Perfect balance - Revision history

Fredg999: Replaced "wheel" with "circle", in case "wheel" wasn't clear enough

2025-05-03T21:57:06Z

Replaced "wheel" with "circle", in case "wheel" wasn't clear enough

← Older revision		Revision as of 21:57, 3 May 2025
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	{{distinguish\|balanced word}}		{{distinguish\|balanced word}}
	A non-empty finite set of real numbers ''S'' in the range <math>[0, 1)</math> is called '''perfectly balanced''' if a ~~wheel{{Term needs definition}}~~ with an equal weight placed at angle <math>2\pi x</math> for each <math>x \in S</math> has its center of gravity exactly at the hub. Mathematically, this is given by the equation <math>\sum_{x \in S} e^{2\pi i x} = 0</math>.		A non-empty finite set of real numbers ''S'' in the range <math>[0, 1)</math> is called '''perfectly balanced''' if a circle with an equal weight placed at angle <math>2\pi x</math> for each <math>x \in S</math> has its center of gravity exactly at the hub. Mathematically, this is given by the equation <math>\sum_{x \in S} e^{2\pi i x} = 0</math>.

	In the context of musical tunings, a perfectly balanced set can be converted to a [[periodic scale]] by taking the frequency ratio <math>2^x</math> for each <math>x</math>, producing a scale that repeats at the [[octave]]. Any other interval of equivalence may be chosen, but for the sake of this article octave-equivalence is assumed. Perfectly balanced sets have been investigated in the context of generating repeating rhythms as well, such as in the freeware app [http://www.dynamictonality.com/xronomorph.htm XronoMorph].		In the context of musical tunings, a perfectly balanced set can be converted to a [[periodic scale]] by taking the frequency ratio <math>2^x</math> for each <math>x</math>, producing a scale that repeats at the [[octave]]. Any other interval of equivalence may be chosen, but for the sake of this article octave-equivalence is assumed. Perfectly balanced sets have been investigated in the context of generating repeating rhythms as well, such as in the freeware app [http://www.dynamictonality.com/xronomorph.htm XronoMorph].

VectorGraphics at 18:45, 3 May 2025

2025-05-03T18:45:13Z

← Older revision		Revision as of 18:45, 3 May 2025
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	{{distinguish\|balanced word}}		{{distinguish\|balanced word}}
	A non-empty finite set of real numbers ''S'' in the range <math>[0, 1)</math> is called '''perfectly balanced''' if a wheel with an equal weight placed at angle <math>2\pi x</math> for each <math>x \in S</math> has its center of gravity exactly at the hub. Mathematically, this is given by the equation <math>\sum_{x \in S} e^{2\pi i x} = 0</math>.		A non-empty finite set of real numbers ''S'' in the range <math>[0, 1)</math> is called '''perfectly balanced''' if a wheel{{Term needs definition}} with an equal weight placed at angle <math>2\pi x</math> for each <math>x \in S</math> has its center of gravity exactly at the hub. Mathematically, this is given by the equation <math>\sum_{x \in S} e^{2\pi i x} = 0</math>.

	In the context of musical tunings, a perfectly balanced set can be converted to a [[periodic scale]] by taking the frequency ratio <math>2^x</math> for each <math>x</math>, producing a scale that repeats at the [[octave]]. Any other interval of equivalence may be chosen, but for the sake of this article octave-equivalence is assumed. Perfectly balanced sets have been investigated in the context of generating repeating rhythms as well, such as in the freeware app [http://www.dynamictonality.com/xronomorph.htm XronoMorph].		In the context of musical tunings, a perfectly balanced set can be converted to a [[periodic scale]] by taking the frequency ratio <math>2^x</math> for each <math>x</math>, producing a scale that repeats at the [[octave]]. Any other interval of equivalence may be chosen, but for the sake of this article octave-equivalence is assumed. Perfectly balanced sets have been investigated in the context of generating repeating rhythms as well, such as in the freeware app [http://www.dynamictonality.com/xronomorph.htm XronoMorph].

