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 (1:2, 2:3, 3:4...).

~~'''Superparticular''' numbers are ratios of~~ the form <math>\frac{n+1}{n} = 1+\frac{1}{n}</math>, where n is a ~~whole number greater than 0~~.

More particularly, the ratio takes the form:

:<math>\frac{n + 1}{n} = 1 + \frac{1}{n}</math> where <math>n</math> is a [[Wikipedia:Positive integer|positive integer]].

~~The word "~~superparticular~~" has Latin etymology~~ and ~~means "above~~ by ~~one part"~~. ~~The equivalent word~~ of ~~Greek origin~~ is ~~"epimoric" (from επιμοριος~~, ~~epimorios)~~.

Superparticular ratios appear frequently in [[just intonation]] and [[harmonic series]] music. Consecutive [[harmonic]]s are separated by superparticular [[interval]]s: for instance, the 20th and 21st by the superparticular ratio [[21/20]]. As the harmonics get closer together, the superparticular intervals get smaller and smaller. Thus, an examination of the superparticular intervals is an examination of some of the simplest small intervals in rational tuning systems. Indeed, many but not all common [[comma]]s are superparticular ratios.

~~These ratios have some peculiar properties:~~

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

* The difference tone of the ~~dyad~~ is also the virtual fundamental.

[[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 ==

The word ''superparticular'' has Latin etymology and means "above by one part". The equivalent word of Greek origin is ''epimoric'' (from επιμοριος, ''epimórios'').

== 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>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.

== Properties ==

Superparticular ratios have some peculiar properties:

* The [[Wikipedia:Difference tone|difference tone]] of the interval is also the [[Wikipedia:Missing fundamental|virtual fundamental]].

* The first 6 such ratios ([[3/2]], [[4/3]], [[5/4]], [[6/5]], [[7/6]], [[8/7]]) are notable [[Harmonic Entropy|harmonic entropy]] minima.

* The logarithmic difference (i.e. quotient) between two successive ~~epimoric~~ ratios is always ~~an epimoric~~ ratio.

* The logarithmic difference (i.e. quotient) between two successive superparticular ratios is always a superparticular ratio.

* The logarithmic sum (i.e. product) of two successive ~~epimoric~~ ratios is either ~~an epimoric~~ ratio or ~~an [[Superpartient|epimeric~~ ratio]].

* The logarithmic sum (i.e. product) of two successive superparticular ratios is either a superparticular ratio or a superpartient ratio.

* Every ~~epimoric~~ ratio can be split into the product of two ~~epimoric~~ ratios. One way is via the identity: <math>1+\frac{1}{n} = (1+\frac{1}{2n})\times(1+\frac{1}{2n+1})</math>~~, but more than one such~~ splitting ~~method may~~ exist.

* Every superparticular ratio can be split into the product of two superparticular ratios.

* If ''a''/''b'' and ''c''/''d'' are Farey neighbors, that is if ''a''/''b'' < ''c''/''d'' and ''bc'' - ''ad'' = 1, then (''c''/''d'')/(''a''/''b'') = ''bc''/''ad'' is ~~epimoric~~.

** One way is via the identity: <math>1+\frac{1}{n} = (1+\frac{1}{2n})\times(1+\frac{1}{2n+1})</math>; e.g. <math>\frac{9}{8} \times \frac{10}{9} = \frac{10}{8} = \frac{5 \times 2}{4 \times 2} = \frac{5}{4}</math>.

* The ~~ratios~~ between successive members of any given [[wikipedia:Farey sequence|Farey sequence]] ~~will be~~ superparticular.

** Other splitting methods exist; e.g. <math>\frac{12}{11} \times \frac{33}{32} = \frac{396}{352} = \frac{9 \times 44}{8 \times 44} = \frac{9}{8}</math>.

* If ''a''/''b'' and ''c''/''d'' are Farey neighbors, that is if ''a''/''b'' < ''c''/''d'' and ''bc'' - ''ad'' = 1, then (''c''/''d'')/(''a''/''b'') = ''bc''/''ad'' is superparticular.

* The ratio between two successive members of any given [[wikipedia:Farey sequence|Farey sequence]] is superparticular.

* [[Wikipedia:Størmer's theorem|Størmer's theorem]] states that, in each limit, there are only a finite number of superparticular ratios.

== Generalizations ==

Taylor describes generalizations of the superparticulars:

* ''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>

~~Curiously enough~~, the ~~ancient Greeks did not consider 2/1~~ to be superparticular ~~because it is~~ a [[~~Harmonic|multiple of the fundamental~~]] (the same ~~rule applies to all natural harmonics~~ in the ~~Greek system). Another explanation for the exclusion~~ of ~~2/1 can be found on the~~ [[~~Generalized superparticulars~~]] ~~page~~.

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.)

