Tenney–Euclidean tuning: Difference between revisions

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== 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.

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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.

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

== Definition ==

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We may call the pure-octave Tenney−Euclidean tuning the ''POTE tuning''. If {{nowrap|''T'' {{=}} ''J''''W''{{subsup|''V''|''W''|+}}''V'' {{=}} ''GV''}} is the TE tuning map, then a corresponding pure-octaves map can be found by {{w|scalar multiplication}}, ''T''/''t''1, where ''t''1, the first entry, is the tuning of 2. ~~The justification for this~~ is ~~that ''T'' does not only define~~ a ~~point, but~~ a ~~line through the origin lying in the subspace defining the temperament~~, ~~or in other words, a point in the linear subspace of projective space corresponding~~ to the ~~temperament,~~ and ~~hence is a projective object. Another way~~ to ~~say this is that ''T'' defines not only the closest point to ''J'', but the closest direction in terms of angular measure between the line through ''T'' and the line through ''J''~~.

We may call the pure-octave Tenney−Euclidean tuning the ''POTE tuning''. If {{nowrap|''T'' {{=}} ''J''''W''{{subsup|''V''|''W''|+}}''V'' {{=}} ''GV''}} is the TE tuning map, then a corresponding pure-octaves map can be found by {{w|scalar multiplication}}, ''T''/''t''1, where ''t''1, the first entry, is the tuning of 2. While POTE is a very simple way to enforce a pure octave, it tends to overtemper the generators for divisive ratios and leads to less than ideal results.

=== Constrained TE tuning ===

Another way to enforce pure octaves is by adding the constraint before the optimization process. This is the ''CTE tuning''. The result, under the constraint of pure octaves, remains TE optimal.

Another way to enforce pure octaves is by adding the constraint before the optimization process. This is the ''CTE tuning''. The result, under the constraint of pure octaves, remains TE optimal. Contrary to POTE, CTE tends to under-optimize for divisive ratios; variations to fix this are considered under [[Constrained tuning]].

== Otherwise normed tunings ==

Tenney weighting gives equal treatment to intervals of equal complexity without choosing a finite set of intervals. While this makes a lot of sense, more specific demands on the field sometimes require it to be altered one way or another.

One common complaint is that it gives an undue weight to very high primes, as evidenced by the fact that you have to choose a limit to get sensible results. It does not converge as you keep adding primes. That it appears to be a limit to infinity of RMS of intervals approaching infinite complexity is psychoacoustically meaningless since the human ear cannot perceive even moderately complex intervals. To fix this, a free parameter ''s'' may be introduced such that each prime ''q'' is weighted by 1/(log2(''q''))s instead of 1/log2(''q''), and we can obtain a steeper weight curve with any {{nowrap| ''s'' > 1 }}:

$$ W = \operatorname {diag} (1/(\log_2 (Q))^s) $$

Another fix is to adopt the Wilson weight in place of Tenney weight, discussed in [[#Benedetti/Wilson–Euclidean tuning]].

The other complaint, contrary to above, is that Tenney weighting gives insufficient weight to higher primes than a weight specifically considering the relevant intervals within the limit. For this, we can use the parameter ''s'' again, as any {{nowrap| ''s'' < 1 }} will give us a flatter weight curve. In particular, we can set ''s'' to 0, which removes the weight entirely and results in equal weight of all primes. This is called Frobenius tuning, discussed right below.

=== Frobenius tuning and Frobenius projection matrix ===

We may ~~also do~~ the same ~~things~~ starting from nonweighted vals. This ~~leads to a different tuning, the '''Frobenius tuning''', which~~ is perfectly functional but has less theoretical justification than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it.

The '''Frobenius tuning''' may be considered as the same RMS tuning but starting from nonweighted vals. This is perfectly functional but has less theoretical justification than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it.

