POTE tuning: Difference between revisions

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'''Destretched tunings''' are tuning [[optimization]] techniques with the [[tuning map]] scaled until a certain interval is just, that is, its stretch introduced in the optimization is removed. '''DTE tuning''' ('''destretched ~~Tenney-Euclidean~~ tuning''') is the most typical instance and will be the focus of this article. Otherwise normed tunings can be defined and computed analogously.

'''Destretched tunings''' are tuning [[optimization]] techniques with the [[tuning map]] scaled until a certain interval is just, that is, its stretch introduced in the optimization is removed. '''DTE tuning''' ('''destretched Tenney–Euclidean tuning''') is the most typical instance and will be the focus of this article. Otherwise normed tunings can be defined and computed analogously.

The most significant form of DTE tuning is pure-octave destretched, which is assumed unless specified otherwise. This has been called the '''POTE tuning''' ('''pure-~~octaves Tenney-Euclidean~~ tuning'''), although there are other ways to enforce a pure octave (→ [[Constrained tuning]]). POTE can be computed from [[TE tuning]] with all primes scaled until 2/1 is just.

The most significant form of DTE tuning is pure-octave destretched, which is assumed unless specified otherwise. This has been called the '''POTE tuning''' ('''pure-octave Tenney–Euclidean tuning'''), although there are other ways to enforce a pure octave (→ [[Constrained tuning]]). POTE can be computed from [[TE tuning]] with all primes scaled until 2/1 is just.

== Approximate Kees optimality ==

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== Computation ==

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:

The TE and POTE tuning for a [[mapping]] such as {{nowrap|'''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 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 }}]

# 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 {{nowrap|[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 {{nowrap|'''V''' {{=}} [{{val| 1 0 2/log25 -1/log27 }}, {{val| 5/log23 1/log25 12/log27 }}]}}

# Find the pseudoinverse of the matrix V~~~~+~~~~ = V~~T~~(VV~~T~~)~~-1~~.

# Find the pseudoinverse of the matrix {{nowrap|'''V'''{{+}} {{=}} '''V'''{{t}}('''VV'''}}t}}){{inv}}}}.

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

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

# Find the TE [[tuning map]]: T = GV.

# Find the TE [[tuning map]]: {{nowrap|'''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 {{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 ~ [{{val| 1 0 0.861 -0.356 }}, {{val| 0 3.155 0.431 4.274 }}]

* G ~ {{val| 1.000902 0.317246 }}

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

* G{{'}} ~ {{val| 1.000000 0.316960 }}

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]].

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

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-'''Destretched tunings''' are tuning [[optimization]] techniques with the [[tuning map]] scaled until a certain interval is just, that is, its stretch introduced in the optimization is removed. '''DTE tuning''' ('''destretched Tenney-Euclidean tuning''') is the most typical instance and will be the focus of this article. Otherwise normed tunings can be defined and computed analogously.
+'''Destretched tunings''' are tuning [[optimization]] techniques with the [[tuning map]] scaled until a certain interval is just, that is, its stretch introduced in the optimization is removed. '''DTE tuning''' ('''destretched Tenney–Euclidean tuning''') is the most typical instance and will be the focus of this article. Otherwise normed tunings can be defined and computed analogously.
-The most significant form of DTE tuning is pure-octave destretched, which is assumed unless specified otherwise. This has been called the '''POTE tuning''' ('''pure-octaves Tenney-Euclidean tuning'''), although there are other ways to enforce a pure octave (→ [[Constrained tuning]]). POTE can be computed from [[TE tuning]] with all primes scaled until 2/1 is just.
+The most significant form of DTE tuning is pure-octave destretched, which is assumed unless specified otherwise. This has been called the '''POTE tuning''' ('''pure-octave Tenney–Euclidean tuning'''), although there are other ways to enforce a pure octave (→ [[Constrained tuning]]). POTE can be computed from [[TE tuning]] with all primes scaled until 2/1 is just.
 == Approximate Kees optimality ==
@@ Line 9: / Line 9: @@
 == Computation ==
-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:
+The TE and POTE tuning for a [[mapping]] such as {{nowrap|'''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 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 }}]
+# 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 {{nowrap|[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 {{nowrap|'''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 pseudoinverse of the matrix {{nowrap|'''V'''{{+}} {{=}} '''V'''{{t}}('''VV'''}}t}}){{inv}}}}.
-# Find the TE generators G = {{val| 1 1 1 1 }}V<sup>+</sup>.
+# Find the TE generators {{nowrap|'''G''' {{=}} {{val| 1 1 1 1 }}'''V'''{{+}}}}.
-# Find the TE [[tuning map]]: T = GV.
+# Find the TE [[tuning map]]: {{nowrap|'''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 {{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
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 * V ~ [{{val| 1 0 0.861 -0.356 }}, {{val| 0 3.155 0.431 4.274 }}]
 * G ~ {{val| 1.000902 0.317246 }}
-* G<nowiki/>' ~ {{val| 1.000000 0.316960 }}
+* G{{'}} ~ {{val| 1.000000 0.316960 }}
-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]].
+The tuning of the POTE [[generator]] corresponding to the mapping '''A''' is therefore 0.31696 octaves, or 380.352{{cent}}. 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 ===