Superparticular ratio: Difference between revisions

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== Definitions ==

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 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>Taylor, Thomas (1816), ''[https://books.google.com.au/books?id=VuY3AAAAMAAJ Theoretic Arithmetic, in Three Books]'', 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 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.

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== Generalizations ==

Taylor~~'s book<ref name="Taylor" /> further~~ describes generalizations of the superparticulars:

Taylor 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)

* ''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).

* one can go on and on, with e.g. ''triple supertriparticulars'', where both the divisions and the remainder are 3 (such as 15/4).<ref>Taylor, Thomas (1816), ''[https://books.google.com.au/books?id=VuY3AAAAMAAJ Theoretic Arithmetic, in Three Books]'', p. 45-50</ref>

== See also ==

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 == Definitions ==
-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 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>Taylor, Thomas (1816), ''[https://books.google.com.au/books?id=VuY3AAAAMAAJ Theoretic Arithmetic, in Three Books]'', 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 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.
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 == Generalizations ==
-Taylor's book<ref name="Taylor" /> further describes generalizations of the superparticulars:
+Taylor 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)
 * ''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).
+* one can go on and on, with e.g. ''triple supertriparticulars'', where both the divisions and the remainder are 3 (such as 15/4).<ref>Taylor, Thomas (1816), ''[https://books.google.com.au/books?id=VuY3AAAAMAAJ Theoretic Arithmetic, in Three Books]'', p. 45-50</ref>
 == See also ==