Constrained tuning - Revision history

FloraC: Oopsie

2026-02-27T17:26:05Z

Oopsie

← Older revision		Revision as of 17:26, 27 February 2026
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	As a standard optimization problem, numerous algorithms exist to solve it, such as {{w\|sequential quadratic programming}}, to name one. [[Flora Canou]]'s [https://github.com/FloraCanou/temperament_evaluator Temperament Evaluator] solves constrained tuning problems in [https://www.python.org Python], using [https://scipy.org/ Scipy]'s [https://www.cobyqa.com/stable/ COBYQA] algorithm. Here is an abridged version of it:		As a standard optimization problem, numerous algorithms exist to solve it, such as {{w\|sequential quadratic programming}}, to name one. [[Flora Canou]]'s [https://github.com/FloraCanou/temperament_evaluator Temperament Evaluator] solves constrained tuning problems in [https://www.python.org Python], using [https://scipy.org/ Scipy]'s [https://www.cobyqa.com/stable/ COBYQA] algorithm. Here is an abridged version of it:

	{{Todo\|rework\| ~~make~~ an absolutely minimal version of it. }}		{{Todo\|rework\|comment=Make an absolutely minimal version of it. }}
	{{Databox\| Code \|		{{Databox\| Code \|
	<syntaxhighlight lang="python">		<syntaxhighlight lang="python">

FloraC: /* Computation */ update and + todo

2026-02-27T17:24:49Z

Computation: update and + todo

Show changes

FloraC: Get rid of the acronym "ettoc"

2026-02-27T17:09:30Z

Get rid of the acronym "ettoc"

← Older revision		Revision as of 17:09, 27 February 2026
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	== Special constraint ==		== Special constraint ==
	The special eigenmonzo ''X'''''j''', where '''j''' is the all-ones monzo, has the effect of removing the weighted–skewed tuning bias. This eigenmonzo is actually proportional to the monzo of the extra dimension introduced by the skew. In other words, it forces the extra dimension to be pure, and therefore, the skew will have no effect with this constrained tuning.		The special eigenmonzo ''X''⋅'''j''', where '''j''' is the all-ones monzo, has the effect of removing the weighted–skewed tuning bias. This eigenmonzo is actually proportional to the monzo of the extra dimension introduced by the skew. In other words, it forces the extra dimension to be pure, and therefore, the skew will have no effect with this constrained tuning.

	It can be regarded as a distinct optimum. In the case of Tenney weighting, it is the '''TOCTE tuning''' ('''Tenney ones constrained Tenney–Euclidean tuning''').		It can be regarded as a distinct optimum. In the case of Tenney weighting, it is the '''TOCTE tuning''' ('''Tenney ones constrained Tenney–Euclidean tuning''').
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	$$		$$

	As a result, the [[~~Relative~~ interval error #Linearity\|relative error space]] is also linear with respect to ''V''.		As a result, the [[relative interval error #Linearity\|relative error space]] is also linear with respect to ''V''.

	For example, the relative errors of ~~12ettoc5 (~~12et in 5-limit TOC) is		For example, the relative errors of 12et in 5-limit TOC is

	$$ \mathcal{E}_\text {r}(12) = \val{-1.55\% & -4.42\% & +10.08\% } $$		$$ \mathcal{E}_\text {r}(12) = \val{-1.55\% & -4.42\% & +10.08\% } $$

	That of ~~19ettoc5~~ is		That of 19et in this tuning is

	$$ \mathcal{E}_\text {r}(19) = \val{+4.08\% & -4.97\% & -2.19\% } $$		$$ \mathcal{E}_\text {r}(19) = \val{+4.08\% & -4.97\% & -2.19\% } $$