Sintel: convert link to ref

2025-04-07T13:14:43Z

convert link to ref

← Older revision		Revision as of 13:14, 7 April 2025
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	These scales along with the evenly spaced scales of [[2edo]], [[3edo]], and [[5edo]] are the full set of "minimal" perfectly balanced scales in 30edo, which cannot be expressed as the union of two disjoint perfectly balanced scales.		These scales along with the evenly spaced scales of [[2edo]], [[3edo]], and [[5edo]] are the full set of "minimal" perfectly balanced scales in 30edo, which cannot be expressed as the union of two disjoint perfectly balanced scales.

	Searching for minimal perfectly balanced scales is nontrivial. Milne et al. ~~[http~~://~~www~~.~~dynamictonality~~.~~com~~/~~perfect_balance_files~~/ computed all such patterns] for products of three distinct primes up to ''N'' = 102.		Searching for minimal perfectly balanced scales is nontrivial. Milne et al.<ref>Milne, A. J., Bulger, D., & Herff, S. A. (2017). Exploring the space of perfectly balanced rhythms and scales. Journal of Mathematics and Music, 11(2–3), 101–133. https://doi.org/10.1080/17459737.2017.1395915</ref> computed all such patterns for products of three distinct primes up to ''N'' = 102.

	== Outside EDOs ==		== Outside EDOs ==
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	Search procedures for perfectly balanced scales under other optimization criteria are conceivable. Minimizing [[harmonic entropy]] is one such approach.		Search procedures for perfectly balanced scales under other optimization criteria are conceivable. Minimizing [[harmonic entropy]] is one such approach.

			== References ==

	[[Category:Scale]]		[[Category:Scale]]

Inthar at 17:47, 21 January 2024

2024-01-21T17:47:23Z

← Older revision		Revision as of 17:47, 21 January 2024
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			{{distinguish\|balanced word}}
	A non-empty finite set of real numbers ''S'' in the range <math>[0, 1)</math> is called '''perfectly balanced''' if a wheel with an equal weight placed at angle <math>2\pi x</math> for each <math>x \in S</math> has its center of gravity exactly at the hub. Mathematically, this is given by the equation <math>\sum_{x \in S} e^{2\pi i x} = 0</math>.		A non-empty finite set of real numbers ''S'' in the range <math>[0, 1)</math> is called '''perfectly balanced''' if a wheel with an equal weight placed at angle <math>2\pi x</math> for each <math>x \in S</math> has its center of gravity exactly at the hub. Mathematically, this is given by the equation <math>\sum_{x \in S} e^{2\pi i x} = 0</math>.

A: /* Within EDOs */

2023-03-20T17:11:54Z

Within EDOs

← Older revision		Revision as of 17:11, 20 March 2023
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	The only perfectly balanced scale within a [[prime EDO]] ''N'' is the equally spaced scale containing every tone in ''N''-EDO. For a composite EDO ''N'', one can construct new perfectly balanced scales by superimposing equally spaced scales of size ''k'' where ''k'' is a divisor of ''N'', such that the scales are transposed so that no pitches coincide. For example, in [[18edo]] the 3edo {0, 6, 12} and the [[2edo]] {1, 10} may be combined to form the perfectly balanced scale {0, 1, 6, 10, 12}.		The only perfectly balanced scale within a [[prime EDO]] ''N'' is the equally spaced scale containing every tone in ''N''-EDO. For a composite EDO ''N'', one can construct new perfectly balanced scales by superimposing equally spaced scales of size ''k'' where ''k'' is a divisor of ''N'', such that the scales are transposed so that no pitches coincide. For example, in [[18edo]] the 3edo {0, 6, 12} and the [[2edo]] {1, 10} may be combined to form the perfectly balanced scale {0, 1, 6, 10, 12}.

