Superparticular ratio: Difference between revisions

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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 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). Another explanation for the exclusion of 2/1 can be found on the [[Generalized superparticulars]] page.

~~According to some sources, such as Thomas Taylor's [https://books.google.com.au/books?id=VuY3AAAAMAAJ Theoretic Arithmetic, in Three Books], define superparticular ratios as those~~ for ~~which the denominator divides into~~ the ~~numerator once, leaving a remainder~~ of ~~one. This is another explanation for why~~ 2/1 does not qualify as superparticular, because 1 divides into 2 twice, leaving a remainder of 0. Taylor's book further describes generalizations of the superparticulars: superbiparticulars are those where the denominator divides into the numerator once, but leaves a remainder of two (such as 5/3), and double superparticulars are those where the denominator divides into the numerator twice, leaving a remainder of one (such as 5/2). One can go on and on, with e.g. triple supertriparticulars where both the divisions and the remainder are 3 (such as 15/4). More details can be found on ~~this forum thread here:~~ [~~http://forum~~.~~sagittal.org/viewtopic.php?f=4&t=410 Generalisation of the terms "epimoric" and "superparticular" as applied to ratios]~~

== See also ==

@@ Line 12: / Line 12: @@
 * 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 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). Another explanation for the exclusion of 2/1 can be found on the [[Generalized superparticulars]] page.
-According to some sources, such as Thomas Taylor's [https://books.google.com.au/books?id=VuY3AAAAMAAJ Theoretic Arithmetic, in Three Books], define superparticular ratios as those for which the denominator divides into the numerator once, leaving a remainder of one. This is another explanation for why 2/1 does not qualify as superparticular, because 1 divides into 2 twice, leaving a remainder of 0. Taylor's book further describes generalizations of the superparticulars: superbiparticulars are those where the denominator divides into the numerator once, but leaves a remainder of two (such as 5/3), and double superparticulars are those where the denominator divides into the numerator twice, leaving a remainder of one (such as 5/2). One can go on and on, with e.g. triple supertriparticulars where both the divisions and the remainder are 3 (such as 15/4). More details can be found on this forum thread here: [http://forum.sagittal.org/viewtopic.php?f=4&t=410 Generalisation of the terms "epimoric" and "superparticular" as applied to ratios]
 == See also ==