Mathematical theory of regular temperaments: Difference between revisions

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For example, using "∨" to represent the interior product, we have W = {{multival| 6 -7 -2 -25 -20 15 }} for the wedgie of 7-limit miracle. Then the interior product W ∨ {{monzo| 1 0 0 0 }} is {{val| 0 -6 7 2 }}, with 15/14 we get W ∨ {{monzo| -1 1 1 -1 }} which is {{val| 1 1 3 3 }}, and with 16/15 we get W ∨ {{monzo| 4 -1 -1 0 }} which is also {{val| 1 1 3 3 }}; {{val| 1 1 3 3 }} tempers out the commas of miracle as well as 15/14 (or equivalently 16/15), sending them to the unison. The interior product forces an additional comma into a multival, lowering the rank by one. When we do this to a wedgie, we obtain a multival of rank one less, which has all the commas of the wedgie plus an additional comma, the interval we want to represent.

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 ~~map~~ 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.

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 ===

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Given a list of vals, we may [[Saturation|saturate]] (or [[defactor]]) 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 ~~map~~ ===

=== Frobenius projection matrix ===

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

{{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 ~~map~~. This corresponds uniquely with an abstract regular temperament. The intervals of the abstract temperament may be defined via multiplication by the projection ~~map~~, leading to [[fractional monzos]] which are actually the tunings of these intervals in [[Fractional monzos|Frobenius tuning]]. However, using the Frobenius projection ~~map~~ to define the abstract temperament by no means commits us to Frobenius tuning.

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 ===

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An alternative explanation of this process is provided here: [[Varianced Exterior Algebra#converting vectorals to matrices]]

=== Frobenius projection ~~maps~~ ===

=== Frobenius projection matrices ===

To translate from the Frobenius matrix to the RREF, simply reduce the matrix to RREF form in the usual way. To translate from RREF to the Frobenius matrix, if E is the RREF form then the matrix is E`E. Here the definition for the pseudoinverse E` using only matrix inverse and transpose can be used.

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=== The normal comma list ===

To translate from the normal comma list to the RREF, find the null space of the matrix of monzos of the normal comma list in form of a matrix, take the transpose of that matrix, and reduce to the RREF. If E is the RREF, to find the normal comma list first find the Frobenius projection ~~map~~ by computing I - M`M, where I is the identity matrix. Clear denominators from this, and saturate. Then reverse rows, reduce to Hermite normal form, reverse rows again, and adjust the result so that the monzos represent commas greater than one.

To translate from the normal comma list to the RREF, find the null space of the matrix of monzos of the normal comma list in form of a matrix, take the transpose of that matrix, and reduce to the RREF. If E is the RREF, to find the normal comma list first find the Frobenius projection matrix by computing I - M`M, where I is the identity matrix. Clear denominators from this, and saturate. Then reverse rows, reduce to Hermite normal form, reverse rows again, and adjust the result so that the monzos represent commas greater than one.

== The Geometry of Regular Temperaments ==

@@ Line 32: / Line 32: @@
 For example, using "∨" to represent the interior product, we have W = {{multival| 6 -7 -2 -25 -20 15 }} for the wedgie of 7-limit miracle. Then the interior product W ∨ {{monzo| 1 0 0 0 }} is {{val| 0 -6 7 2 }}, with 15/14 we get W ∨ {{monzo| -1 1 1 -1 }} which is {{val| 1 1 3 3 }}, and with 16/15 we get W ∨ {{monzo| 4 -1 -1 0 }} which is also {{val| 1 1 3 3 }}; {{val| 1 1 3 3 }} tempers out the commas of miracle as well as 15/14 (or equivalently 16/15), sending them to the unison. The interior product forces an additional comma into a multival, lowering the rank by one. When we do this to a wedgie, we obtain a multival of rank one less, which has all the commas of the wedgie plus an additional comma, the interval we want to represent.
-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 map 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.
+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 ===
@@ Line 39: / Line 39: @@
 Given a list of vals, we may [[Saturation|saturate]] (or [[defactor]]) 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 map ===
+=== Frobenius projection matrix ===
-{{main| Tenney-Euclidean Tuning #Frobenius tuning and Frobenius projection map }}
+{{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 map. This corresponds uniquely with an abstract regular temperament. The intervals of the abstract temperament may be defined via multiplication by the projection map, leading to [[fractional monzos]] which are actually the tunings of these intervals in [[Fractional monzos|Frobenius tuning]]. However, using the Frobenius projection map to define the abstract temperament by no means commits us to Frobenius tuning.
+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 ===
@@ Line 78: / Line 78: @@
 An alternative explanation of this process is provided here: [[Varianced Exterior Algebra#converting vectorals to matrices]]
-=== Frobenius projection maps ===
+=== Frobenius projection matrices ===
 To translate from the Frobenius matrix to the RREF, simply reduce the matrix to RREF form in the usual way. To translate from RREF to the Frobenius matrix, if E is the RREF form then the matrix is E`E. Here the definition for the pseudoinverse E` using only matrix inverse and transpose can be used.
@@ Line 88: / Line 88: @@
 === The normal comma list ===
-To translate from the normal comma list to the RREF, find the null space of the matrix of monzos of the normal comma list in form of a matrix, take the transpose of that matrix, and reduce to the RREF. If E is the RREF, to find the normal comma list first find the Frobenius projection map by computing I - M`M, where I is the identity matrix. Clear denominators from this, and saturate. Then reverse rows, reduce to Hermite normal form, reverse rows again, and adjust the result so that the monzos represent commas greater than one.
+To translate from the normal comma list to the RREF, find the null space of the matrix of monzos of the normal comma list in form of a matrix, take the transpose of that matrix, and reduce to the RREF. If E is the RREF, to find the normal comma list first find the Frobenius projection matrix by computing I - M`M, where I is the identity matrix. Clear denominators from this, and saturate. Then reverse rows, reduce to Hermite normal form, reverse rows again, and adjust the result so that the monzos represent commas greater than one.
 == The Geometry of Regular Temperaments ==