	This leads to the question of whether every perfectly balanced scale within an EDO is decomposable into a union of one or more equally spaced disjoint scales. This turns out to be false, and counterexamples can occur when ''N'' has three or more distinct prime factors. The smallest EDO with three distinct prime factors is [[30edo]], and ~~indeed~~ it has six ~~such~~ scales up to transposition:		This leads to the question of whether every perfectly balanced scale within an EDO is decomposable into a union of one or more equally spaced disjoint scales. This turns out to be false, and counterexamples can occur when ''N'' has three or more distinct prime factors. The smallest EDO with three distinct prime factors is [[30edo]], and it has six scales up to transposition:

	{0, 1, 7, 13, 19, 20}		{0, 1, 7, 13, 19, 20}
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	{0, 1, 2, 8, 12, 14, 18, 20, 24}		{0, 1, 2, 8, 12, 14, 18, 20, 24}

	Searching for ~~these~~ scales is nontrivial. Milne et al. [http://www.dynamictonality.com/perfect_balance_files/ computed all such patterns] for products of three distinct primes up to ''N'' = 102.		These scales along with the evenly spaced scales of [[2edo]], [[3edo]], and [[5edo]] are the full set of "minimal" perfectly balanced scales in 30edo, which cannot be expressed as the union of two disjoint perfectly balanced scales.

			Searching for minimal perfectly balanced scales is nontrivial. Milne et al. [http://www.dynamictonality.com/perfect_balance_files/ computed all such patterns] for products of three distinct primes up to ''N'' = 102.

	== Outside EDOs ==		== Outside EDOs ==

Fredg999 category edits: Removing from Category:Theory using Cat-a-lot

2023-03-05T05:32:47Z

Removing from Category:Theory using Cat-a-lot

← Older revision		Revision as of 05:32, 5 March 2023
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	[[Category:Scale]]		[[Category:Scale]]
	~~[[Category:Theory]]~~

A at 04:40, 11 February 2023

2023-02-11T04:40:56Z

← Older revision		Revision as of 04:40, 11 February 2023
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	It is easy to construct perfectly balanced scales that are not a subset of any EDO, by superimposing two scales within EDOs transposed by an irrational amount. There also exists a continuum of perfectly balanced scales that have no such decomposition. The space of perfectly balanced scales of size ''K'' > 1 forms a ''K''-dimensional manifold, which is in general complex and poorly understood.		It is easy to construct perfectly balanced scales that are not a subset of any EDO, by superimposing two scales within EDOs transposed by an irrational amount. There also exists a continuum of perfectly balanced scales that have no such decomposition. The space of perfectly balanced scales of size ''K'' > 1 forms a ''K''-dimensional manifold, which is in general complex and poorly understood.

	Milne at al. ~~showed that an efficient convex optimization~~ procedure ~~exists~~ that, given an arbitrary scale, computes the closest perfectly balanced scale according to a simple squared-difference metric. This is accomplished by the following steps: place the scale on a circle in 2D space about the origin, translate the ~~points by a vector (''u'', ''v'')~~, project the points back onto the ~~original~~ circle by dividing by the norm, then compute the cost function <math>\left(\sum \mathbf{x}\right)^2 + \left(\sum \mathbf{y}\right)^2</math> where <math>\mathbf{x}</math> and <math>\mathbf{y}</math> are vectors of the ''x''- and ''y''-coordinates. Use any standard unconstrained optimization procedure to ~~find ''u'' and ''v'' so that the cost function is minimized~~. It can be ~~seen that the cost is 0 iff perfect balance is achieved~~.		Milne at al. found a procedure that, given an arbitrary scale, computes the closest perfectly balanced scale according to a simple squared-difference metric. This is accomplished by the following steps: place the scale on a circle in 2D space about the origin, translate the point set so that its [https://en.wikipedia.org/wiki/Geometric_median geometric median] coincides with the origin, and project the points back onto the circle by dividing each point by its distance to produce a perfectly balanced scale. The geometric median does not have a closed-form expression in general, but can be efficiently computed with a convex optimization procedure. This procedure fails if the geometric median exactly coincides with one of the points, which can happen if the original scale is "too unbalanced" (e.g. a three-tone scale whose triangle has an angle exceeding 120 degrees and therefore has a [https://en.wikipedia.org/wiki/Fermat_point Fermat point] located at a vertex).

	~~Due to~~ the ~~convexity~~ of the ~~problem the minimum is guaranteed global~~, ~~but it may not always exist~~ if the original scale is too unbalanced. ~~It is unclear from sources whether the minimal cost is always 0 if it exists, but this seems to be the case in practice~~.