	As 31 = 12 + 19, the relative errors of ~~31ettoc5~~ is		As 31 = 12 + 19, the relative errors of 31et in this tuning is

	$$		$$
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	== Systematic name ==		== Systematic name ==
	In [[D&D's guide\|D&D's guide to RTT]], the [[Dave Keenan & Douglas Blumeyer's guide to RTT/Alternative complexities#Naming\|systematic name]] for the CTE tuning scheme is ''[[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/All-interval tuning schemes #Held-octave minimax-.28E.29S\|held-octave minimax-ES]]'', and the systematic name for the CTWE tuning scheme is ''[[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Tuning fundamentals #Held-intervals\|held-octave]] [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Alternative complexities #Tunings used in 7\|minimax-E-lils-S]]''.		In [[D&D's guide\|D&D's guide to RTT]], the [[Dave Keenan & Douglas Blumeyer's guide to RTT/Alternative complexities #Naming\|systematic name]] for the CTE tuning scheme is ''[[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/All-interval tuning schemes #Held-octave minimax-.28E.29S\|held-octave minimax-ES]]'', and the systematic name for the CTWE tuning scheme is ''[[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Tuning fundamentals #Held-intervals\|held-octave]] [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Alternative complexities #Tunings used in 7\|minimax-E-lils-S]]''.

	== Open problems ==		== Open problems ==

BudjarnLambeth: /* Defense of CTE */

2025-09-21T09:10:42Z

Defense of CTE

← Older revision		Revision as of 09:10, 21 September 2025
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	This is one of the reasons why the tuning of 1-5/4-3/2 is so skewed in CTE blackwood. This problem doesn't happen with the TE tuning: the extra degree of freedom in adjusting the octave, and different weights, tend to even this kind of thing out. TE blackwood has 1-5/4-3/2 tuned to approximately 0-398-717 cents, which does seem to evenly split the error between the 5/4 and 6/5. We can see that something good about the way that TE tunes compact triads has not quite translated to CTE. Another way to look at this situation is that with CTE, 5/4 is prioritized more strongly than 6/5, and also 1-3-5 is tuned as nicely as possible, instead of 1-5/4-3/2.		This is one of the reasons why the tuning of 1-5/4-3/2 is so skewed in CTE blackwood. This problem doesn't happen with the TE tuning: the extra degree of freedom in adjusting the octave, and different weights, tend to even this kind of thing out. TE blackwood has 1-5/4-3/2 tuned to approximately 0-398-717 cents, which does seem to evenly split the error between the 5/4 and 6/5. We can see that something good about the way that TE tunes compact triads has not quite translated to CTE. Another way to look at this situation is that with CTE, 5/4 is prioritized more strongly than 6/5, and also 1-3-5 is tuned as nicely as possible, instead of 1-5/4-3/2.

	==== Defense of CTE ====		=== Defense of CTE ===
	Anyone who performs tuning optimization has [[octave reduction]] to unlearn. It is tempting to optimize for close-voiced chords such as 1–5/4–3/2 without much consideration, since textbooks often present harmony in this way. The close-voiced chord, 1-5/4-3/2, is an octave-reduced version of 1-3-5, with the latter being the simplest voicing possible in the [[chord of nature]] and nontrivially being the simplest such chord containing the fundamental (the 1st harmonic/true root). It is thus important to recognize that all octave-reductions are but simplifications for our cognitive processes.		Anyone who performs tuning optimization has [[octave reduction]] to unlearn. It is tempting to optimize for close-voiced chords such as 1–5/4–3/2 without much consideration, since textbooks often present harmony in this way. The close-voiced chord, 1-5/4-3/2, is an octave-reduced version of 1-3-5, with the latter being the simplest voicing possible in the [[chord of nature]] and nontrivially being the simplest such chord containing the fundamental (the 1st harmonic/true root). It is thus important to recognize that all octave-reductions are but simplifications for our cognitive processes.

