Tenney–Euclidean tuning - Revision history

FloraC: + actually needed contents

2025-12-11T13:21:21Z

+ actually needed contents

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 There are a number of methods to solve least squares problems. One common way is to use the [[Moore–Penrose pseudoinverse]].
-== Computation using pseudoinverse ==
+== Computation ==
 The Moore–Penrose pseudoinverse, denoted ''A''{{+}}, is a generalization of the inverse matrix with which it shares a lot of properties. In this method, the (not necessarily independent) TE generator map ''G'', which correspond to the rows of ''V'' are given by
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 We may find the same projection matrix starting from a list of weighted monzos rather than vals. If ''M''<sub>''W''</sub> is a rank-''n'' matrix whose columns are weighted monzos, and ''I'' is the ''n''×''n'' identity matrix, then {{nowrap| ''P''<sub>''W''</sub> {{=}} ''I'' − ''M''<sub>''W''</sub>{{subsup|''M''|''W''|+}} }} is the same projection matrix as {{subsup|''V''|''W''|+}}''V''<sub>''W''</sub> so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.
 == Enforcement ==

FloraC: Restore an accidentally removed link

2025-12-08T16:24:53Z

Restore an accidentally removed link

← Older revision		Revision as of 16:24, 8 December 2025
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	=== Benedetti/Wilson–Euclidean tuning ===		=== 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

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

FloraC: Rework (see talk). This also introduces the free parameter "s" as I used in my parametric norm (but the idea ultimately comes from Mike's formulation of the Wilson norm)

2025-12-08T12:41:45Z

Rework (see talk). This also introduces the free parameter "s" as I used in my parametric norm (but the idea ultimately comes from Mike's formulation of the Wilson norm)

← Older revision		Revision as of 12:41, 8 December 2025
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	== Motivation ==		== 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 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.		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 ==		== Definition ==
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	{{Main\| POTE tuning }}		{{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 ===		=== Constrained TE tuning ===
	{{Main\| Constrained 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 ==		== 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 ===		=== 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		The weighting matrix ''W'' is given by
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	=== Benedetti/Wilson–Euclidean tuning ===		=== 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

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

FloraC: Note on "Benedetti" tuning

2025-09-03T13:54:13Z

Note on "Benedetti" tuning

← Older revision		Revision as of 13:54, 3 September 2025
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	=== Benedetti/Wilson–Euclidean tuning ===		=== 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

	$$ W = \operatorname{diag} (1/Q) $$		$$ W = \operatorname{diag} (1/Q) $$
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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.		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:Terms]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:Regular temperament tuning]]		[[Category:Regular temperament tuning]]

X31eq: /* Motivation */ add the most important motivations to a new paragraph at the top

2025-07-18T12:14:34Z

Motivation: add the most important motivations to a new paragraph at the top

← Older revision		Revision as of 12:14, 18 July 2025
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	== Motivation ==		== Motivation ==

	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. Tenney weighting weights each prime according to its complexity and if you consider an average over an infinite set of primes, it means similarly sized intervals get similar weight and arbitrarily complex intervals get an arbitrarily small weight.		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.		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.		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 ==		== Weaknesses ==

X31eq: Octave tempering as a weakness

2025-07-11T20:17:40Z

Octave tempering as a weakness

← Older revision		Revision as of 20:17, 11 July 2025
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	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.		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.

X31eq: Weaknesses

2025-07-11T20:15:23Z

Weaknesses

← Older revision		Revision as of 20:15, 11 July 2025
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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.		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.

X31eq: Motivation

2025-07-11T19:59:34Z

Motivation

← Older revision		Revision as of 19:59, 11 July 2025
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	In [[D&D's guide]], the systematic name for TE tuning is ''minimax-ES''.		In [[D&D's guide]], the systematic name for TE tuning is ''minimax-ES''.

			== Motivation ==

			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. Tenney weighting weights each prime according to its complexity and if you consider an average over an infinite set of primes, it means similarly sized intervals get similar weight and arbitrarily complex intervals get an arbitrarily small weight.

			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.


