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 (→ [[~~CTE~~ 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-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.

== Approximate Kees optimality ==

The POTE tuning is very close, but not exactly equal to the [[~~Weil Norms, Tenney-Weil Norms, and TWp Interval and Tuning Space #Kees-Euclidean Seminorm|Kees-Euclidean~~ tuning]].

The POTE tuning is very close, but not exactly equal to the [[KE tuning]].

According to a conjecture of Graham Breed, these tunings also approximately minimize the squared error of all intervals weighted by [[Kees height]], at least for full prime-limits. Graham showed this empirically in his [http://x31eq.com/composite.pdf composite.pdf] paper by measuring the results for different temperaments and of different prime limits. It remains open how closely these approximations hold in all cases.

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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 (→ [[CTE 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-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.
 == Approximate Kees optimality ==
-The POTE tuning is very close, but not exactly equal to the [[Weil Norms, Tenney-Weil Norms, and TWp Interval and Tuning Space #Kees-Euclidean Seminorm|Kees-Euclidean tuning]].
+The POTE tuning is very close, but not exactly equal to the [[KE tuning]].
 According to a conjecture of Graham Breed, these tunings also approximately minimize the squared error of all intervals weighted by [[Kees height]], at least for full prime-limits. Graham showed this empirically in his [http://x31eq.com/composite.pdf composite.pdf] paper by measuring the results for different temperaments and of different prime limits. It remains open how closely these approximations hold in all cases.