Superparticular ratio: Difference between revisions

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* 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.

Generalisation in the "meta" direction gives rise to [[square superparticular]]s and then [[Square superparticular #Sk/S(k + 1) (ultraparticulars)|ultraparticulars]], 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 and an ultraparticular is the difference between two adjacent square superparticulars. This gives rise to descriptions of infinite comma families of which many known commas are examples. A notable property is that just as "all [[superpartient ratio]]s can be constructed as products of [consecutive] superparticular numbers", all ratios between two superparticular intervals (e.g ([[8/7]])/([[11/10]]) = 80/77) can be constructed as a product of consecutive [[square superparticular]] numbers (e.g [[64/63]] * [[81/80]] * [[100/99]] = S8 * S9 * S10), for the same algebraic reason as in the corresponding case of [[superpartient ratio]]s. (There is a corresponding analogy with ultraparticulars too, for the same reason.)

== See also ==

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 * 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.
+Generalisation in the "meta" direction gives rise to [[square superparticular]]s and then [[Square superparticular #Sk/S(k + 1) (ultraparticulars)|ultraparticulars]], 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 and an ultraparticular is the difference between two adjacent square superparticulars. This gives rise to descriptions of infinite comma families of which many known commas are examples. A notable property is that just as "all [[superpartient ratio]]s can be constructed as products of [consecutive] superparticular numbers", all ratios between two superparticular intervals (e.g ([[8/7]])/([[11/10]]) = 80/77) can be constructed as a product of consecutive [[square superparticular]] numbers (e.g [[64/63]] * [[81/80]] * [[100/99]] = S8 * S9 * S10), for the same algebraic reason as in the corresponding case of [[superpartient ratio]]s. (There is a corresponding analogy with ultraparticulars too, for the same reason.)
 == See also ==