Superparticular ratio: Difference between revisions

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== Definitions ==

In ~~Thomas Taylor's ''Theoretic Arithmetic~~, in ~~Three Books''~~<ref name="Taylor">''[https://books.google.com.au/books?id=VuY3AAAAMAAJ Theoretic Arithmetic, in Three Books]'' ~~on Google Books~~</ref> ~~(1816)~~, superparticular ~~ratios are defined~~ as ~~those for which~~ the denominator divides into the numerator once~~, leaving~~ a remainder of ~~one~~.

In ancient Greece and until around the 19th century, superparticular ratios are defined as follows: "When one number contains the whole of another in itself, and some part of it besides, it is called superparticular."<ref name="Taylor">''[https://books.google.com.au/books?id=VuY3AAAAMAAJ Theoretic Arithmetic, in Three Books]'' (1816), p. 37</ref> In other words, a ratio is superparticular if, when expressed as an irreducible fraction, the denominator divides into the numerator once and leaves a remainder of 1.

In almost every case, this ~~checks out with~~ the ~~popular~~ definition of superparticular, i.e. ratios of the form <math>\frac{n + 1}{n}</math>. In only one case does it deviate: that of [[2/1]]. According to ~~Taylor~~, 2/1 is not superparticular, ~~because~~ 1 divides into 2 twice, leaving a remainder of 0.

In almost every case, this matches the modern definition of superparticular, i.e. ratios of the form <math>\frac{n + 1}{n}</math>, where <math>n</math> is a positive integer. In only one case does it deviate: that of [[2/1]]. According to traditional Greek arithmetic, 2/1 is not a superparticular ratio, but rather a ''multiple'': 1 divides into 2 twice, leaving a remainder of 0. Multiples and superparticulars are considered as distinct categories of numbers from that perspective. In musical terms, this would imply considering that 2/1 is not superparticular because it describes a [[Harmonic|multiple of the fundamental]], which other superparticular ratios do not.

== Properties ==

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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 superparticular.

* The ratio between two successive members of any given [[wikipedia:Farey sequence|Farey sequence]] is 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.

== Generalizations ==

@@ Line 15: / Line 15: @@
 == Definitions ==
-In Thomas Taylor's ''Theoretic Arithmetic, in Three Books''<ref name="Taylor">''[https://books.google.com.au/books?id=VuY3AAAAMAAJ Theoretic Arithmetic, in Three Books]'' on Google Books</ref> (1816), superparticular ratios are defined as those for which the denominator divides into the numerator once, leaving a remainder of one.
+In ancient Greece and until around the 19th century, superparticular ratios are defined as follows: "When one number contains the whole of another in itself, and some part of it besides, it is called superparticular."<ref name="Taylor">''[https://books.google.com.au/books?id=VuY3AAAAMAAJ Theoretic Arithmetic, in Three Books]'' (1816), p. 37</ref> In other words, a ratio is superparticular if, when expressed as an irreducible fraction, the denominator divides into the numerator once and leaves a remainder of 1.
-In almost every case, this checks out with the popular definition of superparticular, i.e. ratios of the form <math>\frac{n + 1}{n}</math>. In only one case does it deviate: that of [[2/1]]. According to Taylor, 2/1 is not superparticular, because 1 divides into 2 twice, leaving a remainder of 0.
+In almost every case, this matches the modern definition of superparticular, i.e. ratios of the form <math>\frac{n + 1}{n}</math>, where <math>n</math> is a positive integer. In only one case does it deviate: that of [[2/1]]. According to traditional Greek arithmetic, 2/1 is not a superparticular ratio, but rather a ''multiple'': 1 divides into 2 twice, leaving a remainder of 0. Multiples and superparticulars are considered as distinct categories of numbers from that perspective. In musical terms, this would imply considering that 2/1 is not superparticular because it describes a [[Harmonic|multiple of the fundamental]], which other superparticular ratios do not.
 == Properties ==
@@ Line 31: / Line 31: @@
 * 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 superparticular.
 * The ratio between two successive members of any given [[wikipedia:Farey sequence|Farey sequence]] is 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.
 == Generalizations ==