Superparticular ratio: Difference between revisions
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Superparticular numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number greater than 0. In ancient Greece they were known as Epimoric (επιμοριοσ, epimorios) ratios, which is literally translated as "above a part." | '''Superparticular''' numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number greater than 0. In ancient Greece they were known as Epimoric (επιμοριοσ, epimorios) ratios, which is literally translated as "above a part." | ||
These ratios have some peculiar properties: | These ratios have some peculiar properties: | ||
* 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 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]]. | |||
* Every epimoric ratio can be split into the product of two epimoric ratios. One way is via the identity 1+1/n = (1+1/(2n))*(1+1/(2n+1)), 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. | |||
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). | 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). | ||
See | == See also == | ||
[[Category: | * [[List of Superparticular Intervals]] | ||
[[Category: | * [http://en.wikipedia.org/wiki/Superparticular_number Superparticular number - Wikipedia]. | ||
[[Category: | |||
[[Category: | [[Category:Term]] | ||
[[Category:Epimoric]] | |||
[[Category:Greek]] | |||
[[Category:Ratio]] | |||
[[Category:Superparticular| ]] <!-- main article --> | |||
Revision as of 17:16, 23 October 2018
Superparticular numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number greater than 0. In ancient Greece they were known as Epimoric (επιμοριοσ, epimorios) ratios, which is literally translated as "above a part."
These ratios have some peculiar properties:
- 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 minima.
- The 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 epimeric ratio.
- Every epimoric ratio can be split into the product of two epimoric ratios. One way is via the identity 1+1/n = (1+1/(2n))*(1+1/(2n+1)), 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.
Curiously enough, the ancient Greeks did not consider 2/1 to be superparticular because it is a multiple of the fundamental (the same rule applies to all natural harmonics in the Greek system).