Target tunings

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By a target tuning is meant a tuning for a regular temperament which has been optimized with respect to a set of target intervals. Usually this set of intervals is derived from a finite set of octave equivalence classes, and octaves are taken to be pure 2s. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.

Least squares tunings

By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to 0<r<1, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning T is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, ∑_i (T(q_i) - log2(q_i))^2, where q_i are the rational intervals of the target set. Note that most commonly, the target set is a tonality diamond, since these are the intervals in an overtone series up to some odd integer d, and hence may be considered the consonances of the d odd limit.

We may find the least squares tuning in various ways, one of which is to start from a matrix R whose rows are the monzos of the target set, and a matrix U whose rows are vals spanning the temperament. From U we form the matrix V by taking the normal val list for U and removing the first ("period") row. A list of eigenmonzos E = VR*R, where the asterisk denotes the matrix transpose, can now be obtained by matrix multiplication, and to this we add a row for the monzo of 2. The projection matrix for the least squares tuning is then the square matrix with fractional monzo rows, the rows of E, including 2, as eigenmonzos, that is left eigenvectors with eigenvalue one, and any basis for the commas of the temperament as left eigenvectors with eigenvalues zero.

Minimax tuning

Starting with the same assumptions as with least squares tuning, minimax tuning works similarly, except instead of minimizing the squares of the errors, we minimize max_i |T(q_i) - log2(q_i)|, the maximum error over all the target intervals. This can be solved by setting the minimization up as a linear programming problem, but another approach leads as before to a projection matrix with fractional monzo rows.

For a rank r temperament, we can form a list of candidate sets of eigenmonzos by taking r-1 elements from the list of target intervals, and adding 2 to the set, leading to an r element set. To avoid duplications and sets with rank less than r, we can take the normal interval list defined by each of these r element sets, discarding any with less than r elements. This means we are compiling a list of rank r just intonation subgroups, which are eigenmonzo subgroups for the corresponding tuning. For each of these these subgroups, we may find a corresponding projection matrix as the matrix with the r generators of the subgroup as eigenmonzos and a basis for the commas as left eigenvectors with eigenvalue zero. These projection matrices define tunings, and the tuning, if unique, with the least maximum error is the minimax tuning. However, this tuning may not be unique, in which case we may break the tie by choosing the tuning, among the set of least maximum error tunings, with the smallest sum of errors squared.

Example

Suppose our set of target intervals is the 5-limit diamond {6/5, 5/4, 4/3, 3/2, 8/5, 5/3}. The corresponding matrix is

R = [|1 1 -1>, |-2 0 1>, |2 -1 0>, |-1 1 0>, |3 0 -1>, |0 -1 1>]

Then R*R = [[19 -2 -6], [-2 4 -2], [-6 -2 4]], a positive-definite symmetric matrix. If U = [<1 0 -9|, <0 1 4|], we may remove the top row and obtain V = [<0 1 4|]; then VR*R is [|-26 -4 14>]; using this along with |1 0 0> as eigenmonzos, and using |-4 4 -1> an a commatic eigenvector with eigenvalue zero, we obtain

[|1 0 0>, |14/13 -1/13 7/26>, |4/13 -4/13 14/13>]

which is the projection matrix for the Woolhouse 7/26-comma meantone, the 5-limit least squares tuning.

Starting from the same six intervals of the 5-limit diamond and adding 2, we find after computing the normal interval lists that the three subgroups are [2, 3], [2, 5], and [2, 5/3]. Computing the projection matrix and from thence the tuning in each case, we find that the minimax tuning is [|1 0 0>, |1 0 1/4>, |0 0 1>], the projection matrix for 1/4-comma meantone.