POTE tuning: Difference between revisions

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== Computation ==

The TE and POTE tuning for a [[mapping~~|map matrix~~]] such as M = [{{val| 1 0 2 -1 }}, {{val| 0 5 1 12 }}] (the ~~map~~ 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 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:

# Form a matrix V from M by multiplying by the diagonal matrix which is zero off the diagonal and 1/log2''p'' on the diagonal; in other words the diagonal is [1 1/log23 1/log25 1/log27]. Another way to say this is that each val is "weighted" by dividing through by the logarithms, so that V = [{{val| 1 0 2/log25 -1/log27 }}, {{val| 5/log23 1/log25 12/log27 }}]

# Form a matrix V from A by multiplying by the diagonal matrix which is zero off the diagonal and 1/log2''p'' on the diagonal; in other words the diagonal is [1 1/log23 1/log25 1/log27]. Another way to say this is that each val is "weighted" by dividing through by the logarithms, so that V = [{{val| 1 0 2/log25 -1/log27 }}, {{val| 5/log23 1/log25 12/log27 }}]

# Find the pseudoinverse of the matrix V+ = VT(VVT)-1.

# Find the TE generators G = {{val| 1 1 1 1 }}V+.

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* G<nowiki/>' ~ {{val| 1.000000 0.316960 }}

The tuning of the POTE [[generator]] corresponding to the mapping M is therefore 0.31696 octaves, or 380.352 cents. Naturally, this only gives the single POTE generator in the rank two case~~, and only when the map M is in period-generator form~~, 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 ~~so long as the group contains 2~~ by [[Lp tuning|POL2 tuning]].

The tuning of the POTE [[generator]] corresponding to the mapping A is therefore 0.31696 octaves, or 380.352 cents. 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 ===

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

Below is a [https://www.python.org/ Python] program that takes a ~~map~~ and gives TE and POTE generators.

Note: this program depends on [https://scipy.org/ Scipy].

@@ Line 5: / Line 5: @@
 == Computation ==
-The TE and POTE tuning for a [[mapping|map matrix]] such as M = [{{val| 1 0 2 -1 }}, {{val| 0 5 1 12 }}] (the map 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 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:
-# Form a matrix V from M 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 [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 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 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 [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 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 }}]
 # Find the pseudoinverse of the matrix V<sup>+</sup> = V<sup>T</sup>(VV<sup>T</sup>)<sup>-1</sup>.
 # Find the TE generators G = {{val| 1 1 1 1 }}V<sup>+</sup>.
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 * G<nowiki/>' ~ {{val| 1.000000 0.316960 }}
-The tuning of the POTE [[generator]] corresponding to the mapping M is therefore 0.31696 octaves, or 380.352 cents. Naturally, this only gives the single POTE generator in the rank two case, and only when the map M is in period-generator form, 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 so long as the group contains 2 by [[Lp tuning|POL2 tuning]].
+The tuning of the POTE [[generator]] corresponding to the mapping A is therefore 0.31696 octaves, or 380.352 cents. 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 ===
+Below is a [https://www.python.org/ Python] program that takes a mapping and gives TE and POTE generators.
-Below is a [https://www.python.org/ Python] program that takes a map and gives TE and POTE generators.
 Note: this program depends on [https://scipy.org/ Scipy].