POTE tuning: Difference between revisions
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'''POTE tuning''' ('''pure-octaves Tenney-Euclidean tuning'''), also known as '''KE tuning''' ('''Kees-Euclidean tuning'''), is a good choice for a standard tuning enforcing a just 2/1 octave. It can be computed from [[TE tuning]] with all primes destretched until 2/1 is just. | '''POTE tuning''' ('''pure-octaves Tenney-Euclidean tuning'''), also known as '''KE tuning''' ('''Kees-Euclidean tuning'''), is a good choice for a standard tuning enforcing a just 2/1 octave. It can be computed from [[TE tuning]] with all primes destretched until 2/1 is just. | ||
== Kees optimality == | == Approximate Kees optimality == | ||
POTE tuning is | 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]]. | ||
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. | |||
== Computation == | == Computation == | ||