Superparticular ratio: Difference between revisions

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== Generalizations ==

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)

* ''superbiparticulars'' (or ''odd-particulars'') are those where the denominator divides into the numerator once, but leaves a remainder of two (such as 5/3)

* ''supertriparticulars'' (or ''throdd-particulars'') are those where the denominator divides into the numerator once, but leaves a remainder of three (such as 25/22)

* ''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).<ref>Taylor, Thomas (1816), ''[https://books.google.com.au/books?id=VuY3AAAAMAAJ Theoretic Arithmetic, in Three Books]'', p. 45-50</ref>

Generalisation in the "meta" direction gives rise to [[square superparticular]]s, under the idea that if a superparticular is the difference between two adjacent harmonics then a square superparticular is the difference between two adjacent superparticulars.

== See also ==

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 == Generalizations ==
 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)
+* ''superbiparticulars'' (or ''odd-particulars'') are those where the denominator divides into the numerator once, but leaves a remainder of two (such as 5/3)
+* ''supertriparticulars'' (or ''throdd-particulars'') are those where the denominator divides into the numerator once, but leaves a remainder of three (such as 25/22)
 * ''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).<ref>Taylor, Thomas (1816), ''[https://books.google.com.au/books?id=VuY3AAAAMAAJ Theoretic Arithmetic, in Three Books]'', p. 45-50</ref>
+Generalisation in the "meta" direction gives rise to [[square superparticular]]s, under the idea that if a superparticular is the difference between two adjacent harmonics then a square superparticular is the difference between two adjacent superparticulars.
 == See also ==