User:Sintel/Zeta working page: Difference between revisions

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To assess how well an equal temperament approximates the [[harmonic series]], we need a function that measures the accuracy of approximation for each overtone.

~~For~~ any ~~overtone ''~~n'', ~~the quantity~~ <math>x \log_2(n)</math> ~~represents how many~~ steps ~~in our equal temperament are needed to approximate that overtone~~. ~~When this value~~ is close to an integer, ~~the approximation is good; when~~ it ~~deviates significantly~~ from an integer, the approximation is poor.

The frequency ratio of the nth harmonic is n:1. In any equal temperament, to reach a ratio of n:1, we need <math>\log_2(n)</math> octaves, which is <math>x \log_2(n)</math> steps.

If <math>x \log_2(n)</math> is very close to an integer, it means our equal temperament system can closely approximate that harmonic. If it's far from an integer, the approximation will be poor.

To quantify this proximity to integers, we can use the cosine function <math>f(x) = \cos(2 \pi x)</math>, which peaks at integer values. This provides a simple metric where values close to 1 indicate good approximations and values close to -1 indicate poor ones.

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 To assess how well an equal temperament approximates the [[harmonic series]], we need a function that measures the accuracy of approximation for each overtone.
-For any overtone ''n'', the quantity <math>x \log_2(n)</math> represents how many steps in our equal temperament are needed to approximate that overtone. When this value is close to an integer, the approximation is good; when it deviates significantly from an integer, the approximation is poor.
+The frequency ratio of the nth harmonic is n:1. In any equal temperament, to reach a ratio of n:1, we need <math>\log_2(n)</math> octaves, which is <math>x \log_2(n)</math> steps.
+If <math>x \log_2(n)</math> is very close to an integer, it means our equal temperament system can closely approximate that harmonic. If it's far from an integer, the approximation will be poor.
 To quantify this proximity to integers, we can use the cosine function <math>f(x) = \cos(2 \pi x)</math>, which peaks at integer values. This provides a simple metric where values close to 1 indicate good approximations and values close to -1 indicate poor ones.