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

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

== See also ==

* [[List of superparticular intervals]]

* [[Square superparticular]]

== References ==

== External links ==

* [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:Terms]]

~~[[Category:Greek]]~~

[[Category:Ratio]]

[[Category:~~Superparticular|~~ ]] ~~~~

[[Category:Ancient Greek music]]

@@ Line 1: / Line 1: @@
-{{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 (1:2, 2:3, 3:4...).
-'''Superparticular''' numbers are ratios of the form <math>\frac{n+1}{n} = 1+\frac{1}{n}</math>, where n is a whole number greater than 0.
+More particularly, the ratio takes the form:
+:<math>\frac{n + 1}{n} = 1 + \frac{1}{n}</math>  where <math>n</math> is a [[Wikipedia:Positive integer|positive integer]].
-The word "superparticular" has Latin etymology and means "above by one part". The equivalent word of Greek origin is "epimoric" (from επιμοριος, epimorios).
+Superparticular ratios appear frequently in [[just intonation]] and [[harmonic series]] music. Consecutive [[harmonic]]s are separated by superparticular [[interval]]s: for instance, the 20th and 21st by the superparticular ratio [[21/20]]. As the harmonics get closer together, the superparticular intervals get smaller and smaller. Thus, an examination of the superparticular intervals is an examination of some of the simplest small intervals in rational tuning systems. Indeed, many but not all common [[comma]]s are superparticular ratios.
-These ratios have some peculiar properties:
+A ratio greater than 1 which is ''not'' superparticular is a [[superpartient ratio]].
-* The difference tone of the dyad is also the virtual fundamental.
+[[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 ==
+The word ''superparticular'' has Latin etymology and means "above by one part". The equivalent word of Greek origin is ''epimoric'' (from επιμοριος, ''epimórios'').
+== 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>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.
+== Properties ==
+Superparticular ratios have some peculiar properties:
+* The [[Wikipedia:Difference tone|difference tone]] of the interval is also the [[Wikipedia:Missing fundamental|virtual fundamental]].
 * The first 6 such ratios ([[3/2]], [[4/3]], [[5/4]], [[6/5]], [[7/6]], [[8/7]]) are notable [[Harmonic Entropy|harmonic entropy]] minima.
-* The logarithmic difference (i.e. quotient) between two successive epimoric ratios is always an epimoric ratio.
+* The logarithmic difference (i.e. quotient) between two successive superparticular ratios is always a superparticular ratio.
-* The logarithmic sum (i.e. product) of two successive epimoric ratios is either an epimoric ratio or an [[Superpartient|epimeric ratio]].
+* The logarithmic sum (i.e. product) of two successive superparticular ratios is either a superparticular ratio or a superpartient ratio.
-* Every epimoric ratio can be split into the product of two epimoric ratios. One way is via the identity: <math>1+\frac{1}{n} = (1+\frac{1}{2n})\times(1+\frac{1}{2n+1})</math>, but more than one such splitting method may exist.
+* Every superparticular ratio can be split into the product of two superparticular ratios.
-* If ''a''/''b'' and ''c''/''d'' are Farey neighbors, that is if ''a''/''b'' &lt; ''c''/''d'' and ''bc'' - ''ad'' = 1, then (''c''/''d'')/(''a''/''b'') = ''bc''/''ad'' is epimoric.
+** One way is via the identity: <math>1+\frac{1}{n} = (1+\frac{1}{2n})\times(1+\frac{1}{2n+1})</math>; e.g. <math>\frac{9}{8} \times \frac{10}{9} = \frac{10}{8} = \frac{5 \times 2}{4 \times 2} = \frac{5}{4}</math>.
-* The ratios between successive members of any given [[wikipedia:Farey sequence|Farey sequence]] will be superparticular.
+** Other splitting methods exist; e.g. <math>\frac{12}{11} \times \frac{33}{32} = \frac{396}{352} = \frac{9 \times 44}{8 \times 44} = \frac{9}{8}</math>.
+* If ''a''/''b'' and ''c''/''d'' are Farey neighbors, that is if ''a''/''b'' &lt; ''c''/''d'' and ''bc'' - ''ad'' = 1, then (''c''/''d'')/(''a''/''b'') = ''bc''/''ad'' is superparticular.
+* The ratio between two successive members of any given [[wikipedia:Farey sequence|Farey sequence]] is superparticular.
+* [[Wikipedia:Størmer's theorem|Størmer's theorem]] states that, in each limit, there are only a finite number of superparticular ratios.
+== Generalizations ==
+Taylor describes generalizations of the superparticulars:
+* ''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>
-Curiously enough, the ancient Greeks did not consider 2/1 to be superparticular because it is a [[Harmonic|multiple of the fundamental]] (the same rule applies to all natural harmonics in the Greek system). Another explanation for the exclusion of 2/1 can be found on the [[Generalized superparticulars]] page.
+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.)
-[[Kite Giedraitis]] has proposed the term delta-1 (where [[delta]] means difference, here the difference between the numerator and the denominator) as a replacement for superparticular, delta-2 for ratios of the form <math>\frac{n+2}{n}</math>, likewise delta-3, delta-4, etc.
+"Subparticular" is a natural generalization of the idea to ratios which are ''n''/(''n'' + 1).
 == See also ==
 * [[List of superparticular intervals]]
+* [[Square superparticular]]
+== References ==
+<references />
+== External links ==
+* [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:Terms]]
-[[Category:Greek]]
 [[Category:Ratio]]
-[[Category:Superparticular| ]] <!-- main article -->
+[[Category:Ancient Greek music]]