Perfect balance: Difference between revisions

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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 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].

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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}.

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}

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{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.<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 ==

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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.

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:

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

== References ==

[[Category:Scale]]

~~[[Category:Theory]]~~

@@ Line 1: / Line 1: @@
-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>.
+{{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 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].
@@ Line 11: / Line 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}
@@ Line 20: / Line 21: @@
      {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.<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 ==
@@ Line 26: / Line 29: @@
 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:
@@ Line 47: / Line 48: @@
 Search procedures for perfectly balanced scales under other optimization criteria are conceivable. Minimizing [[harmonic entropy]] is one such approach.
+== References ==
 [[Category:Scale]]
-[[Category:Theory]]