	For example, a perfectly balanced approximation to Ptolemy's intense diatonic scale [1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8] displaces by the following cent values: [0, +8.61, +20.00, +23.60, +20.03, +9.97, +0.72]. The resulting scale is given by the following [[Scala]] file:		For example, a perfectly balanced approximation to Ptolemy's intense diatonic scale [1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8] displaces by the following cent values: [0, +8.61, +20.00, +23.60, +20.03, +9.97, +0.72]. The resulting scale is given by the following [[Scala]] file:

A at 06:58, 8 February 2023

2023-02-08T06:58:41Z

← Older revision		Revision as of 06:58, 8 February 2023
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	A non-empty set of real numbers ''S'' in the range <math>[0, 1)</math> is called '''perfectly balanced''' if a wheel with an equal weight placed at angle <math>2\pi x</math> for each <math>x \in S</math> has its center of gravity exactly at the hub. Mathematically, this is given by the equation <math>\sum_{x \in S} e^{2\pi i x} = 0</math>.		A non-empty finite set of real numbers ''S'' in the range <math>[0, 1)</math> is called '''perfectly balanced''' if a wheel with an equal weight placed at angle <math>2\pi x</math> for each <math>x \in S</math> has its center of gravity exactly at the hub. Mathematically, this is given by the equation <math>\sum_{x \in S} e^{2\pi i x} = 0</math>.

	In the context of musical tunings, a perfectly balanced set can be converted to a [[periodic scale]] by taking the frequency ratio <math>2^x</math> for each <math>x</math>, producing a scale that repeats at the [[octave]]~~, but any~~ other interval of equivalence may be chosen. Perfectly balanced sets have been ~~particularly~~ investigated in the context of generating repeating rhythms.		In the context of musical tunings, a perfectly balanced set can be converted to a [[periodic scale]] by taking the frequency ratio <math>2^x</math> for each <math>x</math>, producing a scale that repeats at the [[octave]]. Any other interval of equivalence may be chosen, but for the sake of this article octave-equivalence is assumed. Perfectly balanced sets have been investigated in the context of generating repeating rhythms as well, such as in the freeware app [http://www.dynamictonality.com/xronomorph.htm XronoMorph].

			In general, the ''balance'' of a set is given by <math>B = 1 - \frac{1}{K}\left\|\sum_{x \in S} e^{2\pi i x}\right\|</math> where <math>K</math> is the size of the set. A set is perfectly balanced iff its balance is the maximum value of 1.

	== Within EDOs ==		== Within EDOs ==
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	== Outside EDOs ==		== Outside EDOs ==

	It is easy to construct perfectly balanced scales that are not a subset of any EDO, by superimposing two scales within EDOs transposed by an irrational amount. There also exists a continuum of perfectly balanced scales that have no such decomposition.		It is easy to construct perfectly balanced scales that are not a subset of any EDO, by superimposing two scales within EDOs transposed by an irrational amount. There also exists a continuum of perfectly balanced scales that have no such decomposition. The space of perfectly balanced scales of size ''K'' > 1 forms a ''K''-dimensional manifold, which is in general complex and poorly understood.

	Milne at al. showed that an efficient convex optimization procedure exists that, given an arbitrary scale, computes the closest perfectly balanced scale according to a simple squared-difference metric. This is accomplished by the following steps: place the scale on a circle in 2D space about the origin, translate the points by a vector (''u'', ''v''), project the points back onto the original circle by dividing by the norm, then compute the cost function <math>\left(\sum \mathbf{x}\right)^2 + \left(\sum \mathbf{y}\right)^2</math> where <math>\mathbf{x}</math> and <math>\mathbf{y}</math> are vectors of the ''x''- and ''y''-coordinates. Use any standard unconstrained optimization procedure to find ''u'' and ''v'' so that the cost function is minimized. It can be seen that the cost is 0 iff perfect balance is achieved.		Milne at al. showed that an efficient convex optimization procedure exists that, given an arbitrary scale, computes the closest perfectly balanced scale according to a simple squared-difference metric. This is accomplished by the following steps: place the scale on a circle in 2D space about the origin, translate the points by a vector (''u'', ''v''), project the points back onto the original circle by dividing by the norm, then compute the cost function <math>\left(\sum \mathbf{x}\right)^2 + \left(\sum \mathbf{y}\right)^2</math> where <math>\mathbf{x}</math> and <math>\mathbf{y}</math> are vectors of the ''x''- and ''y''-coordinates. Use any standard unconstrained optimization procedure to find ''u'' and ''v'' so that the cost function is minimized. It can be seen that the cost is 0 iff perfect balance is achieved.
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	Due to the convexity of the problem the minimum is guaranteed global, but it may not always exist if the original scale is too unbalanced. It is unclear from sources whether the minimal cost is always 0 if it exists, but this seems to be the case in practice.		Due to the convexity of the problem the minimum is guaranteed global, but it may not always exist if the original scale is too unbalanced. It is unclear from sources whether the minimal cost is always 0 if it exists, but this seems to be the case in practice.