BudjarnLambeth: Reduce use of the word "historically" in response to Graham Breed's comment on Facebook where he said the use of "historically" implies this is a settled debate, when it's actually still ongoing

2025-09-21T09:09:54Z

Reduce use of the word "historically" in response to Graham Breed's comment on Facebook where he said the use of "historically" implies this is a settled debate, when it's actually still ongoing

FloraC: /* CTE tuning vs POTE tuning vs CWE tuning vs CTWE tuning */ todo

2025-09-01T16:04:45Z

CTE tuning vs POTE tuning vs CWE tuning vs CTWE tuning: todo

← Older revision		Revision as of 16:04, 1 September 2025
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	</pre>		</pre>

	== ~~CTE tuning vs POTE tuning vs CWE tuning vs CTWE tuning~~ ==		== Comparison of tunings ==
			{{Todo\|inline=1\| rework \|comment=More properly and concisely summarize each side's POV. }}

	=== Criticism of CTE ===		=== Criticism of CTE ===
	People have long noted, since the early days of the tuning list, that the CTE tuning, despite having very nice qualities on paper, can give surprisingly strange results.{{citation needed}} One good example is blackwood, where the 4:5:6 chord is tuned to 0–386–720 cents, so that the error is not even close to evenly divided between the 5/4, 6/5, and 3/2. The reasons for this are subtle.		People have long noted, since the early days of the tuning list, that the CTE tuning, despite having very nice qualities on paper, can give surprisingly strange results.{{citation needed}} One good example is blackwood, where the 4:5:6 chord is tuned to 0–386–720 cents, so that the error is not even close to evenly divided between the 5/4, 6/5, and 3/2. The reasons for this are subtle.
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	So, one simple solution is to interpolate between the two, giving the '''Tenney–Weil–Euclidean norm''': a weighted average of the TE and WE norms, with free weighting parameter k. This can be thought of as adjusting how much we care about the span: {{nowrap\|k {{=}} 0}} is the TE norm, {{nowrap\|k {{=}} 1}} is the WE norm, and in between we have intermediate norms. This also gives a '''Constrained Tenney–Weil–Euclidean''' or '''CTWE''' tuning as a result, which interpolates between CTE and CKE.		So, one simple solution is to interpolate between the two, giving the '''Tenney–Weil–Euclidean norm''': a weighted average of the TE and WE norms, with free weighting parameter k. This can be thought of as adjusting how much we care about the span: {{nowrap\|k {{=}} 0}} is the TE norm, {{nowrap\|k {{=}} 1}} is the WE norm, and in between we have intermediate norms. This also gives a '''Constrained Tenney–Weil–Euclidean''' or '''CTWE''' tuning as a result, which interpolates between CTE and CKE.

	=== ~~Comparison~~ ===		=== Examples ===
	These tunings can be very different from each other.		These tunings can be very different from each other.

FloraC: /* Definition */ clarify

2025-09-01T15:51:17Z

Definition: clarify

← Older revision		Revision as of 15:51, 1 September 2025
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	The problem is feasible if		The problem is feasible if
	# rank(''M'') ≤ rank(''V''), and		# rank(''M'') ≤ rank(''V''), and
	# The subgroups of ''M'' and N (''V'') are {{w\|linear independence\|linearly independent}}.		# The subgroups of ''M'' and nullspace(''V'') are {{w\|linear independence\|linearly independent}}.

	== Computation ==		== Computation ==

FloraC: /* Computation */

2025-09-01T15:38:50Z

Computation

← Older revision		Revision as of 15:38, 1 September 2025
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	Notice we introduced the vector of lagrange multipliers ''Λ'', with length equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be discarded.		Notice we introduced the vector of lagrange multipliers ''Λ'', with length equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be discarded.

	=== Simple fast closed-form algorithm ===		==== Simple fast closed-form algorithm ====
	Another way to compute the CTE and CWE tunings, and the CTWE tuning in general, is to use the pseudoinverse.		Another way to compute the CTE and CWE tunings, and the CTWE tuning in general, is to use the pseudoinverse.

