Superparticular ratio: Difference between revisions
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'''Superparticular''' numbers are ratios of the form <math>\frac{n+1}{n}</math>, or <math>1+\frac{1}{n}</math>, where n is a whole number greater than 0. | '''Superparticular''' numbers are ratios of the form <math>\frac{n+1}{n}</math>, or <math>1+\frac{1}{n}</math>, where n is a whole number greater than 0. | ||
The word "superparticular" has Latin etymology and means "above by one part". The equivalent word of Greek origin is "epimoric" (from επιμοριοσ, epimorios). | The word "superparticular" has Latin etymology and means "above by one part". The equivalent word of Greek origin is "epimoric" (from επιμοριοσ, epimorios). | ||
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* Every epimoric ratio can be split into the product of two epimoric ratios. One way is via the identity: <math>1+\frac{1}{n} = (1+\frac{1}{2n})\times(1+\frac{1}{2n+1})</math>, but more than one such splitting method may exist. | * Every epimoric ratio can be split into the product of two epimoric ratios. One way is via the identity: <math>1+\frac{1}{n} = (1+\frac{1}{2n})\times(1+\frac{1}{2n+1})</math>, but more than one such splitting method may exist. | ||
* 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 epimoric. | * 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 epimoric. | ||
* The ratios between successive members of any given Farey sequence will be superparticular. | |||
Curiously enough, the ancient Greeks did not consider 2/1 to be superparticular because it is a [[Harmonic|multiple of the fundamental]] (the same rule applies to all natural harmonics in the Greek system). Another explanation for the exclusion of 2/1 can be found on the [[Generalized superparticulars]] page. | Curiously enough, the ancient Greeks did not consider 2/1 to be superparticular because it is a [[Harmonic|multiple of the fundamental]] (the same rule applies to all natural harmonics in the Greek system). Another explanation for the exclusion of 2/1 can be found on the [[Generalized superparticulars]] page. | ||
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== See also == | == See also == | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
* [http://en.wikipedia.org/wiki/Superparticular_number Superparticular number - Wikipedia]. | * [http://en.wikipedia.org/wiki/Superparticular_number Superparticular number - Wikipedia]. [[Category:Term]] [[Category:Epimoric]] [[Category:Greek]] [[Category:Ratio]] [[Category:Superparticular| ]] <!-- main article --> | ||
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