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.

~~'''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)~~.

In music, superparticular ratios describe [[interval]]s between consecutive [[harmonic]]s in the [[harmonic series]].

~~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.

== Etymology ==

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

[[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.

== Definitions ==

In Thomas Taylor's ''Theoretic Arithmetic, in Three Books''<ref name="Taylor">''[https://books.google.com.au/books?id=VuY3AAAAMAAJ Theoretic Arithmetic, in Three Books]'' on Google Books</ref> (1816), superparticular ratios are defined as those for which the denominator divides into the numerator once, leaving a remainder of one.

In almost every case, this checks out with the popular definition of superparticular, i.e. ratios of the form <math>\frac{n + 1}{n}</math>. In only one case does it deviate: that of [[2/1]]. According to Taylor, 2/1 is not superparticular, because 1 divides into 2 twice, leaving a remainder of 0.

== Properties ==

Superparticular ratios have some peculiar properties:

* The [[Wikipedia:Difference tone|difference tone]] of the dyad 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.

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.

~~[[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~~.

== Generalizations ==

Taylor's book<ref name="Taylor" /> further 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)

* ''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).

== See also ==

* [[List of superparticular intervals]]

== 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:Superparticular| ]]

[[Category:Terms]]

[[Category:Greek]]

[[Category:Ratio]]

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

@@ 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.
-'''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).
+In music, superparticular ratios describe [[interval]]s between consecutive [[harmonic]]s in the [[harmonic series]].
-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.
+== Etymology ==
+The word ''superparticular'' has Latin etymology and means "above by one part". The equivalent word of Greek origin is ''epimoric'' (from επιμοριος, ''epimórios'').
+[[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.
+== Definitions ==
+In Thomas Taylor's ''Theoretic Arithmetic, in Three Books''<ref name="Taylor">''[https://books.google.com.au/books?id=VuY3AAAAMAAJ Theoretic Arithmetic, in Three Books]'' on Google Books</ref> (1816), superparticular ratios are defined as those for which the denominator divides into the numerator once, leaving a remainder of one.
+In almost every case, this checks out with the popular definition of superparticular, i.e. ratios of the form <math>\frac{n + 1}{n}</math>. In only one case does it deviate: that of [[2/1]]. According to Taylor, 2/1 is not superparticular, because 1 divides into 2 twice, leaving a remainder of 0.
+== Properties ==
+Superparticular ratios have some peculiar properties:
+* The [[Wikipedia:Difference tone|difference tone]] of the dyad 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.
 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.
-[[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.
+== Generalizations ==
+Taylor's book<ref name="Taylor" /> further 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)
+* ''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).
 == See also ==
 * [[List of superparticular intervals]]
+== 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:Superparticular| ]] <!-- main article -->
 [[Category:Terms]]
 [[Category:Greek]]
 [[Category:Ratio]]
-[[Category:Superparticular| ]] <!-- main article -->