Mathematical theory of regular temperaments: Difference between revisions
Cmloegcmluin (talk | contribs) →Translation between methods of specifying temperaments: add link to Maple code to the top so it more clearly applies to all subsections and is easier found; also add link to alternative explanation inspired by it |
m →Characterizing a regular temperament: update some links |
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== Characterizing a regular temperament == | == Characterizing a regular temperament == | ||
=== Wedgie === | === Wedgie === | ||
{{main| Wedgies and | {{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. | 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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=== Frobenius projection map === | === Frobenius projection map === | ||
{{main| Tenney-Euclidean Tuning #Frobenius projection map }} | {{main| Tenney-Euclidean Tuning #Frobenius tuning and Frobenius projection map }} | ||
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 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. | ||
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=== Normal comma lists === | === Normal comma lists === | ||
{{main| Normal lists #Normal | {{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. | 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. | ||