Tenney-Euclidean tuning is a variant of TOP tuning which employs the TE norm. Just as the TOP tuning minimizes the maximum Tenney-weighted (L1) error of any interval, the TE tuning minimizes the maximum TE-weighted error of any interval.
If we have k linearly independent vals of dimension n, they will span a subspace of tuning space. This subspace defines a regular temperament of rank k in the prime limit p, where p is the nth prime. Similarly, starting from n-k independent commas for the same regular temperament, the corresponding monzos span an n-k dimensional subspace of 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 RMS tuning.
If we put the (weighted) Euclidean metric on tuning space, leading to TE tuning space in weighted coordinates, it is easy to find the nearest point in the subspace to the JI point <1 1 ... 1|, and this closest point will define a tuning map which is called TE tuning (or TOP-RMS), a tuning which has been extensively studied by Graham Breed. We may also keep unweighted coordinates and use the TE norm on tuning space; in these coordinates the JI point is <1 log2(3) ... log2(p)|. The two approaches are equivalent.
One way to find this tuning is to use k parameters times the vals, leading to a parametrization of the subspace, and then to find the nearest point by least squares, differentiating the square of the distance to the JI point and solving the resulting linear equations. Another is to use the Moore-Penrose pseudoinverse.
If A is an m×n matrix with real entries, and if we denote the pseudoinverse by A`, then it is defined as the n×m matrix such that
- AA`A = A. Hence, AA` maps the rows of A to itself and A`A the columns of A to itself.
- A`AA` = A`
- A`A and AA` are symmetric matrices
From these properties it can be deduced that
- If A is invertible, its inverse is A`
- If A has rational entries, so does A`
- A`` = A
- The pseudoinverse of the transpose is the transpose of the pseudoinverse
- AA` is the orthogonal projection map onto the space spanned by the columns of A
- A`A is the orthogonal projection map onto the space spanned by the rows of A
- I - A`A, where I is the identity matrix, is the orthogonal projection map onto the kernel, or null space, of A
- If the rows of A are linearly independent, then A` = A*(AA*)^(-1), where A* is the transpose of A. This means the pseudoinverse can be found in this important special case by people who don't have a pseudoinverse routine available by using a matrix inverse routine.
- uA` is the nearest point to u in the subspace spanned by the rows of A; A`v is the nearest point to v in the space spanned by the columns of A.
Computing TE tuning using pseudoinverses
Suppose V is a matrix whose rows consist of vals in the weighted basis. No assumption need be made that the rows are linearly independent or that the vals are free of contorsion. If J is the JI point, <1 1 ... 1|, then JV` gives the TE tuning in the sense that it gives (not necessarily independent) generators which correspond to the rows of V. How many of each generator to take to map a rational number contained in the prime limit in question is determined by applying the val corresponding to the generator to the rational number.
We may also obtain the TE tuning from a projection map. P = V`V is the orthogonal projection map 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 map starting from a list of weighted monzos rather than vals. If M is a rank n matrix whose rows are weighted monzos, and I is the nxn identity matrix, then P = I - M`M is the same projection map 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.
Pure octaves TE tuning
(alias POTE tuning)
If T = JP is the TE tuning map, then a corresponding pure-octaves map can be found by scalar multiplication, T/T, where T, 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. We may call pure-octaves Tenney-Euclidean tuning the POTE tuning.
The Frobenius projection map
We may also do the same things starting from unweighted vals. This leads to a different tuning, the Frobenius tuning, which is perfectly functional but has less theoretical justification than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are fractional monzos. The Frobenius projection map therefore, like the wedgie, defines a completely canonical object not depending on any arbitrary definition (eg how Hermite normal form or LLL reduction is specifically defined) which corresponds 1-1 with temperaments, and which does not depend on whether the monzos or vals from which it is computed are saturated. It may be found starting either from a set of vals or a set of commas, since if Q is the projection map found by treating monzos in the same way as vals, P = I-Q is the same projection map as would be found if starting from a set of vals defining the same temperament.
Spelling this out, if V is a matrix whose rows are vals, then P = V`V is a positive-semidefinite symmetric matrix with rational matrix entries, which exactly specifies the regular temperament defined by the vals of V. If M is a matrix with rows of monzos which spans the subspace of interval space containing the commas, then this same matrix P is given by I - M`M.
If the vals defining V are linearly independent, then P = V*(VV*)^(-1)V. If the rows of M are independent, then we likewise have P = I - M*(MM*)^(-1)M.
The val for 5-limit 12et is <12 19 28|. In weighted coordinates, that becomes v12 = <12 19/log2(3) 28/log2(5)| ~ <12.0 11.988 12.059|. If we take this to be a 1x3 matrix and take the pseudoinverse, we get the 3x1 matrix v12` ~ [|0.027706 0.027677 0.027842>]. Then P = v12`v12 is a projection map onto the one-dimensional subspace whose single basis vector is v12. We find that v12P equals v12; on the other hand, if we take the monzo for 81/80, which is |-4 4 -1> and monzo-weight it to <-4 4log2(3) - log2(5)| and multiply (either side, the matrix is symmetric) by P, we get the zero vector, corresponding to the unison.
Now consider pajara, the 7-limit temperament tempering out both 50/49 and 64/63. Two possible equal temperament tunings for pajara are 12edo and 22edo. We may define a 2x4 matrix with rows equal to the vals for 12, and 22; in weighted coordinates this would be
V ~ [<12 11.988 12.059 12.111|, <22 22.083 21.965 22.085|]
Then the pseudoinverse V` is approximately
which we may also write as [|-1.81029 -6.68250 4.83496 3.6765175829982166001>, |1.00052 3.66285 -2.63063 -1.99757>]
Paj = V`V is a 4x4 symmetrical matrix which projects weighted vals in TE tuning space, or weighted monzos in TE interval space, to a subspace defined by pajara. It therefore projects the weighted monzos for 50/49, 64/63, 225/224, 2048/2025 and etc to the zero vector, whereas it leaves pajara vals such as 10edo in weighted coordinates unchanged.
If we use unweighted coordinates we get the Frobenius projection map instead, whose rows are fractional monzos. The unweighted pseudoinverse u12` of the 5-limit val u12 for 12 equal is the column matrix u12*/1289, where the * denotes the transpose; that is, the 1x3 matrix with column |12/1289 19/1289 28/1289>. Then u12`u12 is the 3x3 Frobenius projection map F:
<144 228 336|, <228 361 532|, <336 532 784|]/1289
Multiplying <12 19 28| by F gives <12 19 28| again. Multiplying the monzos for 81/80, 128/125, 648/625 and etc gives the zero monzo, corresponding to a unison. Multiplying the val for 5-limit 19 equal, <19 30 44|, by F gives <24360 38570 56840|/1289, which is approximately the 19 equal val. Multiplying the 5-limit monzo for 3/2, which is |-1 1 0> times F gives the fractional monzo corresponding to (2^84 3^133 5^196)^(1/1289), which equates to 698.121 cents, the tempering of 3/2 in Frobenius tuning for 5-limit 12et, the tuning with octave defined by the top row of F, which is to say by |1 0 0>F, of 1196.778 cents.
We can do the same thing with a matrix U with rows consisting of the vals for 7-limit 12 and 22 equal; then U`U, the Frobenius projection map for pajara, is
[|36 92 14 32>, |92 269 -32 14>, |14 -32 141 148>, |32 14 148 64>]/305
This sends monzos for 50/49, 64/63 etc to the unison monzo, and vals for 10et, 12et and 22et to themselves.