	== Definition ==		== Definition ==

FloraC: Actual formulation of the problem. Display style

2025-06-12T15:26:24Z

Actual formulation of the problem. Display style

Show changes

FloraC: /* Examples */ further cleanup

2025-06-12T13:29:43Z

Examples: further cleanup

← Older revision		Revision as of 13:29, 12 June 2025
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	=== Using a Frobenius projection matrix ===		=== Using a Frobenius projection matrix ===
	{{nowrap\|''P'' {{=}} {{subsup\|''V''\|''W''\|+}}''V''<sub>''W''</sub>}} is a 4×4 symmetrical matrix which projects weighted vals in TE tuning space, or weighted monzos in TE interval space, to a subspace defined by pajara. It therefore projects the weighted monzos for 50/49, 64/63, 225/224, 2048/2025 etc. to the zero vector, whereas it leaves pajara vals such as [[10edo]] in weighted coordinates unchanged.		Using the values provided above, {{nowrap\|''P''<sub>''W''</sub> {{=}} {{subsup\|''V''\|''W''\|+}}''V''<sub>''W''</sub>}} is a 4×4 symmetrical matrix which projects weighted vals in TE tuning space, or weighted monzos in TE interval space, to a subspace defined by pajara. It therefore projects the weighted monzos for 50/49, 64/63, 225/224, 2048/2025 etc. to the zero vector, whereas it leaves pajara vals such as [[10edo]] in weighted coordinates unchanged.

	If we use unweighted coordinates we get the Frobenius projection matrix instead, whose rows are [[fractional monzos]]. ~~The~~ unweighted pseudoinverse {{subsup\|''V''\|12\|+}} of the 5-limit val ''V''<sub>12</sub> for 12 equal is the column matrix {{subsup\|''V''\|12\|T}}/1289; that is, the 1×3 matrix with column {{monzo\| 12/1289 19/1289 28/1289 }}. Then {{subsup\|''V''\|12\|+}}''V''<sub>12</sub> is the 3×3 Frobenius projection matrix ''P''<sub>''F''</sub>:		If we use unweighted coordinates we get the Frobenius projection matrix instead, whose rows are [[fractional monzos]]. For instance, the unweighted pseudoinverse {{subsup\|''V''\|12\|+}} of the 5-limit val ''V''<sub>12</sub> for 12 equal is the column matrix {{subsup\|''V''\|12\|T}}/1289; that is, the 1×3 matrix with column {{monzo\| 12/1289 19/1289 28/1289 }}. Then {{subsup\|''V''\|12\|+}}''V''<sub>12</sub> is the 3×3 Frobenius projection matrix ''P''<sub>F</sub>:

	<math>\displaystyle		<math>\displaystyle
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	</math>		</math>

	Multiplying {{val\| 12 19 28 }} by ''P''<sub>''F''</sub> gives {{val\| 12 19 28 }} again. Multiplying the monzos for 81/80, 128/125, 648/625 etc. gives the zero monzo, corresponding to a unison. Multiplying the val for 5-limit 19 equal, {{val\| 19 30 44 }}, by P<sub>F</sub> gives {{val\| 24360 38570 56840 }}/1289, which is approximately the 19 equal val. Multiplying the 5-limit monzo for 3/2, which is {{monzo\| -1 1 0 }}; times F gives the fractional monzo corresponding to (2<sup>84</sup> 3<sup>133</sup> 5<sup>196</sup>)<sup>1/1289</sup>, which equates to 698.121 cents, the tempering of 3/2 in Frobenius tuning for 5-limit 12et, the tuning with octave defined by the top row of P<sub>F</sub>, which is to say by {{monzo\| 1 0 0 }}P<sub>F</sub>, of 1196.778 cents.		Multiplying {{val\| 12 19 28 }} by ''P''<sub>F</sub> gives {{val\| 12 19 28 }} again. Multiplying the monzos for 81/80, 128/125, 648/625 etc. gives the zero monzo, corresponding to a unison. Multiplying the val for 5-limit 19 equal, {{val\| 19 30 44 }}, by ''P''<sub>F</sub> gives {{val\| 24360 38570 56840 }}/1289, which is approximately the 19 equal val. Multiplying the 5-limit monzo for 3/2, which is {{monzo\| -1 1 0 }}; times ''P''<sub>F</sub> gives the fractional monzo corresponding to (2<sup>84</sup> 3<sup>133</sup> 5<sup>196</sup>)<sup>1/1289</sup>, which equates to 698.121 cents, the tempering of 3/2 in Frobenius tuning for 5-limit 12et, the tuning with octave defined by the top row of ''P''<sub>F</sub>, which is to say by {{monzo\| 1 0 0 }}''P''<sub>F</sub>, of 1196.778 cents.

	We can do the same thing with a matrix ''V'' ~~with rows consisting of the vals~~ for ~~7-limit 12 and 22 equal~~; then ''~~V''{{+}}''V~~'', the Frobenius projection matrix ~~for pajara~~, is		We can do the same thing with a matrix ''V'' for pajara; then ''P''<sub>F</sub>, the Frobenius projection matrix, is

	<math>\displaystyle		<math>\displaystyle
	P_\text{F} = \frac{1}{305}		P_\text{F} = V^+V = \frac{1}{305}
	\begin{bmatrix}		\begin{bmatrix}
	36 & 92 & 14 & 32 \\		36 & 92 & 14 & 32 \\