The weighting matrix ''W'' is given by

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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 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

'''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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 == 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.
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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.
-Requiring octaves to be tempered is inconsistent with some electronic intervals with octave-repeating tuning tables.
 == Definition ==
@@ Line 67: / Line 51: @@
 {{Main| POTE tuning }}
-We may call the pure-octave Tenney−Euclidean tuning the ''POTE tuning''. If {{nowrap|''T'' {{=}} ''J''<sub>''W''</sub>{{subsup|''V''|''W''|+}}''V'' {{=}} ''GV''}} is the TE tuning map, then a corresponding pure-octaves map can be found by {{w|scalar multiplication}}, ''T''/''t''<sub>1</sub>, where ''t''<sub>1</sub>, the first entry, is the tuning of 2. The justification for this is that ''T'' does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that ''T'' defines not only the closest point to ''J'', but the closest direction in terms of angular measure between the line through ''T'' and the line through ''J''.
+We may call the pure-octave Tenney−Euclidean tuning the ''POTE tuning''. If {{nowrap|''T'' {{=}} ''J''<sub>''W''</sub>{{subsup|''V''|''W''|+}}''V'' {{=}} ''GV''}} is the TE tuning map, then a corresponding pure-octaves map can be found by {{w|scalar multiplication}}, ''T''/''t''<sub>1</sub>, where ''t''<sub>1</sub>, the first entry, is the tuning of 2. While POTE is a very simple way to enforce a pure octave, it tends to overtemper the generators for divisive ratios and leads to less than ideal results.
 === Constrained TE tuning ===
 {{Main| Constrained tuning }}
-Another way to enforce pure octaves is by adding the constraint before the optimization process. This is the ''CTE tuning''. The result, under the constraint of pure octaves, remains TE optimal.
+Another way to enforce pure octaves is by adding the constraint before the optimization process. This is the ''CTE tuning''. The result, under the constraint of pure octaves, remains TE optimal. Contrary to POTE, CTE tends to under-optimize for divisive ratios; variations to fix this are considered under [[Constrained tuning]].
 == Otherwise normed tunings ==
+Tenney weighting gives equal treatment to intervals of equal complexity without choosing a finite set of intervals. While this makes a lot of sense, more specific demands on the field sometimes require it to be altered one way or another.
+One common complaint is that it gives an undue weight to very high primes, as evidenced by the fact that you have to choose a limit to get sensible results. It does not converge as you keep adding primes. That it appears to be a limit to infinity of RMS of intervals approaching infinite complexity is psychoacoustically meaningless since the human ear cannot perceive even moderately complex intervals. To fix this, a free parameter ''s'' may be introduced such that each prime ''q'' is weighted by 1/(log<sub>2</sub>(''q''))<sup>s</sup> instead of 1/log<sub>2</sub>(''q''), and we can obtain a steeper weight curve with any {{nowrap| ''s'' > 1 }}:
+$$ W = \operatorname {diag} (1/(\log_2 (Q))^s) $$
+Another fix is to adopt the Wilson weight in place of Tenney weight, discussed in [[#Benedetti/Wilson–Euclidean tuning]].
+The other complaint, contrary to above, is that Tenney weighting gives insufficient weight to higher primes than a weight specifically considering the relevant intervals within the limit. For this, we can use the parameter ''s'' again, as any {{nowrap| ''s'' < 1 }} will give us a flatter weight curve. In particular, we can set ''s'' to 0, which removes the weight entirely and results in equal weight of all primes. This is called Frobenius tuning, discussed right below.
 === Frobenius tuning and Frobenius projection matrix ===
-We may also do the same things starting from nonweighted vals. This leads to a different tuning, the '''Frobenius tuning''', which is perfectly functional but has less theoretical justification than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it.
+The '''Frobenius tuning''' may be considered as the same RMS tuning but starting from nonweighted vals. This is perfectly functional but has less theoretical justification than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it.
 The weighting matrix ''W'' is given by
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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 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
+'''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
 $$ W = \operatorname{diag} (1/Q) $$