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. ~~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 <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).

These ratios have some peculiar properties:

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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).

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 ==

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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. 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 <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).
 These ratios have some peculiar properties:
@@ Line 11: / Line 13: @@
 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).
+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 ==