Mathematical theory of regular temperaments: Difference between revisions

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

=== ~~Wedgie~~ ===

=== Plücker coordinates ===

This uses ~~{{w|~~exterior algebra~~}} and {{w|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.~~

This uses [[exterior algebra]] to give unique coordinates associated to the abstract regular temperament.

For example, using "∨" to represent the interior product, we have {{nowrap|''W'' {{=}} {{multival| 6 -7 -2 -25 -20 15 }}}} for the wedgie of 7-limit miracle. Then {{nowrap|''W'' ∨ {{monzo| 1 0 0 0 }} {{=}} {{val| 0 -6 7 2 }}}}, with 15/14 we get {{nowrap|''W'' ∨ {{monzo| -1 1 1 -1 }} {{=}} {{val| 1 1 3 3 }}}}, and with 16/15 we get {{nowrap|''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 a mapping for the temperament can be defined via an ({{nowrap|''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 {{nowrap|(''W'' ∨ ''q'') · ''V''}}.

=== Frobenius projection matrix ===

@@ Line 41: / Line 41: @@
 When specifying a temperament by the list of commas it tempers out, the list should be [[defactoring|defactored]] so it presents the intervals in their simplest, most direct form.
-=== Wedgie ===
+=== Plücker coordinates ===
-{{Main| Wedgies and multivals }}
+{{Main| Plücker coordinates }}
-This uses {{w|exterior algebra}} and {{w|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.
+This uses [[exterior algebra]] to give unique coordinates associated to the abstract regular temperament.
-For example, using "∨" to represent the interior product, we have {{nowrap|''W'' {{=}} {{multival| 6 -7 -2 -25 -20 15 }}}} for the wedgie of 7-limit miracle. Then {{nowrap|''W'' ∨ {{monzo| 1 0 0 0 }} {{=}} {{val| 0 -6 7 2 }}}}, with 15/14 we get {{nowrap|''W'' ∨ {{monzo| -1 1 1 -1 }} {{=}} {{val| 1 1 3 3 }}}}, and with 16/15 we get {{nowrap|''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 a mapping for the temperament can be defined via an ({{nowrap|''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 {{nowrap|(''W'' ∨ ''q'') · ''V''}}.
 === Frobenius projection matrix ===