Superparticular ratio - Revision history

Xenjacob: added example in first sentence. added Kyle Gann's "Superparticular Woman" (may belong lower in page)

2025-01-15T22:08:55Z

added example in first sentence. added Kyle Gann's "Superparticular Woman" (may belong lower in page)

← Older revision		Revision as of 22:08, 15 January 2025
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	{{Wikipedia\|Superparticular ratio}}		{{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:		More particularly, the ratio takes the form:
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	[[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.		[[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 ==		== Etymology ==

FloraC: Style and link improvements

2024-12-05T08:30:46Z

Style and link improvements

← Older revision		Revision as of 08:30, 5 December 2024
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	A ratio greater than 1 which is ''not'' superparticular is a [[superpartient ratio]].		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.

	== Etymology ==		== Etymology ==

FloraC: /* Generalizations */ style

2024-12-05T08:22:33Z

Generalizations: style

← Older revision		Revision as of 08:22, 5 December 2024
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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>		* 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 [[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.)		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).		"Subparticular" is a natural generalization of the idea to ratios which are ''n''/(''n'' + 1).

	== See also ==		== See also ==

Cmloegcmluin: /* Generalizations */

2024-09-12T01:35:08Z

Generalizations

← Older revision		Revision as of 01:35, 12 September 2024
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	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.)		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 ==		== See also ==

Fredg999: Categories

2024-06-10T15:00:27Z

FloraC: This page doesn't need an eponymous category

2023-08-29T09:24:58Z

This page doesn't need an eponymous category

← Older revision		Revision as of 09:24, 29 August 2023
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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		* [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 ratio\| ]] <!-- main article -->~~
	[[Category:Terms]]		[[Category:Terms]]
			[[Category:Ratio]]
	[[Category:Greek music]]		[[Category:Greek music]]

FloraC: Recategorize

2023-08-29T09:22:12Z

Recategorize

← Older revision		Revision as of 09:22, 29 August 2023
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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		* [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:Superparticular ratio\| ]] <!-- main article -->
	<!-- main article -->
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Greek]]		[[Category:Greek music]]

Godtone: /* Generalizations */

2023-07-31T19:07:18Z

Generalizations

← Older revision		Revision as of 19:07, 31 July 2023
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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>		* 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 [[~~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.)

	== See also ==		== See also ==

Godtone: /* Generalizations */

2023-07-31T19:06:58Z

Generalizations

← Older revision		Revision as of 19:06, 31 July 2023
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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>		* 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 [[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 ==		== See also ==

Godtone: added analogy to superpartient ratios and a summary on why square superparticulars might be interesting

2023-06-02T19:13:00Z

added analogy to superpartient ratios and a summary on why square superparticulars might be interesting

← Older revision		Revision as of 19:13, 2 June 2023
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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>		* 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 ==		== See also ==

← Older revision		Revision as of 15:00, 10 June 2024
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	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Ratio]]		[[Category:Ratio]]
	[[Category:Greek music]]		[[Category:Ancient Greek music]]