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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
ArrowHead294 (talk | contribs)
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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 way of prioritizing smaller intervals.
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.


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== Weaknesses ==
== Weaknesses ==
* POTE tuning inherits problems of TE in being chosen for mathematical simplicity rather than a sound psychoacoustic basis.
* 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]]
* 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]]
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# 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 [[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 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''.  
# 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
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* ''V''<sub>''W''</sub> ~ {{mapping| 1.000 0 0.861 -0.356 | 0.000 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 ''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]].  
The tuning of the POTE [[generator]] corresponding to the mapping ''V'' is therefore 0.31696&nbsp;octaves, or 380.352{{c}}. 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 ===
=== Computer program ===
Below is a [https://www.python.org/ Python] script that takes a mapping and gives TE and POTE generators, using [https://scipy.org/ Scipy].  
Based on the [https://www.python.org/ Python] script in [[Tenney–Euclidean tuning #Computer program]], here is a variant that takes a mapping and gives 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 (mapping, subgroup):
def te (mapping, subgroup_basis):
     just_tuning_map = np.log2 (subgroup)
     just_tuning_map = 1200*np.log2 (subgroup_basis)
     te_weight = np.diag (1/np.log2 (subgroup))
     te_weight = np.diag (1/np.log2 (subgroup_basis))
     mapping = mapping @ te_weight
     mapping_w = mapping @ te_weight
     just_tuning_map = just_tuning_map @ te_weight
     just_tuning_map_w = just_tuning_map @ te_weight


     te_generators = linalg.lstsq (np.transpose (mapping), just_tuning_map)[0]
     te_generators = linalg.lstsq (np.transpose (mapping_w), just_tuning_map_w)[0]
     te_tuning_map = te_generators @ mapping
     te_tuning_map = te_generators @ mapping
     print (1200*te_generators)
     return te_generators, te_tuning_map
     pote_generators = te_generators/te_tuning_map[0]
 
     print (1200*pote_generators)
def pote (mapping, subgroup_basis):
    te_generators, te_tuning_map = te (mapping, subgroup_basis)
     pote_generators = te_generators/(te_tuning_map[0]/1200)
     pote_tuning_map = te_tuning_map/(te_tuning_map[0]/1200)
    return pote_generators, pote_tuning_map
</syntaxhighlight>


<syntaxhighlight lang="python">
# taking 7-limit magic as an example ...
# taking 7-limit magic as an example ...
seven_limit = [2, 3, 5, 7]
mapping = np.array ([[1, 0, 2, -1], [0, 5, 1, 12]])
mapping_magic = [[1, 0, 2, -1], [0, 5, 1, 12]]
subgroup_basis = np.array ([2, 3, 5, 7])


# to find TE and POTE you enter
# to find the POTE tuning you enter
find_te (mapping_magic, seven_limit)
pote (mapping, subgroup_basis)
</syntaxhighlight>
</syntaxhighlight>


Output:  
Output:  
<pre>
<pre>
[1201.08240941 380.695113  ]
[1200.       , 380.35203249]
[1200.         380.35203249]
[1200.       , 1901.76016243, 2780.35203249, 3364.22438984]
</pre>
</pre>


Retrieved from "https://en.xen.wiki/w/POTE_tuning"