals and tuning space|val]] lists (a.k.a. [[mapping]]s) or [[monzos and interval space|monzo]] lists (a.k.a. [[comma basis|comma bases]]), it can be helpful – e.g. when comparing or cataloguing temperaments – to choose a single one of these equivalent lists to use as its unique identifier. A set of rules that are consistently able to narrow the full set of equivalent lists down to a single list for each temperament may be called a '''normal form''', and accordingly, a list which uniquely identifies a temperament in this way may be called a '''normal list'''.
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 === 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.
+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 norm]], 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.
+{{Note| This is the form shown in the "comma lists" of each temperament on this wiki. }}
 === Antitransposed defactored Hermite form ===