	For example, a perfectly balanced approximation to Ptolemy's diatonic scale [1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8] displaces by the following cent values: [0, +8.61, +20.00, +23.60, +20.03, +9.97, +0.72]. The resulting scale is given by the following [[Scala]] file:		For example, a perfectly balanced approximation to Ptolemy's intense diatonic scale [1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8] displaces by the following cent values: [0, +8.61, +20.00, +23.60, +20.03, +9.97, +0.72]. The resulting scale is given by the following [[Scala]] file:

	<pre>		<pre>
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	The effect here is rather subtle, as the diatonic scale is already close to balanced. As a more dramatic example, the perfectly balanced version of the 12edo harmonic minor scale displaces it by the cent values [0.0, +22.96, +40.23, +57.08, +44.32, +32.06, +0.59].		The effect here is rather subtle, as the diatonic scale is already close to balanced. As a more dramatic example, the perfectly balanced version of the 12edo harmonic minor scale displaces it by the cent values [0.0, +22.96, +40.23, +57.08, +44.32, +32.06, +0.59].

			Search procedures for perfectly balanced scales under other optimization criteria are conceivable. Minimizing [[harmonic entropy]] is one such approach.

	[[Category:Scale]]		[[Category:Scale]]
	[[Category:Theory]]		[[Category:Theory]]

A: /* Outside EDOs */

2023-02-07T23:31:06Z

Outside EDOs

← Older revision		Revision as of 23:31, 7 February 2023
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	It is easy to construct perfectly balanced scales that are not a subset of any EDO, by superimposing two scales within EDOs transposed by an irrational amount. There also exists a continuum of perfectly balanced scales that have no such decomposition.		It is easy to construct perfectly balanced scales that are not a subset of any EDO, by superimposing two scales within EDOs transposed by an irrational amount. There also exists a continuum of perfectly balanced scales that have no such decomposition.

	Milne at al. showed that an efficient convex optimization procedure exists that, given an arbitrary scale, computes the closest perfectly balanced scale according to a simple squared-difference metric. ~~Briefly, this~~ is accomplished by the following steps: place the scale on a circle in 2D space about the origin, translate the points by a vector (''u'', ''v''), project the points back onto the original circle by dividing by the norm, then compute the cost function <math>\left(\sum \mathbf{x}\right)^2 + \left(\sum \mathbf{y}\right)^2</math> where <math>\mathbf{x}</math> and <math>\mathbf{y}</math> are vectors of the ''x''- and ''y''-coordinates. Use any standard unconstrained optimization procedure to find ''u'' and ''v'' so that the cost function is minimized. It can be seen that the cost is 0 iff perfect balance is achieved. It~~'s not clear~~ from sources whether the ~~minimum~~ always ~~exists and has a cost of~~ 0 ~~(although~~ it ~~is always a global minimum due to convexity)~~, but it seems to be in practice.		Milne at al. showed that an efficient convex optimization procedure exists that, given an arbitrary scale, computes the closest perfectly balanced scale according to a simple squared-difference metric. This is accomplished by the following steps: place the scale on a circle in 2D space about the origin, translate the points by a vector (''u'', ''v''), project the points back onto the original circle by dividing by the norm, then compute the cost function <math>\left(\sum \mathbf{x}\right)^2 + \left(\sum \mathbf{y}\right)^2</math> where <math>\mathbf{x}</math> and <math>\mathbf{y}</math> are vectors of the ''x''- and ''y''-coordinates. Use any standard unconstrained optimization procedure to find ''u'' and ''v'' so that the cost function is minimized. It can be seen that the cost is 0 iff perfect balance is achieved.

