Mathematical theory of regular temperaments
- This article focuses on the mathematical tools used to describe a regular temperament. For an introduction to regular temperaments, see Regular temperament.
A regular temperament is a homomorphism that maps an abelian group of target/pure intervals to another abelian group of tempered intervals. Typically, the source set is assumed to be a multiplicative subgroup of the rational numbers (aka just intonation), and 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 tempered interval — in this case we say that the two JI intervals are tempered together.
For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals located by four-dimensional coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, intervals can be located with a set of one-, two-, or three-dimensional coordinates depending on the number of commas that have been tempered out. The dimensionality is the rank of the temperament.
To use a concrete example, if 7-limit meantone temperament is a function M, then M(6/5) = M(32/27) = "minor third". The difference between these, 81/80 (syntonic comma), is tempered out in meantone temperament. M(81/80) = M(1/1) = "unison".
A regular temperament is abstract, and has no preferred exact tuning. There are ways to compute an optimal tuning for any given temperament, but there are multiple definitions of optimality that disagree with each other, so in general we can consider a regular temperament as having a range of possible tunings of the generators. Once a tuning of each generator is provided the tuning of any interval can be computed as an integer linear combination of generator tunings. This property that all intervals are linear combinations of the generators is in fact what makes a temperament regular.
Dimensionality, or rank
A rank r regular temperament in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r vals. An abstract regular temperament can be defined in various ways, for instance by giving a set of commas tempered out by the temperament, or a set of r independent vals defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the comma pumps of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.
Rank-1 (equal) temperaments
Equal temperaments (abbreviated ET or tET) and equal divisions of the octave (abbreviated edo or ed2) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator, making it a rank-1 temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an equal division of any interval, let alone the octave, and in fact many ETs do not divide the pure octave at all. On the other hand, an n-edo is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An edo can be treated as an ET by applying a temperament mapping to the intervals of the edo, typically by using a val for a temperament supported by that edo, although one can also use unsupported vals or poorly-supported vals to achieve fun results. The familiar 12-note equal temperament, or 12edo, reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12et.
Rank-2 (including linear) temperaments
A p-limit rank-2 temperament maps all intervals of p-limit JI using a set of 2-dimensional coordinates, thus a rank-2 temperament is said to have two generators, though it may have any number of step-sizes. This means that a rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a linear temperament. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12et by tempering out the Pythagorean comma.
Regular temperaments of ranks two and three are cataloged on the Optimal patent val page. Rank-2 temperaments are also listed at Proposed names for rank 2 temperaments by their generator mappings, and at Map of rank-2 temperaments by their generator size. See also the pergens page. There is also Graham Breed's giant list of regular temperaments.
Characterizing a regular temperament
Normal val lists
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].
Normal comma 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.
When specifying a temperament by the list of commas it tempers out, the list should be defactored so it presents the intervals in their simplest, most direct form.
Wedgie
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 W = ⟨⟨ 6 -7 -2 -25 -20 15 ]] for the wedgie of 7-limit miracle. Then the interior product W ∨ [1 0 0 0⟩ is ⟨0 -6 7 2], with 15/14 we get W ∨ [-1 1 1 -1⟩ which is ⟨1 1 3 3], and with 16/15 we get W ∨ [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 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.
Frobenius projection matrix
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 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
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-2 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.
Translation between methods of specifying temperaments
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.
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 n = pi(p) is the number of primes in the p-limit) take a wedge product of basis vectors involving r-1 basis elements (i.e., 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.
An alternative explanation of this process is provided here: Intro_to_exterior_algebra_for_RTT#Converting_varianced_multivectors_to_matrices
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.
Normal val lists
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.
Normal comma lists
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.
Geometry of regular temperaments
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 Rn. 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 (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.
