View source for 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}} 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 ([https://terhardt.userweb.mwn.de/ter/top/scalestretch.html 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 [https://en.wikipedia.org/wiki/Concert%20pitch upward drift of pitch reference] in European orchestras: [https://pianotuninginyork.blogspot.com/2018/11/a-history-of-pitch-standards-in-piano.html 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_tuning#CTWE_tuning|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 [http://x31eq.com/composite.pdf 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 {{nowrap| ''V'' {{=}} {{mapping| 1 0 2 -1 | 0 5 1 12 }} }} (the mapping for 7-limit [[magic]], which consists of a linearly independent list of [[val]]s defining magic) can be found as follows:

# Form a matrix ''V''<sub>''W''</sub> from ''V'' by multiplying by the diagonal matrix which is zero off the diagonal and 1/log<sub>2</sub>''p'' on the diagonal; in other words the diagonal is {{nowrap|[1 1/log<sub>2</sub>3 1/log<sub>2</sub>5 1/log<sub>2</sub>7]}}. Another way to say this is that each val is weighted by dividing through by the logarithms, so that {{nowrap| ''V''<sub>''W''</sub> {{=}} {{mapping| 1 0 2/log<sub>2</sub>5 -1/log<sub>2</sub>7 | 5/log<sub>2</sub>3 1/log<sub>2</sub>5 12/log<sub>2</sub>7 }} }}
# Find the pseudoinverse of the matrix {{nowrap| {{subsup|''V''|''W''|{{+}}}} {{=}} {{subsup|''V''|''W''|T}}(''V''<sub>''W''</sub>{{subsup|''V''|''W''|T}}){{inv}} }}. 
# Find the TE [[generator tuning map|generator map]] {{nowrap| ''G'' {{=}} ''J''<sub>''W''</sub>{{subsup|''V''|''W''|{{+}}}} }}, where {{nowrap| ''J''<sub>''W''</sub> {{=}} {{val| 1 1 1 1 }} }}. 
# Find the TE [[tuning map]] {{nowrap| ''T'' {{=}} ''GV''<sub>''W''</sub> }}. 
# Find the POTE generator map {{nowrap|''G''{{'}} {{=}} ''G''/''t''<sub>1</sub>}}; in other words ''G'' divided by the first entry of ''T''. 

If you carry out these operations, you should find

* ''V''<sub>''W''</sub> ~ {{mapping| 1.000 0 0.861 -0.356 | 0.000 3.155 0.431 4.274 }}
* ''G'' ~ {{val| 1.000902 0.317246 }}
* ''G''{{'}} ~ {{val| 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 [https://www.python.org/ Python] script that takes a mapping and gives TE and POTE generators, using [https://scipy.org/ Scipy]. 

<syntaxhighlight lang="python">
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)
</syntaxhighlight>

Output: 
<pre>
[1201.08240941  380.695113  ]
[1200.          380.35203249]
</pre>

== Systematic name ==
In D&D's guide to RTT, the [[Dave Keenan & Douglas Blumeyer's guide to RTT/Alternative complexities#Naming|systematic name]] for the POTE tuning scheme is '''[[Dave Keenan & Douglas Blumeyer's guide to RTT/All-interval tuning schemes#Destretched-octave_minimax-.28E.29S|destretched-octave minimax-ES]]'''.

[[Category:Terms]]
[[Category:Acronyms]]
[[Category:Regular temperament tuning]]