Tenney–Euclidean tuning

Tenney–Euclidean tuning (TE tuning), also known as TOP–RMS tuning, is a tuning technique for regular temperaments which leads to the least sum of squared errors of the Tenney-weighted basis.

If we have r linearly independent vals of dimension n, they will span a subspace of tuning space. This subspace defines a regular temperament of rank r in the prime limit p, where p is the n-th prime. Similarly, starting from n − r independent commas for the same regular temperament, the corresponding monzos span an n − r 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 the weighted RMS (root-mean-squared) tuning discussed right here.

TE tuning can be viewed as a variant of TOP tuning since it employs the TE norm in place of the Tenney height as in TOP tuning. Just as TOP tuning minimizes the maximum Tenney-weighted L₁ error of any interval, TE tuning minimizes the maximum Tenney-weighted L₂ error of any interval.

In D&D's guide, the systematic name for TE tuning is minimax-ES.

Motivation

TE tuning combines the ideas of Tenney weighting of prime limits and optimizing the root mean square (RMS) of weighted error/damage. Prime limits are a way of choosing musically useful intervals without being too specific about which intervals you think are useful. Tenney weighting is a way of giving equal treatment to intervals of equal complexity without choosing a finite set of intervals. RMS optimization (least squares) implies optimizing for the average sensory dissonance of an interval, and so allows for good consonances to balance weak dissonances and well tuned intervals to balance poorly tuned intervals.

TE shares with TOP tuning the insight that adding octaves to the optimization simplifies the calculation. Allowing for scale stretch balances intervals of different sizes.

As an RMS measure, TE error is optimized by the least squares method, which is well known and simple and efficient to calculate.

TE tuning is uniquely optimized for a given prime limit. There are no free parameters determining the weighting of different intervals or the balance of wide and narrow intervals: all follow from the definition of Tenney weighting of primes.

Definition

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 JIP ⟨1 1 … 1], and this closest point will define a tuning map which is called TE tuning, 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 log₂3 … log₂p]. The two approaches are equivalent.

To start with, suppose W is the weighting matrix. For the prime basis Q = ⟨2 3 5 … p],

$$ W = \operatorname {diag} (1/\log_2 (Q)) $$

If V is the mapping of the abstract temperament whose rows are (not necessarily independent) vals, then V_W = VW is the mapping in the weighted space. If J is the row vector of targeted JI intervals (i.e. the JIP), then J_W = JW is the JI intervals in the weighted space, in the case of Tenney-weighting it is ⟨1 1 … 1]. Let us also denote the row vector of TE generators G. TE tuning then defines a least squares problem of the following overdetermined system of linear equations: GV_W = J_W, which simply says that the sum of (v_w)_kl steps of generator g_k for all k's should equal the l-th targeted JI interval (j_w)_l. That is to say we want to

$$ \begin{align} & \text{find} && G \\ & \text{that minimizes} && \lVert GV_W - J_W \rVert_2 \end{align} $$

There are a number of methods to solve least squares problems. One common way is to use the Moore–Penrose pseudoinverse.

Computation

Using pseudoinverse

The Moore–Penrose pseudoinverse, denoted A⁺, is a generalization of the inverse matrix with which it shares a lot of properties. In this method, the (not necessarily independent) TE generator map G, which correspond to the rows of V are given by

$$ G = J_W V_W^+ $$

Applying the val list to the generators, The TE tuning map is given by

$$ T = GV = J_W V_W^+ V $$

We may also obtain the TE tuning from a projection matrix. P_W = V +
W V_W is the orthogonal projection matrix that maps onto the space spanned by the rows of V_W. This space corresponds to the temperament, and so does P_W. However, P_W 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 weighted tuning map T_W = TW giving the weighted tuning of each prime number is found by multiplying by the JIP: T_W = J_WP_W = J_WV +
W V_W, which is the nearest point in the subspace corresponding to the temperament to J_W, so that T = J_WP_WW⁻¹ = J_WV +
W V.

We may find the same projection matrix starting from a list of weighted monzos rather than vals. If M_W is a rank-n matrix whose columns are weighted monzos, and I is the n×n identity matrix, then P_W = I − M_WM +
W  is the same projection matrix as V +
W V_W 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.

Computer program

Here is a minimalistic Python script that takes a mapping and gives TE generators and tuning maps, using Scipy.

import numpy as np
from scipy import linalg

def te (mapping, subgroup_basis):
    just_tuning_map = 1200*np.log2 (subgroup_basis)
    te_weight = np.diag (1/np.log2 (subgroup_basis))
    mapping_w = mapping @ te_weight
    just_tuning_map_w = just_tuning_map @ te_weight

    te_generators = linalg.lstsq (np.transpose (mapping_w), just_tuning_map_w)[0]
    te_tuning_map = te_generators @ mapping
    return te_generators, te_tuning_map

# taking septimal magic as an example ...
mapping = np.array ([[1, 0, 2, -1], [0, 5, 1, 12]])
subgroup_basis = np.array ([2, 3, 5, 7])

# to find the TE tuning you enter
te (mapping, subgroup_basis)

Output:

[1201.08240941  380.695113  ]
[1201.08240941, 1903.47556502, 2782.85993183, 3367.25894662]

Enforcement

Pure-octave TE tuning

Main article: POTE tuning

We may call the pure-octave Tenney−Euclidean tuning the POTE tuning. If T = J_WV +
W V = GV is the TE tuning map, then a corresponding pure-octaves map can be found by scalar multiplication, T/t₁, where t₁, the first entry, is the tuning of 2. While POTE is a very simple way to enforce a pure octave, it tends to overtemper the generators for divisive ratios and leads to less than ideal results.

Constrained TE tuning

Main article: Constrained tuning

Another way to enforce pure octaves 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. Contrary to POTE, CTE tends to under-optimize for divisive ratios; variations to fix this are considered under Constrained tuning.

Otherwise normed tunings

Tenney weighting gives equal treatment to intervals of equal complexity without choosing a finite set of intervals. While this makes a lot of sense, more specific demands on the field sometimes require it to be altered one way or another.

One common complaint is that it gives an undue weight to very high primes, as evidenced by the fact that you have to choose a limit to get sensible results. It does not converge as you keep adding primes. That it appears to be a limit to infinity of RMS of intervals approaching infinite complexity is psychoacoustically meaningless since the human ear cannot perceive even moderately complex intervals. To fix this, a free parameter s may be introduced such that each prime q is weighted by 1/(log₂(q))^s instead of 1/log₂(q), and we can obtain a steeper weight curve with any s > 1:

$$ W = \operatorname {diag} (1/(\log_2 (Q))^s) $$

Another fix is to adopt the Wilson weight in place of Tenney weight, discussed in #Benedetti/Wilson–Euclidean tuning.

The other complaint, contrary to above, is that Tenney weighting gives insufficient weight to higher primes than a weight specifically considering the relevant intervals within the limit. For this, we can use the parameter s again, as any s < 1 will give us a flatter weight curve. In particular, we can set s to 0, which removes the weight entirely and results in equal weight of all primes. This is called Frobenius tuning, discussed right below.

Frobenius tuning and Frobenius projection matrix

The Frobenius tuning may be considered as the same RMS tuning but starting from nonweighted vals. This 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.

The weighting matrix W is given by

$$ W = I $$

That simplifies the list of Frobenius generators, G, to

$$ G = JV^+ $$

where J is the non-weighted JIP and V is the nonweighted mapping.

The Frobenius tuning map, T, becomes

$$ T = GV = JV^+V $$

However, the main value of unweighted vals is that the pseudoinverse and projection matrix have rational entries, so that the rows of the matrix are fractional monzos. The Frobenius projection matrix therefore defines a completely canonical object not depending on any arbitrary definition (e.g. how Hermite normal form or LLL reduction is specifically defined) which corresponds one-to-one 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 matrix found by treating monzos in the same way as vals, P = I − Q is the same projection matrix 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 columns of monzos which spans the subspace of interval space containing the commas, then this same matrix P is given by I − MM⁺.

If the vals defining V are linearly independent, then P = V^T(VV^T)⁻¹V. If the columns of M are independent, then we likewise have P = I − M(M^TM)⁻¹M^T.

In D&D's guide, the systematic name for Frobenius tuning is minimax-E-copfr-S.

Benedetti/Wilson–Euclidean tuning

Benedetti/Wilson–Euclidean tuning (BE tuning) adopts the Benedetti or Wilson weight in place of Tenney weight, based on the dual norm of Wilson norm. For Q = ⟨2 3 5 …], the weighting matrix has the form

$$ W = \operatorname{diag} (1/Q) $$

In D&D's guide, the systematic name for BE tuning is minimax-E-sopfr-S.

Examples

For practical help, see POTE tuning.

Optimizing a temperament

Consider pajara, the 7-limit temperament tempering out 50/49 and 64/63. Its mapping V is

$$ V = \begin{bmatrix} 2 & 0 & 11 & 12 \\ 0 & 1 & -2 & -2 \end{bmatrix} $$

This can be found by taking two equal temperaments supporting pajara (e.g. 12edo and 22edo), defining a 2×4 matrix formed by the vals and then canonicalizing it. In weighted coordinates this would be

$$ V_W = \begin{bmatrix} 2.000000 & 0.000000 & 4.737442 & 4.274486 \\ 0.000000 & 0.630930 & -0.861353 & -0.712414 \end{bmatrix} $$

Then the pseudoinverse V⁺ is approximately

$$ V_W^+ = \begin{bmatrix} 0.143942 & 0.622569 \\ 0.196399 & 1.232405 \\ 0.072833 & -0.207801 \\ 0.085876 & -0.060988 \end{bmatrix} $$

so we obtain the generator tuning map G = J_WV +
W :

$$ G = \begin{bmatrix} 0.499049 & 1.586185 \end{bmatrix} $$

and the tuning map T = GV which shows the value each prime harmonic is tuned to:

$$ T = \begin{bmatrix} 0.998099 & 1.586185 & 2.317174 & 2.816223 \end{bmatrix} $$

The results are in octaves. Multiply them by 1200 to obtain the tuning in cents.

Using a Frobenius projection matrix

Using the values provided above, P_W = V +
W V_W is a 4×4 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 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 matrix instead, whose rows are fractional monzos. For instance, the unweighted pseudoinverse V +
12  of the 5-limit val V₁₂ for 12 equal is the column matrix V T
12 /1289; that is, the 1×3 matrix with column [12/1289 19/1289 28/1289⟩. Then V +
12 V₁₂ is the 3×3 Frobenius projection matrix P_F:

$$ P_\text{F} = \frac{1}{1289} \begin{bmatrix} 144 & 228 & 336 \\ 228 & 361 & 532 \\ 336 & 532 & 784 \end{bmatrix} $$

Multiplying ⟨12 19 28] by P_F gives ⟨12 19 28] again. Multiplying the monzos for 81/80, 128/125, 648/625 etc. gives the zero monzo, corresponding to a unison. Multiplying the val for 5-limit 19 equal, ⟨19 30 44], by P_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 P_F gives the fractional monzo corresponding to (2⁸⁴ 3¹³³ 5¹⁹⁶)^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 P_F, which is to say by [1 0 0⟩P_F, of 1196.778 cents.

We can do the same thing with a matrix V for pajara; then P_F, the Frobenius projection matrix, is

$$ P_\text{F} = V^+V = \frac{1}{305} \begin{bmatrix} 36 & 92 & 14 & 32 \\ 92 & 269 & -32 & 14 \\ 14 & -32 & 141 & 148 \\ 32 & 14 & 148 & 64 \end{bmatrix} $$

This sends monzos for 50/49, 64/63 etc. to the unison monzo, and vals for 10et, 12et, and 22et to themselves.