POTE tuning: Difference between revisions
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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''{{ | # 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''{{ | * ''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{{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]]. | The tuning of the POTE [[generator]] corresponding to the mapping ''V'' is therefore 0.31696 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]]. | ||