Superparticular ratio: Difference between revisions

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In mathematics, a '''superparticular ratio''', also called an '''epimoric ratio''' or '''delta-1 ratio''', is the [[ratio]] of two consecutive integer numbers.

In mathematics, a '''superparticular ratio''', also called an '''epimoric ratio''' or '''delta-1 ratio''', is the [[ratio]] of two consecutive integer numbers (1:2, 2:3, 3:4...).

More particularly, the ratio takes the form:

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A ratio greater than 1 which is ''not'' superparticular is a [[superpartient ratio]].

[[Kite Giedraitis]] has proposed a [[delta-N]] terminology (where [[delta]] means difference, here the difference between the numerator and the denominator). Thus delta-1 is an alternative term for superparticular, delta-2 is for ratios of the form <math>\frac{n+2}{n}</math>, likewise delta-3, delta-4, etc.

[[Kite Giedraitis]] has proposed a [[delta-N ratio|delta-''N'']] terminology (where ''delta'' means difference, here the difference between the numerator and the denominator). Thus delta-1 is an alternative term for superparticular, delta-2 is for ratios of the form <math>\frac{n+2}{n}</math>, likewise delta-3, delta-4, etc.

[[Kyle Gann]]'s 1992 composition ''[https://www.kylegann.com/Super.html Superparticular Woman]'' bears the namesake of this term, and indeed originated from a melody which uses superparticular ratios 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, and 5/4, "seven pitches which lie within only 221 cents (a slightly large whole-step)"<ref>https://www.kylegann.com/Super.html</ref>.

== Etymology ==

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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 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.)

Generalisation in the "meta" direction gives rise to [[square superparticular]]s and then [[ultraparticular]]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 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.)

"Subparticular" is a natural generalization of the idea to ratios which are ''n''/(''n'' + 1).

== See also ==

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* [http://forum.sagittal.org/viewtopic.php?f=4&t=410 Generalisation of the terms "epimoric" and "superparticular" as applied to ratios] on the Sagittal forum

~~[[Category:superparticular ratios| ]]~~

~~~~

[[Category:Terms]]

[[Category:Greek]]

[[Category:Ratio]]

[[Category:Ancient Greek music]]

@@ Line 1: / Line 1: @@
 {{Wikipedia|Superparticular ratio}}
-In mathematics, a '''superparticular ratio''', also called an '''epimoric ratio''' or '''delta-1 ratio''', is the [[ratio]] of two consecutive integer numbers.
+In mathematics, a '''superparticular ratio''', also called an '''epimoric ratio''' or '''delta-1 ratio''', is the [[ratio]] of two consecutive integer numbers (1:2, 2:3, 3:4...).
 More particularly, the ratio takes the form:
@@ Line 9: / Line 9: @@
 A ratio greater than 1 which is ''not'' superparticular is a [[superpartient ratio]].
-[[Kite Giedraitis]] has proposed a [[delta-N]] terminology (where [[delta]] means difference, here the difference between the numerator and the denominator). Thus delta-1 is an alternative term for superparticular, delta-2 is for ratios of the form <math>\frac{n+2}{n}</math>, likewise delta-3, delta-4, etc.
+[[Kite Giedraitis]] has proposed a [[delta-N ratio|delta-''N'']] terminology (where ''delta'' means difference, here the difference between the numerator and the denominator). Thus delta-1 is an alternative term for superparticular, delta-2 is for ratios of the form <math>\frac{n+2}{n}</math>, likewise delta-3, delta-4, etc.
+[[Kyle Gann]]'s 1992 composition ''[https://www.kylegann.com/Super.html Superparticular Woman]'' bears the namesake of this term, and indeed originated from a melody which uses superparticular ratios 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, and 5/4, "seven pitches which lie within only 221 cents (a slightly large whole-step)"<ref>https://www.kylegann.com/Super.html</ref>.
 == Etymology ==
@@ Line 40: / Line 42: @@
 * 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 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.)
+Generalisation in the "meta" direction gives rise to [[square superparticular]]s and then [[ultraparticular]]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 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.)
+"Subparticular" is a natural generalization of the idea to ratios which are ''n''/(''n'' + 1).
 == See also ==
@@ Line 52: / Line 56: @@
 * [http://forum.sagittal.org/viewtopic.php?f=4&t=410 Generalisation of the terms "epimoric" and "superparticular" as applied to ratios] on the Sagittal forum
-[[Category:superparticular ratios| ]]
-<!-- main article -->
 [[Category:Terms]]
-[[Category:Greek]]
+[[Category:Ratio]]
+[[Category:Ancient Greek music]]