Tenney–Euclidean tuning: Difference between revisions

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In [[D&D's guide]], the systematic name for TE tuning is ''minimax-ES''.

== Motivation ==

TE tuning combines the ideas of Tenney weighting of prime limits and optimizing the root mean square (RMS) of weighted error/damage. Prime limits are a way of choosing musically useful intervals without being too specific about which intervals you think are useful. Tenney weighting is a way of giving equal treatment to intervals of equal complexity without choosing a finite set of intervals. RMS optimization (least squares) implies optimizing for the average sensory dissonance of an interval, and so allows for good consonances to balance weak dissonances and well tuned intervals to balance poorly tuned intervals.

TE shares with TOP tuning the insight that adding octaves to the optimization simplifies the calculation. Allowing for scale stretch balances intervals of different sizes.

As an RMS measure, TE error is optimized by the least squares method, which is well known and simple and efficient to calculate.

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.

Requiring octaves to be tempered is inconsistent with some electronic intervals with octave-repeating tuning tables.

== Definition ==

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=== Benedetti/Wilson–Euclidean tuning ===

'''Benedetti/Wilson–Euclidean tuning''' ('''BE tuning''') adopts the Benedetti or Wilson weight in place of Tenney weight, based on the dual norm of [[Wilson height]]. For {{nowrap|''Q'' {{=}} {{val| 2 3 5 … }}}}, the weighting matrix has the form

'''Benedetti/Wilson–Euclidean tuning''' ('''BE tuning''') adopts the Benedetti or Wilson weight in place of Tenney weight, based on the dual norm of [[Wilson norm]]<ref group="note">Technically, the [[Benedetti height]] is not a norm, and tunings that minimize the maximum Benedetti-weighted damage can be different from those based on the Wilson norm for certain subgroups. However, it is almost always more convenient to simply use the Wilson norm in these cases.</ref>. For {{nowrap|''Q'' {{=}} {{val| 2 3 5 … }}}}, the weighting matrix has the form

$$ W = \operatorname{diag} (1/Q) $$

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This sends monzos for 50/49, 64/63 etc. to the unison monzo, and vals for 10et, 12et, and 22et to themselves.

== Notes ==

[[Category:Terms]]

[[Category:Math]]

[[Category:Regular temperament tuning]]

@@ Line 6: / Line 6: @@
 In [[D&D's guide]], the systematic name for TE tuning is ''minimax-ES''.
+== Motivation ==
+TE tuning combines the ideas of Tenney weighting of prime limits and optimizing the root mean square (RMS) of weighted error/damage.  Prime limits are a way of choosing musically useful intervals without being too specific about which intervals you think are useful.  Tenney weighting is a way of giving equal treatment to intervals of equal complexity without choosing a finite set of intervals.  RMS optimization (least squares) implies optimizing for the average sensory dissonance of an interval, and so allows for good consonances to balance weak dissonances and well tuned intervals to balance poorly tuned intervals.
+TE shares with TOP tuning the insight that adding octaves to the optimization simplifies the calculation.  Allowing for scale stretch balances intervals of different sizes.
+As an RMS measure, TE error is optimized by the least squares method, which is well known and simple and efficient to calculate.
+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.
+Requiring octaves to be tempered is inconsistent with some electronic intervals with octave-repeating tuning tables.
 == Definition ==
@@ Line 76: / Line 101: @@
 === Benedetti/Wilson–Euclidean tuning ===
-'''Benedetti/Wilson–Euclidean tuning''' ('''BE tuning''') adopts the Benedetti or Wilson weight in place of Tenney weight, based on the dual norm of [[Wilson height]]. For {{nowrap|''Q'' {{=}} {{val| 2 3 5 … }}}}, the weighting matrix has the form
+'''Benedetti/Wilson–Euclidean tuning''' ('''BE tuning''') adopts the Benedetti or Wilson weight in place of Tenney weight, based on the dual norm of [[Wilson norm]]<ref group="note">Technically, the [[Benedetti height]] is not a norm, and tunings that minimize the maximum Benedetti-weighted damage can be different from those based on the Wilson norm for certain subgroups. However, it is almost always more convenient to simply use the Wilson norm in these cases.</ref>. For {{nowrap|''Q'' {{=}} {{val| 2 3 5 … }}}}, the weighting matrix has the form
 $$ W = \operatorname{diag} (1/Q) $$
@@ Line 156: / Line 181: @@
 This sends monzos for 50/49, 64/63 etc. to the unison monzo, and vals for 10et, 12et, and 22et to themselves.
+== Notes ==
+<references group="note"/>
 [[Category:Terms]]
 [[Category:Math]]
 [[Category:Regular temperament tuning]]