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	}}		}}

	=== Interpolating TE/CTE ===		==== Interpolating TE/CTE ====
	We can also interpolate between the TE and CTE tunings, if we want. To do this, we modify the TE tuning so that the weighting of the 2's coefficient is very large. As the weighting goes to infinity, we get the CTE tuning. Thus, we can set it to some sufficiently large number, so that we get whatever numerical precision we want, and compute the result in closed-form using the pseudoinverse. Without comments, docstrings, etc, the calculation is only about five lines of python code:		We can also interpolate between the TE and CTE tunings, if we want. To do this, we modify the TE tuning so that the weighting of the 2's coefficient is very large. As the weighting goes to infinity, we get the CTE tuning. Thus, we can set it to some sufficiently large number, so that we get whatever numerical precision we want, and compute the result in closed-form using the pseudoinverse. Without comments, docstrings, etc, the calculation is only about five lines of python code:

FloraC: /* Interpolating TE/CTE */

2025-09-01T15:38:09Z

Interpolating TE/CTE

← Older revision		Revision as of 15:38, 1 September 2025
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	J = 1200*np.log2(limit)		J = 1200*np.log2(limit)

	# Main calculation: get the generator map g such that g@M@W ≈ J@W. Use pinv		# Main calculation: get the generator map g such that g @ M @ W ≈ J @ W. Use pinv
	G = (J@W) @ np.linalg.pinv(M@W)		G = (J @ W) @ np.linalg.pinv(M @ W)
	T = G @ M		T = G @ M

FloraC: /* Simple fast closed-form algorithm */ cleanup: unify variable names with the other sections. Fix markup. Also important clarifications on the dimensionalities of these variables.

2025-09-01T15:36:52Z

Simple fast closed-form algorithm: cleanup: unify variable names with the other sections. Fix markup. Also important clarifications on the dimensionalities of these variables.

← Older revision		Revision as of 15:36, 1 September 2025
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	=== Simple fast closed-form algorithm ===		=== Simple fast closed-form algorithm ===
	~~{{Todo\|inline=1\|cleanup\|comment=Unify variable names with the other sections. }}~~

	Another way to compute the CTE and CWE tunings, and the CTWE tuning in general, is to use the pseudoinverse.		Another way to compute the CTE and CWE tunings, and the CTWE tuning in general, is to use the pseudoinverse.

	The basic idea is that the set of all pure-octave tuning maps of some temperament will be the intersection of a linear subspace and a shifted hyperplane, and thus will be a shifted subspace. This means that any pure-octave tuning map can be expressed as the sum of some arbitrary "reference" pure-octave tuning map for the temperament, plus some other one also in the temperament whose octave-coordinate is 0. The set of all such tuning maps of the latter category form a linear subspace.		The basic idea is that the set of all pure-octave tuning maps of some temperament will be the intersection of a linear subspace and a shifted hyperplane, and thus will be a shifted subspace. This means that any pure-octave tuning map can be expressed as the sum of some arbitrary "reference" pure-octave tuning map for the temperament, plus some other one also in the temperament whose octave-coordinate is 0. The set of all such tuning maps of the latter category form a linear subspace.

	We have the same thing with generator maps, meaning that any pure-octave generator map ~~<math>g</math>~~ can be expressed as:		We have the same thing with generator maps, meaning that any pure-octave generator map ''G'' can be expressed as:

	$$		$$ G = HB + G_0 $$
	g = hB + x
	$$

	where		where
			* ''G''<sub>0</sub> is any random generator map giving pure octaves;
			* ''B'' is an (''r'' - 1)×''r'' matrix whose rows are a basis for the subspace of generator maps with octave coordinate set to 0;
			* ''H'' is an (''r'' - 1)-element covector free variable.

	* ''x'' is any random generator map giving pure octaves		Given that, and assuming ''V'' is our mapping matrix, ''X'' our transformation matrix, and ''J'' our just tuning map, we can solve for the best possible ''G'' in closed form:
	* ''B'' is a matrix whose rows are a basis for the subspace of generator maps with octave coordinate set to 0
	* ''h'' is a free variable.

