Perfect balance: Difference between revisions

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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 ==

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== 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.

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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.

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>

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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].

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

[[Category:Scale]]

[[Category:Theory]]

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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 ==
@@ Line 22: / Line 24: @@
 == 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.
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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.
-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>
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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].
+Search procedures for perfectly balanced scales under other optimization criteria are conceivable. Minimizing [[harmonic entropy]] is one such approach.
 [[Category:Scale]]
 [[Category:Theory]]