POTE tuning: Difference between revisions

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# Find the TE [[generator tuning map|generator map]] {{nowrap| ''G'' {{=}} ''J''''W''{{subsup|''V''|''W''|{{+}}}} }}, where {{nowrap| ''J''''W'' {{=}} {{val| 1 1 1 1 }} }}.

# Find the TE [[tuning map]] {{nowrap| ''T'' {{=}} ''GV''''W'' }}.

# Find the POTE generator map {{nowrap|''G''{{'}} {{=}} ''G''/''t''1}}; in other words ''G'' divided by the first entry of ''T''.

# Find the POTE generator map {{nowrap|''G''{{,}} {{=}} ''G''/''t''1}}; in other words ''G'' divided by the first entry of ''T''.

If you carry out these operations, you should find

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* ''V''''W'' ~ {{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 }}

* ''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]].

@@ Line 33: / Line 33: @@
 # 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''.
+# 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
@@ Line 39: / Line 39: @@
 * ''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 }}
+* ''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]].