POTE tuning (pure-octaves Tenney-Euclidean tuning), also known as KE tuning (Kees-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.
POTE tuning is alternatively called KE tuning because it is the optimal tuning by Kees metric, that is, it minimizes the squared error of all intervals weighted by Kees height.
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:
- 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]]
- Find the pseudoinverse of the matrix V+ = VT(VVT)-1.
- Find the TE generators G = ⟨1 1 1 1]V+.
- Find the TE tuning map: T = GV.
- 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) te_map = te_gen @ map print (1200*te_gen) pote_gen = te_gen/te_map 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)
[1201.08240941 380.695113 ] [1200. 380.35203249]