Superparticular ratio: Difference between revisions
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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. | * 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. | ||