Superparticular ratio: Difference between revisions
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{{Wikipedia| Superparticular ratio }} | |||
'''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. | ||
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* The difference tone of the dyad is also the virtual fundamental. | * The difference tone of the dyad is also the virtual fundamental. | ||
* The first 6 such ratios ([[3/2]], [[4/3]], [[5/4]], [[6/5]], [[7/6]], [[8/7]]) are notable [[Harmonic Entropy|harmonic entropy]] minima. | * The first 6 such ratios ([[3/2]], [[4/3]], [[5/4]], [[6/5]], [[7/6]], [[8/7]]) are notable [[Harmonic Entropy|harmonic entropy]] minima. | ||
* The difference (i.e. quotient) between two successive epimoric ratios is always an epimoric ratio. | * The logarithmic difference (i.e. quotient) between two successive epimoric ratios is always an epimoric ratio. | ||
* The sum (i.e. product) of two successive epimoric ratios is either an epimoric ratio or an [[Superpartient|epimeric ratio]]. | * The logarithmic sum (i.e. product) of two successive epimoric ratios is either an epimoric ratio or an [[Superpartient|epimeric ratio]]. | ||
* 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 [[wikipedia: | * The ratios between successive members of any given [[wikipedia:Farey sequence|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. | ||
== See also == | == See also == | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Greek]] | [[Category:Greek]] | ||
[[Category:Ratio]] | [[Category:Ratio]] | ||
[[Category:Superparticular| ]] <!-- main article --> | [[Category:Superparticular| ]] <!-- main article --> | ||