Superparticular ratio: Difference between revisions

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:<math>\frac{n + 1}{n} = 1 + \frac{1}{n}</math> where <math>n</math> is a [[Wikipedia:Positive integer|positive integer]].

~~In music, superparticular~~ ratios ~~describe~~ [[~~interval~~]]~~s between consecutive~~ [[harmonic]]s in ~~the~~ [[~~harmonic series~~]].

Superparticular ratios appear frequently in [[just intonation]] and [[harmonic series]] music. Consecutive [[harmonic]]s are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio [[21/20]]. As the harmonics get closer together, the superparticular intervals get smaller and smaller. Thus, an examination of the superparticular intervals is an examination of some of the simplest small intervals in rational tuning systems. Indeed, many but not all common [[comma]]s are superparticular ratios.

A ratio greater than 1 which is not superparticular is a [[superpartient ratio]].

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* If ''a''/''b'' and ''c''/''d'' are Farey neighbors, that is if ''a''/''b'' < ''c''/''d'' and ''bc'' - ''ad'' = 1, then (''c''/''d'')/(''a''/''b'') = ''bc''/''ad'' is superparticular.

* The ratio between two successive members of any given [[wikipedia:Farey sequence|Farey sequence]] is superparticular.

* [[Wikipedia:Størmer's theorem|Størmer's theorem]] states that, in each limit, there are only a finite number of superparticular ratios.

== Generalizations ==

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 :<math>\frac{n + 1}{n} = 1 + \frac{1}{n}</math>  where <math>n</math> is a [[Wikipedia:Positive integer|positive integer]].
-In music, superparticular ratios describe [[interval]]s between consecutive [[harmonic]]s in the [[harmonic series]].
+Superparticular ratios appear frequently in [[just intonation]] and [[harmonic series]] music. Consecutive [[harmonic]]s are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio [[21/20]]. As the harmonics get closer together, the superparticular intervals get smaller and smaller. Thus, an examination of the superparticular intervals is an examination of some of the simplest small intervals in rational tuning systems. Indeed, many but not all common [[comma]]s are superparticular ratios.
 A ratio greater than 1 which is not superparticular is a [[superpartient ratio]].
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 * If ''a''/''b'' and ''c''/''d'' are Farey neighbors, that is if ''a''/''b'' &lt; ''c''/''d'' and ''bc'' - ''ad'' = 1, then (''c''/''d'')/(''a''/''b'') = ''bc''/''ad'' is superparticular.
 * The ratio between two successive members of any given [[wikipedia:Farey sequence|Farey sequence]] is superparticular.
+* [[Wikipedia:Størmer's theorem|Størmer's theorem]] states that, in each limit, there are only a finite number of superparticular ratios.
 == Generalizations ==