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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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 ==

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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]]

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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 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 23: / Line 23: @@
 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 ==
@@ Line 48: / Line 48: @@
 Search procedures for perfectly balanced scales under other optimization criteria are conceivable. Minimizing [[harmonic entropy]] is one such approach.
+== References ==
 [[Category:Scale]]