Regular temperament
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Characterizing a regular temperament[edit]
The wedgie[edit]
This uses 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.
For example, using "∨" to represent the interior product, we have mir = <<6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir ∨ |1 0 0 0> is <0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1> which is <1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0> which is also <1 1 3 3|; <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 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.
Normal val lists[edit]
Given a list of vals, we may saturate it and reduce it using the 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 [<1 1 3 3|, <0 6 -7 -2|] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].
The Frobenius projection map[edit]
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 Frobenius tuning. However, using the Frobenius projection map to define the abstract temperament by no means commits us to Frobenius tuning.
Just intonation subgroups and transversals[edit]
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 miracle temperament [2, 15/14] defines a rank two 7-limit subgroup whose 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[edit]
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.
Reduced row echelon form[edit]
If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix.
For example, if we feed [<22 35 51 62|, <31 49 72 87|, <84 133 195 236|] into a reduced row echelon form routine, we obtain [<1 0 3 1|, <0 1 -3/7 8/7|, <0 0 0 0|]. Stripping off the zero val in the final row, we get E = [<1 0 3 1|, <0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1>, and |-1 -1 0 1>E* = [0 1/7]. Multiply by |1 0 0 0>, the val for 2, and the result is |1 0 0 0>E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.
Translation between methods of specifying temperaments[edit]
The various methods for specifying an abstract regular temperament can be translated from one to another. Below we explain how to translate to and from reduced row echelon form (RREF.) The point of using RREF as the transportation hub is that while it in some ways is not a very good system for musical purposes, it is quick and easy to compute, with no requirement to use Smith or Hermite normal forms or to make use of the pseudoinverse in its full generality.
Wedgies[edit]
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 n = pi(p) is the number of primes in the p-limit) take a wedge product of basis vectors involving r-1 basis elements (ie, the wedge product of 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 r-1 basis elements to obtain n choose 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.
Frobenius projection maps[edit]
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.
The normal val list[edit]
To translate from the normal val list to the RREF, simply reduce the normal val list. To obtain the normal val list from the RREF, clear denominators from the rows of the RREF, saturate the result, reduce that to Hermite normal form and make the adjustment to normal val form.
The normal comma list[edit]
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.
Maple code for the parts of this 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.
The Geometry of Regular Temperaments[edit]
Abstract regular temperaments can be identified with rational points on an algebraic variety known as a 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, n) has points identified with the r-dimensional subspaces of the n-dimensional real vector space R^n. 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, 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, 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, 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.
Grassmannians have the structure of a smooth, homogenous metric space, and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian Gr(2, 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.