	Given that, and assuming ''M'' is our mapping matrix, ''W'' our ~~weighting~~ matrix, and ''j'' our ~~JIP~~, we can solve for the best possible ''g'' in closed form:

	$$		$$ GV_X \approx J_X $$
	~~gMW ≈ jW~~
	$$

	which becomes		which becomes

	$$		$$
	\left(hB + x\right)~~MW ≈ jW~~ \\		\begin{align}
	h = \left(j - xM\right)W \cdot \left(~~BMW~~\right)^\~~dagger~~		\left( HB + G_x \right) V_X &\approx J_X \\
			H &= \left( J_X - G_0 V_X \right) \cdot \left( BV_X \right)^+
			\end{align}
	$$		$$

	We note that this also works for any ~~weighting~~ matrix, and so we can use this to compute an arbitrary TWE norm very quickly in closed-form. Here is some Python code:		We note that this also works for any transformation matrix, and so we can use this to compute an arbitrary TWE norm very quickly in closed-form. Here is some Python code:

	{{Databox\|Code\|		{{Databox\|Code\|
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	# All pure-octave generator maps are just pure_octave_start + something in		# All pure-octave generator maps are just pure_octave_start + something in
	# the above row space. Now we have to solve		# the above row space. Now we have to solve
	# (h@B + x)@M@W ≈ j@W		# (h @ B + x) @ M @ W ≈ j @ W
	# which, solving for h and doing the algebra out gives:		# which, solving for h and doing the algebra out gives:
	h = (j - x@M)@W @ pinv(B@M@W)		h = (j - x @ M) @ W @ pinv(B @ M @ W)
	g = h@B + x		g = h @ B + x
	t = g@M		t = g @ M
	return g, t		return g, t

← Older revision		Revision as of 09:09, 21 September 2025
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	=== Criticism of CTE ===		=== Criticism of CTE ===
	People have long noted, since the early days of the tuning ~~list~~, that the CTE tuning, despite having very nice qualities on paper, can give surprisingly strange results.{{citation needed}} One good example is blackwood, where the 4:5:6 chord is tuned to 0–386–720 cents, so that the error is not even close to evenly divided between the 5/4, 6/5, and 3/2. The reasons for this are subtle.		People have long noted, since the early days of the [[Yahoo Groups tuning lists]], that the CTE tuning, despite having very nice qualities on paper, can give surprisingly strange results.{{citation needed}} One good example is blackwood, where the 4:5:6 chord is tuned to 0–386–720 cents, so that the error is not even close to evenly divided between the 5/4, 6/5, and 3/2. The reasons for this are subtle.

	This sort of thing was important ~~historically~~ when looking at optimal tunings for meantone, and is ultimately the motivation for advanced tuning methods such as TOP, TE, etc. to begin with. Thus, if our goal is to extend this principle in an elegant way to all intervals (and hopefully, triads and large chords), it would seem to defeat the purpose if we use a tuning optimization that doesn't also have this property,		This sort of thing was important to early [[regular temperament theory]] pioneers when looking at optimal tunings for meantone, and is ultimately the motivation for advanced tuning methods such as TOP, TE, etc. to begin with. Thus, if our goal is to extend this principle in an elegant way to all intervals (and hopefully, triads and large chords), it would seem to defeat the purpose if we use a tuning optimization that doesn't also have this property,

	As a result of this, ~~historically,~~ the POTE tuning ~~was used instead~~, which tunes it to a result that is approximately [[delta-rational]] 0-400-720 cents. People have also suggested using the Kees-Euclidean or KE tuning, also known as the constrained-Weil-Euclidean or CWE tuning. Here is a summary of the math involved and the ~~historical~~ reasoning behind this.		As a result of this, many use the POTE tuning, which tunes it to a result that is approximately [[delta-rational]]: 0-400-720 cents. People have also suggested using the Kees-Euclidean or KE tuning, also known as the constrained-Weil-Euclidean or CWE tuning. Here is a summary of the math involved and the reasoning behind this:

