POTE tuning

Destretched tunings are tuning optimization techniques with the tuning map scaled until a certain interval is just, that is, its stretch introduced in the optimization is removed. Destretched Tenney–Euclidean tuning^{[idiosyncratic term]} is the most typical instance and will be the focus of this article. Otherwise normed tunings can be defined and computed analogously.

The most significant form of these tunings is pure-octave destretched, which is assumed unless specified otherwise. This has been called the POTE tuning (pure-octave Tenney–Euclidean tuning), although there are other ways to enforce a pure octave (→ Constrained tuning). POTE can be computed from TE tuning with all primes scaled until 2/1 is just.

Motivation

POTE is the same as TE in the limit of very small intervals. This means it is most similar to TE for intervals smaller than an octave, and most divergent for intervals of several octaves. As a tuning for the full audible range, the logic is that smaller intervals are more common in chords and so more important to optimize for. There are other ways to do this. POTE is the simplest way of prioritizing smaller intervals.

POTE can stand in for TE where a pure-octave tuning is convenient for implementation constraints, like when a synthesizer has pure octave tuning tables. POTE is close to TE for melodic steps, so melodies can be translated between POTE and TE with minimal damage.

POTE has the conceptual advantage that it is a simple deformation of TE, itself a simple measure, and introduces no more free parameters. POTE can also be used to give a feel for how a tuning damages different odd primes and other simple intervals without requiring the mental arithmetic of juggling multiples of the damage of 2:1. (TE with a basis of 2:1, 3:2, 5:4, etc. would also do this.)

POTE has practical advantages for tuning instruments constrained to pure octaves as part of a band targeting TE. You can set the absolute pitch reference for each instrument so that it agrees with the TE background for a target register. Guitars (or other fretted string instruments) can implement this within themselves by having the frets assuming pure octaves and the open strings following the TE stretch.

Psychoacoustics shows that many bands are tuned according to stretched octaves even when the instruments are producing harmonic timbres (Terhardt: Stretch of the musical tone scale). This might be with each instrument having a stretched scale, or high-pitched instruments having a slightly sharp pitch reference. The magnitude of this stretch often swamps the optimal stretch for TE (which can be in either direction). So, if you are not going to observe the TE stretch, you might as well simplify it out. There are other reasons for putting instruments deliberately out of tune, for example solo instruments can be tuned slightly sharp to make them stand out. This leads to an upward drift of pitch reference in European orchestras: pianos are tuned slightly sharp to make them sound bright, and then the orchestra sharpens up to follow them.

Weaknesses

• POTE tuning inherits problems of TE in being chosen for mathematical simplicity rather than a sound psychoacoustic basis.
• Like Kees height, POTE agrees with TE for arbitrarily small intervals, which means it puts less emphasis on actually audible intervals, particularly those larger than an octave. This tendency is mediated by Constrained Tenney–Weil–Euclidean tuning
• Something must be lost by not tempering the octaves

Approximate Kees optimality

The POTE tuning is very close, but not exactly equal to the KE 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 V = [⟨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_W from V 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_W = [⟨1 0 2/log₂5 -1/log₂7], ⟨5/log₂3 1/log₂5 12/log₂7]]
2. Find the pseudoinverse of the matrix V ⁺
   W  = V T
   W (V_WV T
   W )⁻¹.
3. Find the TE generator map G = J_WV ⁺
   W , where J_W = ⟨1 1 1 1].
4. Find the TE tuning map T = GV_W.
5. Find the POTE generator map G' = G/t₁; in other words G divided by the first entry of T.

If you carry out these operations, you should find

• V_W ~ [⟨1.000 0 0.861 -0.356], ⟨0.000 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 V is therefore 0.31696 octaves, or 380.352 cents. Naturally, this only gives the single POTE generator in the rank-2 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, treating the formal prime represented by the first column as the equave.

Computer program for TE and POTE

Below is a Python script that takes a mapping and gives TE and POTE generators, using Scipy.

import numpy as np
from scipy import linalg

def find_te (mapping, subgroup):
    just_tuning_map = np.log2 (subgroup)
    te_weight = np.diag (1/np.log2 (subgroup))
    mapping = mapping @ te_weight
    just_tuning_map = just_tuning_map @ te_weight

    te_generators = linalg.lstsq (np.transpose (mapping), just_tuning_map)[0]
    te_tuning_map = te_generators @ mapping
    print (1200*te_generators)
    pote_generators = te_generators/te_tuning_map[0]
    print (1200*pote_generators)

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

# to find TE and POTE you enter
find_te (mapping_magic, seven_limit)

Output:

[1201.08240941  380.695113  ]
[1200.          380.35203249]

Systematic name

In D&D's guide to RTT, the systematic name for the POTE tuning scheme is destretched-octave minimax-ES.