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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 {{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].