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{{Expert|Regular temperament}}
{{Expert|Regular temperament}}


A '''regular temperament''' is a homomorphism that maps an abelian group of target/just intervals to another abelian group of [[tempering out|tempered]] intervals. In other words, it is a function from a group of just intervals to another simpler group of intervals that preserves the operation of [[stacking]]. Typically, the source set is assumed to be a multiplicative subgroup of the rational numbers (aka [[just intonation]]). Musically, tempering is done by deliberately mistuning some of the ratios such that a [[comma]] or set of commas vanishes by becoming a unison (it is ''tempered out'' in the temperament). The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the dimensionality of JI, thereby simplifying the pitch relationships.
A '''regular temperament''' is a homomorphism that maps an abelian group of target/just intervals to another abelian group of [[tempering out|tempered]] intervals. In other words, it is a function from a group of (usually just) intervals to another simpler group of intervals that preserves the operation of [[stacking]]. Typically, the source set is assumed to be a multiplicative subgroup of the rational numbers (aka [[just intonation]]). Musically, tempering is done by deliberately mistuning some of the ratios such that a [[comma]] or set of commas vanishes by becoming a unison (it is ''tempered out'' in the temperament). The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the dimensionality of JI, thereby simplifying the pitch relationships.


In mathematical terms, it is a function whose domain is a target tuning we wish to approximate, and its range is the intervals of the temperament. In general, this mapping is many-to-one, and two different rational numbers may be mapped to the same [[mapped interval|tempered interval]] — in this case we say that the two JI intervals are ''tempered together''.
In mathematical terms, it is a function whose domain is a target tuning we wish to approximate, and its range is the intervals of the temperament. In general, this mapping is many-to-one, and two different rational numbers may be mapped to the same [[mapped interval|tempered interval]] — in this case we say that the two JI intervals are ''tempered together''.
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== Characterizing a regular temperament ==
== Characterizing a regular temperament ==
=== Normal val lists ===
=== Normal mapping matrices ===
{{Main| Normal lists #Normal val lists }}
{{Main| Temperament mapping matrix }}
 
{{See also| Normal forms #Normal forms for mappings }}
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 ===
Since an abstract temperament corresponds to some linear map, we can represent it as a matrix. We can [[Mathematical theory of saturation|saturate]] it and reduce it to the [[Hermite normal form]], which gives a unique representation. Applying this map to the vector representation of a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal form for 7-limit miracle is
{{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.
$$
\begin{bmatrix}
1 & 1 & 3 & 3 \\
0 & 6 & -7 & -2 \\
\end{bmatrix}
$$


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.
and applying this to the vector for either 16/15 or 15/14 leads to [0 1].


=== Wedgie ===
=== Normal comma bases ===
{{Main| Wedgies and multivals }}
{{Main| Comma basis }}
{{See also| Normal forms #Normal forms for commas }}


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.
A temperament may also be defined by a list of commas. By putting these into a normal form, the representation is also unique. Using commas has the advantage of showing family relationships more clearly.


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.
=== Plücker coordinates ===
{{Main| Plücker coordinates }}


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


=== Frobenius projection matrix ===
=== Frobenius projection matrix ===
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=== Just intonation subgroups and transversals ===
=== Just intonation subgroups and transversals ===
{{Main| Just intonation subgroups | Transversal }}
{{Main| Just intonation subgroups | Transversal }}
{{See also| Gencom }}


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.
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.
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Maple code for the parts of these which do not call the Maple functions for Hermite normal form, Smith normal form or the pseudoinverse can be found in the article [[Basic abstract temperament translation code]].
Maple code for the parts of these which do not call the Maple functions for Hermite normal form, Smith normal form or the pseudoinverse can be found in the article [[Basic abstract temperament translation code]].
=== Wedgies ===
To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank ''r'' in ''n'' dimensions (where {{nowrap|''n'' {{=}} π(''p'')}} is the number of primes in the ''p''-limit) take a wedge product of basis vectors involving {{nowrap|''r'' − 1}} basis elements (i.e., the wedge product of {{nowrap|''r'' − 1}} elements representing primes) and wedge these with the basis element for each prime, obtaining either 0 or an ''r''-fold wedge product with sign ±1. Take the corresponding element of the wedgie times the ±1 sign (which is computed from the parity of the permutation of the ''r'' elements.) This gives a val; do this for every combination of {{nowrap|''r'' − 1}} basis elements to obtain ''n'' choose {{nowrap|''r'' − 1}} vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arithmetic, not floating point numbers.
An alternative explanation of this process is provided here: [[Intro to exterior algebra for RTT#Converting varianced multivectors to matrices]]


=== Frobenius projection matrices ===
=== Frobenius projection matrices ===
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== Geometry of regular temperaments ==
== Geometry of regular temperaments ==
Abstract regular temperaments can be identified with {{w|rational point}}s on an {{w|algebraic variety}} known as a {{w|Grassmannian}}. In particular, if the number of primes in the ''p''-limit is ''n'', and the rank of the temperament is ''r'', then the real Grassmannian '''Gr'''(''r'',&nbsp;''n'') has points identified with the ''r''-dimensional subspaces of the ''n''-dimensional real vector space '''R'''<sup>''n''</sup>. This has an embedding into a real vector space known as the [[Plücker embedding]], which if the point in question corresponds to a temperament is the wedgie of the temperament. Regular temperaments of rank ''r'' in the ''p''-limit may be defined as rational points on '''Gr'''(''r'',&nbsp;''n''), though we should note that most of these do not correspond to anything worth much as a temperament. In matrix terms, the real Grassmannian '''Gr'''(''r'',&nbsp;''n'') can be identified with real symmetric projection matrices with trace ''r''. The rational symmetric projection matrices with trace ''r'' are precisely the Frobenius projections, so under this identification it is clear they represent rational points on '''Gr'''(''r'',&nbsp;''n''). A rational projection matrix of trace ''r'' which is not symmetric is still a [[tuning map]]; minimax and least squares tunings provide examples of this.
{{Main|Plucker coordinates}}
 
Grassmannians have the structure of a smooth, homogenous {{w|metric space}}, and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian '''Gr'''(2,&nbsp;3), consisting of the planes through the origin in three dimensional space, may be identified with the projective plane, and hence 5-limit rank two temperaments may be pictured as points in a projective plane, as below (known as "projective tone space").
 
See also [[equivalence continuum]] for a description of the space of rank-''r'' temperaments supported by a given temperament, such as a rank-1 temperament, as an algebraic variety.
See also [[equivalence continuum]] for a description of the space of rank-''r'' temperaments supported by a given temperament, such as a rank-1 temperament, as an algebraic variety.


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