	The CTE tuning can be thought of as a modified TE tuning in which the weighting (in monzo space) on the 2/1 coordinate has been changed to 0, making it a kind of seminorm rather than a norm. As a result, all elements in the same octave-equivalence class are weighted identically: they are all given complexity equal to the ''representative'' in each equivalence class in which all factors of 2 have been removed. Thus 5/4 is given the same complexity as 5/1, 13/8 as 13/1, and so on.		The CTE tuning can be thought of as a modified TE tuning in which the weighting (in monzo space) on the 2/1 coordinate has been changed to 0, making it a kind of seminorm rather than a norm. As a result, all elements in the same octave-equivalence class are weighted identically: they are all given complexity equal to the ''representative'' in each equivalence class in which all factors of 2 have been removed. Thus 5/4 is given the same complexity as 5/1, 13/8 as 13/1, and so on.
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	This is one of the reasons why the tuning of 1-5/4-3/2 is so skewed in CTE blackwood. This problem doesn't happen with the TE tuning: the extra degree of freedom in adjusting the octave, and different weights, tend to even this kind of thing out. TE blackwood has 1-5/4-3/2 tuned to approximately 0-398-717 cents, which does seem to evenly split the error between the 5/4 and 6/5. We can see that something good about the way that TE tunes compact triads has not quite translated to CTE. Another way to look at this situation is that with CTE, 5/4 is prioritized more strongly than 6/5, and also 1-3-5 is tuned as nicely as possible, instead of 1-5/4-3/2.		This is one of the reasons why the tuning of 1-5/4-3/2 is so skewed in CTE blackwood. This problem doesn't happen with the TE tuning: the extra degree of freedom in adjusting the octave, and different weights, tend to even this kind of thing out. TE blackwood has 1-5/4-3/2 tuned to approximately 0-398-717 cents, which does seem to evenly split the error between the 5/4 and 6/5. We can see that something good about the way that TE tunes compact triads has not quite translated to CTE. Another way to look at this situation is that with CTE, 5/4 is prioritized more strongly than 6/5, and also 1-3-5 is tuned as nicely as possible, instead of 1-5/4-3/2.

	=== Defense of CTE ===		==== Defense of CTE ====
	Anyone who performs tuning optimization has [[octave reduction]] to unlearn. It is tempting to optimize for close-voiced chords such as 1–5/4–3/2 without much consideration, since textbooks often present harmony in this way. The close-voiced chord, 1-5/4-3/2, is an octave-reduced version of 1-3-5, with the latter being the simplest voicing possible in the [[chord of nature]] and nontrivially being the simplest such chord containing the fundamental (the 1st harmonic/true root). It is thus important to recognize that all octave-reductions are but simplifications for our cognitive processes.		Anyone who performs tuning optimization has [[octave reduction]] to unlearn. It is tempting to optimize for close-voiced chords such as 1–5/4–3/2 without much consideration, since textbooks often present harmony in this way. The close-voiced chord, 1-5/4-3/2, is an octave-reduced version of 1-3-5, with the latter being the simplest voicing possible in the [[chord of nature]] and nontrivially being the simplest such chord containing the fundamental (the 1st harmonic/true root). It is thus important to recognize that all octave-reductions are but simplifications for our cognitive processes.

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	Another way to think of it is that as POTE destretches the equave, it keeps the angle in the tuning space unchanged, and thus can be thought of as sacrificing multiplicative (typically very large) ratios for divisive (typically very small) ratios, whereas CTE sticks to the original design book of TE-optimality without worrying about that.		Another way to think of it is that as POTE destretches the equave, it keeps the angle in the tuning space unchanged, and thus can be thought of as sacrificing multiplicative (typically very large) ratios for divisive (typically very small) ratios, whereas CTE sticks to the original design book of TE-optimality without worrying about that.

	~~Historically, there was also an observation~~ that ~~the~~ POTE tuning can be thought of as an approximation to the CWE/KE tuning, which we will talk about below.		Observe that POTE tuning can be thought of as an approximation to the CWE/KE tuning, which we will talk about below.

	=== Using the Weil norm or Kees expressibility ===		=== Using the Weil norm or Kees expressibility ===