Tenney–Euclidean tuning: Difference between revisions

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TE tuning is uniquely optimized for a given prime limit. There are no free parameters determining the weighting of different intervals or the balance of wide and narrow intervals: all follow from the definition of Tenney weighting of primes.

== Weaknesses ==

TE must give an undue weight to extremely large intervals, as evidenced by the fact that you have to choose a prime limit to get sensible results. It doesn't converge as you keep adding primes.

The optimization with octaves constrained to be pure (CTE) is controversial, and many believe the implied TE error function being minimized to be incorrect in this case and so generally invalid. Variations to fix this are considered under [[Constrained_tuning]].

Weighting intervals according to their size gives less weight to higher primes than an RMS specifically considering audible ratios within the prime limit.

That TE tuning appears to be a limit to infinity of RMS of intervals approaching infinite complexity is meaningless. The human ear can't perceive even moderately complex intervals and the convergence is too slow to be psychoacoustically meaningful.

Optimizing for an average rather than a minimax means intolerably mistuned intervals are balanced by needlessly pure intervals, rather than ensuring all intervals get tempered to within tolerable bounds.

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 TE tuning is uniquely optimized for a given prime limit.  There are no free parameters determining the weighting of different intervals or the balance of wide and narrow intervals: all follow from the definition of Tenney weighting of primes.
+== Weaknesses ==
+TE must give an undue weight to extremely large intervals, as evidenced by the fact that you have to choose a prime limit to get sensible results.  It doesn't converge as you keep adding primes.
+The optimization with octaves constrained to be pure (CTE) is controversial, and many believe the implied TE error function being minimized to be incorrect in this case and so generally invalid.  Variations to fix this are considered under [[Constrained_tuning]].
+Weighting intervals according to their size gives less weight to higher primes than an RMS specifically considering audible ratios within the prime limit.
+That TE tuning appears to be a limit to infinity of RMS of intervals approaching infinite complexity is meaningless.  The human ear can't perceive even moderately complex intervals and the convergence is too slow to be psychoacoustically meaningful.
+Optimizing for an average rather than a minimax means intolerably mistuned intervals are balanced by needlessly pure intervals, rather than ensuring all intervals get tempered to within tolerable bounds.