Tenney–Euclidean tuning: Difference between revisions

If we have ''k'' linearly independent [[Vals and tuning space|vals]] of dimension ''n'', they will span a subspace of [[Vals and tuning space|tuning space]]. This subspace defines a regular temperament of rank ''k'' in the prime limit ''p'', where ''p'' is the ''n''-th prime. Similarly, starting from ''n'' - ''k'' independent commas for the same regular temperament, the corresponding monzos span an ''n'' - ''k'' dimensional subspace of [[Monzos and interval space|interval space]]. Both the subspace of tuning space and the subspace of interval space characterize the temperament completely. A question then arises as to how to choose a specific tuning for this temperament, which is the same as asking how to choose a point (vector) in this subspace of tuning space which provides a good tuning. One answer to this is the weighted RMS ([[Wikipedia: Root mean square|root mean squared]]) tuning discussed right here.

If A is the mapping of the [[Regular temperament|abstract temperament]] whose rows are (not necessarily independent) vals, then V = AW is the mapping in the weighted space. If J₀ is the row vector of targeted JI intervals (i.e. the [[JIP]]), then J = J₀W is the JI intervals in the weighted space, in the case of Tenney-weighting it is {{val| 1 1 … 1 }}. Let us also denote the row vector of TE generators G. TE tuning then defines a [[Wikipedia: Least squares|least squares problem]] of the following overdetermined linear equation system:  

<math>GV = J</math>

The system simply says that the sum of ''v''_''kl'' steps of generator ''g''_''k'' for all ''k'''s should equal the ''l''-th targeted JI interval ''j''_''l''.  

<math>G = JV^+</math>

<math>T = GV = JV^+V</math>

We may also obtain the TE tuning from a projection matrix. P = V⁺V is the orthogonal projection matrix that maps onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether [[contorsion]] has been removed. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.

We may find the same projection matrix starting from a list of weighted monzos rather than vals. If M is a rank ''n'' matrix whose columns are weighted monzos, and I is the ''n''×''n'' identity matrix, then P = I - MM⁺ is the same projection matrix as V⁺V so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.

{{See also| POTE tuning }}

We may call pure-octaves Tenney-Euclidean tuning the '''POTE tuning'''. If T = JP = GV is the TE tuning map, then a corresponding pure-octaves map can be found by [[wikipedia: Scalar multiplication|scalar multiplication]], T/T₁, where T₁, the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.  

{{Main| CTE tuning }}

Another way to enforce the pure octave is by adding the constraint before the optimization process. This is the '''CTE tuning'''. The result, under the constraint of pure octaves, remains TE optimal.  

The list of Frobenius generators, G_F, is given by:  

<math>G_\text{F} = J_0 A^+</math>

where J₀ is the nonweighted JIP and A is the nonweighted mapping.  

The Frobenius tuning map, T_F, is given by:  

<math>T_\text{F} = G_\text{F} A = J_0 A^+A</math>

Spelling this out, if A is a matrix whose rows are vals, then P = A⁺A is a [[wikipedia: Positive-definite matrix|positive-semidefinite]] [[wikipedia: Symmetric matrix|symmetric matrix]] with rational matrix entries, which exactly specifies the regular temperament defined by the vals of A. If B is a matrix with columns of monzos which spans the subspace of interval space containing the commas, then this same matrix P is given by I - BB⁺.

If the vals defining A are linearly independent, then P = A^T(AA^T)^-1A. If the columns of B are independent, then we likewise have P = I - B(B^TB)^-1B^T.