Mathematical theory of regular temperaments: Difference between revisions

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== Characterizing a regular temperament ==

=== Normal val list ===

Given a list of vals, we may [[Mathematical theory of saturation|saturate]] it and reduce it using the [[Normal lists|Hermite normal form]] to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [{{val| 1 1 3 3 }}, {{val| 0 6 -7 -2 }}] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].

=== Normal comma lists ===

The normal comma list uniquely defines the abstract temperament, and has the advantage of showing family relationships even more clearly than the normal val list. Intervals of the temperament may be defined after computing another means of representing the temperament such as the normal val list.

When specifying a temperament by the list of commas it tempers out, the list should be [[defactored]] so it presents the intervals in their simplest, most direct form.

=== Wedgie ===

This uses [[Wikipedia: Exterior algebra|multilinear algebra]] to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the [[interior product]] of a [[wedgie]] for a ''p''-limit temperament with the ''p''-limit monzos.

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As explained on the [[Interior product #Applications|interior product]] page, if W is the ''r''-wedgie defining the rank-''r'' temperament, then the tuning of a mapping for the temperament can be defined via an (''r'' - 1)-multimonzo V which has the property that for every JI interval ''q'', the tempered value of ''q'' is given by the dot product (W∨''q'')·V.

~~=== Normal val list ===~~

~~{{main| Normal lists #Normal val list }}~~

Given a list of vals, we may [[Mathematical theory of saturation|saturate]] it and reduce it using the [[Normal lists|Hermite normal form]] to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [{{val| 1 1 3 3 }}, {{val| 0 6 -7 -2 }}] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].

=== Frobenius projection matrix ===

{{~~main~~| Tenney-Euclidean Tuning #Frobenius tuning and Frobenius projection matrix }}

{{Main| Tenney-Euclidean Tuning #Frobenius tuning and Frobenius projection matrix }}

Given any list of monzos, or any list of vals, we may compute the associated Frobenius projection matrix. This corresponds uniquely with an abstract regular temperament. The intervals of the abstract temperament may be defined via multiplication by the projection matrix, leading to [[fractional monzos]] which are actually the tunings of these intervals in [[Fractional monzos|Frobenius tuning]]. However, using the Frobenius projection matrix to define the abstract temperament by no means commits us to Frobenius tuning.

=== Just intonation subgroups and transversals ===

A relatively concrete approach, but one which is not canonically defined, is to define a [[transversal]] for the temperament by giving generators for a just intonation subgroup which when tempered becomes the notes of the temperament.

For example, for [[Gamelismic clan #Miracle|miracle temperament]] [2, 15/14] defines a rank-2 7-limit subgroup whose [[Normal lists #Normal interval list|normal interval list]] is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament.

~~=== Normal comma lists ===~~

~~{{main| Normal lists #Normal interval list }}~~

The normal comma list uniquely defines the abstract temperament, and has the advantage of showing family relationships even more clearly than the normal val list. Intervals of the temperament may be defined after computing another means of representing the temperament such as the normal val list.

~~When specifying a temperament by the list of commas it tempers out, the list should be [[defactored]] so it presents the intervals in their simplest, most direct form.~~

== Translation between methods of specifying temperaments ==

@@ Line 25: / Line 25: @@
 == Characterizing a regular temperament ==
+=== Normal val list ===
+{{Main| Normal lists #Normal val lists }}
+Given a list of vals, we may [[Mathematical theory of saturation|saturate]] it and reduce it using the [[Normal lists|Hermite normal form]] to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [{{val| 1 1 3 3 }}, {{val| 0 6 -7 -2 }}] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].
+=== Normal comma lists ===
+{{Main| Normal lists #Normal interval lists }}
+The normal comma list uniquely defines the abstract temperament, and has the advantage of showing family relationships even more clearly than the normal val list. Intervals of the temperament may be defined after computing another means of representing the temperament such as the normal val list.
+When specifying a temperament by the list of commas it tempers out, the list should be [[defactored]] so it presents the intervals in their simplest, most direct form.
 === Wedgie ===
-{{main| Wedgies and multivals }}
+{{Main| Wedgies and multivals }}
 This uses [[Wikipedia: Exterior algebra|multilinear algebra]] to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the [[interior product]] of a [[wedgie]] for a ''p''-limit temperament with the ''p''-limit monzos.
@@ Line 33: / Line 45: @@
 As explained on the [[Interior product #Applications|interior product]] page, if W is the ''r''-wedgie defining the rank-''r'' temperament, then the tuning of a mapping for the temperament can be defined via an (''r'' - 1)-multimonzo V which has the property that for every JI interval ''q'', the tempered value of ''q'' is given by the dot product (W∨''q'')·V.
-=== Normal val list ===
-{{main| Normal lists #Normal val list }}
-Given a list of vals, we may [[Mathematical theory of saturation|saturate]] it and reduce it using the [[Normal lists|Hermite normal form]] to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [{{val| 1 1 3 3 }}, {{val| 0 6 -7 -2 }}] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].
 === Frobenius projection matrix ===
-{{main| Tenney-Euclidean Tuning #Frobenius tuning and Frobenius projection matrix }}
+{{Main| Tenney-Euclidean Tuning #Frobenius tuning and Frobenius projection matrix }}
 Given any list of monzos, or any list of vals, we may compute the associated Frobenius projection matrix. This corresponds uniquely with an abstract regular temperament. The intervals of the abstract temperament may be defined via multiplication by the projection matrix, leading to [[fractional monzos]] which are actually the tunings of these intervals in [[Fractional monzos|Frobenius tuning]]. However, using the Frobenius projection matrix to define the abstract temperament by no means commits us to Frobenius tuning.
 === Just intonation subgroups and transversals ===
-{{main| Just intonation subgroups }}
+{{Main| Just intonation subgroups }}
-{{main| Transversal }}
+{{Main| Transversal }}
 A relatively concrete approach, but one which is not canonically defined, is to define a [[transversal]] for the temperament by giving generators for a just intonation subgroup which when tempered becomes the notes of the temperament.
 For example, for [[Gamelismic clan #Miracle|miracle temperament]] [2, 15/14] defines a rank-2 7-limit subgroup whose [[Normal lists #Normal interval list|normal interval list]] is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament.
-=== Normal comma lists ===
-{{main| Normal lists #Normal interval list }}
-The normal comma list uniquely defines the abstract temperament, and has the advantage of showing family relationships even more clearly than the normal val list. Intervals of the temperament may be defined after computing another means of representing the temperament such as the normal val list.
-When specifying a temperament by the list of commas it tempers out, the list should be [[defactored]] so it presents the intervals in their simplest, most direct form.
 == Translation between methods of specifying temperaments ==