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|map matrix]] such as M = [{{val|1 0 2 -1}}, {{val|0 5 1 12}}] (the map for 7-limit [[~~Magic family|~~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 }}]

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

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

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

# Find the TE tuning map: T = gV.

# Find the TE tuning map: T = GV.

# Find the POTE generators g<nowiki/>' = g/T1; in other words g scalar divided by the first entry of T.

# Find the POTE generators G<nowiki/>' = G/T1; in other words G scalar divided by the first entry of T.

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 }}]

* G ~ {{val| 1.000902 0.317246 }}

* g ~ {{val| 1.000902 0.317246 }}

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

* 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]].

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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|map matrix]] such as M = [{{val|1 0 2 -1}}, {{val|0 5 1 12}}] (the map for 7-limit [[Magic family|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 }}]
 # 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>.
+# Find the TE generators G = {{val| 1 1 1 1 }}V<sup>+</sup>.
-# Find the TE tuning map: T = gV.
+# Find the TE tuning map: T = GV.
-# Find the POTE generators g<nowiki/>' = g/T<sub>1</sub>; in other words g scalar divided by the first entry of T.
+# Find the POTE generators G<nowiki/>' = G/T<sub>1</sub>; in other words G scalar divided by the first entry of T.
 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 }}]
+* G ~ {{val| 1.000902 0.317246 }}
-* g ~ {{val| 1.000902 0.317246 }}
+* G<nowiki/>' ~ {{val| 1.000000 0.316960 }}
-* 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]].