			Due to the convexity of the problem the minimum is guaranteed global, but it may not always exist if the original scale is too unbalanced. It is unclear from sources whether the minimal cost is always 0 if it exists, but this seems to be the case in practice.

	For example, a perfectly balanced approximation to Ptolemy's diatonic scale [1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8] displaces by the following cent values: [0, +8.61, +20.00, +23.60, +20.03, +9.97, +0.72]. The resulting scale is given by the following [[Scala]] file:		For example, a perfectly balanced approximation to Ptolemy's diatonic scale [1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8] displaces by the following cent values: [0, +8.61, +20.00, +23.60, +20.03, +9.97, +0.72]. The resulting scale is given by the following [[Scala]] file:
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	</pre>		</pre>

			The effect here is rather subtle, as the diatonic scale is already close to balanced. As a more dramatic example, the perfectly balanced version of the 12edo harmonic minor scale displaces it by the cent values [0.0, +22.96, +40.23, +57.08, +44.32, +32.06, +0.59].

	[[Category:Scale]]		[[Category:Scale]]
	[[Category:Theory]]		[[Category:Theory]]

A at 23:21, 7 February 2023

2023-02-07T23:21:44Z

← Older revision		Revision as of 23:21, 7 February 2023
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	It is easy to construct perfectly balanced scales that are not a subset of any EDO, by superimposing two scales within EDOs transposed by an irrational amount. There also exists a continuum of perfectly balanced scales that have no such decomposition.		It is easy to construct perfectly balanced scales that are not a subset of any EDO, by superimposing two scales within EDOs transposed by an irrational amount. There also exists a continuum of perfectly balanced scales that have no such decomposition.

	Milne at al. showed that an efficient convex optimization procedure exists that, given an arbitrary scale, computes the closest perfectly balanced scale according to a simple squared-difference metric. Briefly, this is accomplished by the following steps: place the scale on a circle in 2D space about the origin, translate the points by a vector (''u'', ''v''), project the points back onto the original circle by dividing by the norm, then compute the ~~squared balance~~ <math>\left(\sum \mathbf{x}\right)^2 + \left(\sum \mathbf{y}\right)^2</math> where <math>\mathbf{x}</math> and <math>\mathbf{y}</math> are vectors of the ''x''- and ''y''-coordinates. Use any standard unconstrained optimization procedure to find ''u'' and ''v'' so that the ~~squared~~ balance is ~~minimized~~. It's not clear from sources whether the minimum (~~guaranteed to be~~ global due to convexity) ~~is perfectly balanced~~, but it seems to be in practice.		Milne at al. showed that an efficient convex optimization procedure exists that, given an arbitrary scale, computes the closest perfectly balanced scale according to a simple squared-difference metric. Briefly, this is accomplished by the following steps: place the scale on a circle in 2D space about the origin, translate the points by a vector (''u'', ''v''), project the points back onto the original circle by dividing by the norm, then compute the cost function <math>\left(\sum \mathbf{x}\right)^2 + \left(\sum \mathbf{y}\right)^2</math> where <math>\mathbf{x}</math> and <math>\mathbf{y}</math> are vectors of the ''x''- and ''y''-coordinates. Use any standard unconstrained optimization procedure to find ''u'' and ''v'' so that the cost function is minimized. It can be seen that the cost is 0 iff perfect balance is achieved. It's not clear from sources whether the minimum always exists and has a cost of 0 (although it is always a global minimum due to convexity), but it seems to be in practice.

	For example, a perfectly balanced approximation to Ptolemy's diatonic scale [1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8] displaces by the following cent values: [0, +8.61, +20.00, +23.60, +20.03, +9.97, +0.72]. The resulting scale is given by the following [[Scala]] file:		For example, a perfectly balanced approximation to Ptolemy's diatonic scale [1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8] displaces by the following cent values: [0, +8.61, +20.00, +23.60, +20.03, +9.97, +0.72]. The resulting scale is given by the following [[Scala]] file: