POTE tuning

POTE tuning (pure-octaves Tenney-Euclidean tuning) is a good choice for a standard tuning enforcing a just 2/1 octave. It can be computed from TE tuning with all primes destretched until 2/1 is just.

Approximate Kees optimality

The POTE tuning is very close, but not exactly equal to the Kees-Euclidean tuning.

According to a conjecture of Graham Breed, these tunings also approximately minimize the squared error of all intervals weighted by Kees height, at least for full prime-limits. Graham showed this empirically in his composite.pdf paper by measuring the results for different temperaments and of different prime limits. It remains open how closely these approximations hold in all cases.

Computation

The TE and POTE tuning for a mapping such as A = [⟨1 0 2 -1], ⟨0 5 1 12]] (the mapping for 7-limit magic, which consists of a linearly independent list of vals defining magic) can be found as follows:

1. Form a matrix V from A by multiplying by the diagonal matrix which is zero off the diagonal and 1/log₂p on the diagonal; in other words the diagonal is [1 1/log₂3 1/log₂5 1/log₂7]. Another way to say this is that each val is "weighted" by dividing through by the logarithms, so that V = [⟨1 0 2/log₂5 -1/log₂7], ⟨5/log₂3 1/log₂5 12/log₂7]]
2. Find the pseudoinverse of the matrix V⁺ = V^T(VV^T)^-1.
3. Find the TE generators G = ⟨1 1 1 1]V⁺.
4. Find the TE tuning map: T = GV.
5. Find the POTE generators G' = G/T₁; in other words G scalar divided by the first entry of T.

If you carry out these operations, you should find

• V ~ [⟨1 0 0.861 -0.356], ⟨0 3.155 0.431 4.274]]
• G ~ ⟨1.000902 0.317246]
• G' ~ ⟨1.000000 0.316960]

The tuning of the POTE generator corresponding to the mapping A is therefore 0.31696 octaves, or 380.352 cents. Naturally, this only gives the single POTE generator in the rank two case, but the POTE tuning can still be found in this way for mappings defining higher-rank temperaments. The method can be generalized to subgroup temperaments by POL2 tuning, treating the formal prime represented by the first column as the equave.

Computer Program for TE and POTE

Below is a Python program that takes a mapping and gives TE and POTE generators.

Note: this program depends on Scipy.

import numpy as np
from scipy import linalg

def find_te (map, subgroup):
    jip = np.log2 (subgroup)
    weighter = np.diag (1/np.log2 (subgroup))
    map = map @ weighter
    jip = jip @ weighter

    te_gen = linalg.lstsq (np.transpose (map), jip)[0]
    te_map = te_gen @ map
    print (1200*te_gen)
    pote_gen = te_gen/te_map[0]
    print (1200*pote_gen)

# taking 7-limit magic as an example ...
seven_limit = [2, 3, 5, 7]
map_magic = [[1, 0, 2, -1], [0, 5, 1, 12]]

# to find TE and POTE you input
find_te (map_magic, seven_limit)

Output:

[1201.08240941  380.695113  ]
[1200.          380.35203249]