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POTE tuning: Difference between revisions

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Links for the side issue of pitch inflation. I think we're allowed to link to Wikipedia. I couldn't find a citation for the specific issue of pianos driving the pitch up although I believe this is the case. It doesn't matter either way for the usefulness of POTE
 
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== Motivation ==
== 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.
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 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.
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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.
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 (Terhardt's website was a good reference for this{{citation needed}}) that many bands are tuned according to stretched octaves even when the instruments are producing harmonic timbres. 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.
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 ==
== Approximate Kees optimality ==
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== Computation ==
== Computation ==
The TE and POTE tuning for a [[mapping]] such as {{nowrap|'''A''' {{=}} [{{val| 1 0 2 -1 }}, {{val| 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:
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''' from '''A''' 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''' {{=}} [{{val| 1 0 2/log<sub>2</sub>5 -1/log<sub>2</sub>7 }}, {{val| 5/log<sub>2</sub>3 1/log<sub>2</sub>5 12/log<sub>2</sub>7 }}]}}
# 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|'''V'''{{+}} {{=}} '''V'''{{t}}('''VV'''{{t}}){{inv}}}}.  
# 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 generators {{nowrap|'''G''' {{=}} {{val| 1 1 1 1 }}'''V'''{{+}}}}.  
# 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'''}}.  
# Find the TE [[tuning map]] {{nowrap| ''T'' {{=}} ''GV''<sub>''W''</sub> }}.  
# Find the POTE generators {{nowrap|'''G'''{{'}} {{=}} '''G'''/'''T'''<sub>1</sub>}}; in other words '''G''' divided by the first entry of '''T'''.  
# 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
If you carry out these operations, you should find


* V ~ [{{val| 1 0 0.861 -0.356 }}, {{val| 0 3.155 0.431 4.274 }}]
* ''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| 1.000902 0.317246 }}
* G{{'}} ~ {{val| 1.000000 0.316960 }}
* ''G''{{'}} ~ {{val| 1.000000 0.316960 }}
 
The tuning of the POTE [[generator]] corresponding to the mapping '''A''' is therefore 0.31696 octaves, or 380.352{{cent}}. 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 [[Lp tuning|POL2 tuning]], treating the formal prime represented by the first column as the [[equave]].


=== Computer Program for TE and POTE ===
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]].  
Below is a [https://www.python.org/ Python] program that takes a mapping and gives TE and POTE generators.  


Note: this program depends on [https://scipy.org/ Scipy].  
=== 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">
<syntaxhighlight lang="python">
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from scipy import linalg
from scipy import linalg


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


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


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


# to find TE and POTE you input
# to find TE and POTE you enter
find_te (map_magic, seven_limit)
find_te (mapping_magic, seven_limit)
</syntaxhighlight>
</syntaxhighlight>


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