1729/1728: Difference between revisions

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1729/1728, the ~~massma~~, is a 19-limit superparticular interval and an unnoticeable comma.

{{Infobox Interval

| Name = ramanujanisma

| Color name = 19o3oz2, nothozo 2nd, Nothozo comma

| Comma = yes

}}

'''1729/1728''', known as the '''ramanujanisma''', is a [[19-limit]] (more accurately, 2.3.7.13.19 [[subgroup]]) [[superparticular]] interval and an [[unnoticeable comma]] that is remarkably close to one cent in size. It forms the difference between the [[octave]] and a stack of [[7/6]], [[13/12]] and [[19/12]], and less likely, the difference between [[19/18]] and [[96/91]], which in turn is [[8/7]] less [[13/12]] or [[16/13]] less [[7/6]].

Both the numerator and denominator of this interval are famous in mathematics. 1728, being 12 to the 3rd power, is also known as mass. 1729 is known for being the first number that can be expressed as a ~~sum~~ of ~~more than~~ 1 ~~integer cubes~~.

Both the numerator and denominator of this interval are famous in mathematics. [[Wikipedia: 1728 (number)|1728]], being 12 to the 3rd power, is also known as mass. [[Wikipedia:1729 (number)|1729]] is known for being Ramanujan's number and the first number that can be expressed as the sum of two cubes in two different ways (1729 = 13 + 123 = 93 + 103).

== Commatic relations ==

This comma is the difference between the following superparticular pairs:

* [[91/90]] and [[96/95]]

* [[133/132]] and [[144/143]]

* [[273/272]] and [[324/323]]

* [[325/324]] and [[400/399]]

* [[361/360]] and [[456/455]]

* [[385/384]] and [[495/494]]

* [[513/512]] and [[729/728]] *

* [[1001/1000]] and [[2376/2375]]

* [[1216/1215]] and [[4096/4095]]

* [[1225/1224]] and [[4200/4199]]

* [[1521/1520]] and [[12636/12635]]

* [[1540/1539]] and [[14080/14079]]

* [[1701/1700]] and [[104976/104975]]

* [[1716/1715]] and [[228096/228095]]

<nowiki>*</nowiki> all is within the 2.3.7.13.19 subgroup

It factors into the following superparticular pairs:

* [[2926/2925]] and [[4225/4224]]

* [[2431/2430]] and [[5985/5984]]

* [[2401/2400]] and [[6175/6174]]

* [[2080/2079]] and [[10241/10240]]

== Temperaments ==

Tempering out this comma in the 19-limit leads to the rank-7 '''ramanujanismic temperament''', or in the 2.3.7.13.19 subgroup, the rank-4 '''ramanujanic temperament'''. In either case it enables the [[ramanujanismic chords]], the [[essentially tempered chord]]s in the 19- or 21-odd-limit. The basic equivalence related to all these chords can be expressed as (7/6)(13/12)(19/12)~2/1, similar to (7/5)(11/10)(13/10)~2/1 as is enabled by the [[1001/1000|sinbadma (1001/1000)]]. Futhermore, [[8/7]] is short of a stack consisting of 19/18 and 13/12, [[16/13]] short of a stack consisting of 19/18 and 7/6, and [[32/19]] short of a stack consisting of 7/6 and 13/9, all by the ramanujanisma, so that any accurate tuning of the 2.3.13.19, 2.3.7.19, or 2.3.7.13 subgroup will naturally have an accurate approximation to [[7/1|7]], [[13/1|13]], or [[19/1|19]], respectively.

== Terminology ==

The name ''ramanujanisma'' was first proposed by [[User:Fredg999|Frédéric Gagné]] in reference to the anecdotal story of [[Wikipedia: Ramanujan|Ramanujan]] finding 1729 an interesting number. Alternative names include ''lesser massma'', proposed by [[User:Eliora|Eliora]], in reference to the number 1728 being known as the ''Maß'' in German, and ''dodecentisma'', proposed by [[User:Godtone|Godtone]], in reference to the size being close to the relative ''cent'' of ''12''edo (''dodeca'') (12 × 100 = 1200 and this comma is a low [[prime limit]] superparticular approximating 1/1200 of an octave) and in reference to 1728 being a power of 12 (''dodeca'').

== See also ==

[[~~Wikipedia~~:~~1728 (number)~~]]

* [[List of superparticular intervals]]

[[~~Wikipedia~~:~~1729 (~~number)]]

* [[Unnoticeable comma]]

[[Category:Ramanujanismic]]

[[Category:Commas named after mathematicians]]

[[Category:Commas referencing a famous use of a number]]

@@ Line 1: / Line 1: @@
-/1728, the massma, is a 19-limit superparticular interval and an unnoticeable comma.
+{{Infobox Interval
+| Name = ramanujanisma
+| Color name = 19o3oz2, nothozo 2nd,<br>Nothozo comma
+| Comma = yes
+}}
+'''1729/1728''', known as the '''ramanujanisma''', is a [[19-limit]] (more accurately, 2.3.7.13.19 [[subgroup]]) [[superparticular]] interval and an [[unnoticeable comma]] that is remarkably close to one cent in size. It forms the difference between the [[octave]] and a stack of [[7/6]], [[13/12]] and [[19/12]], and less likely, the difference between [[19/18]] and [[96/91]], which in turn is [[8/7]] less [[13/12]] or [[16/13]] less [[7/6]].
-Both the numerator and denominator of this interval are famous in mathematics. 1728, being 12 to the 3rd power, is also known as mass. 1729 is known for being the first number that can be expressed as a sum of more than 1 integer cubes.
+Both the numerator and denominator of this interval are famous in mathematics. [[Wikipedia: 1728 (number)|1728]], being 12 to the 3rd power, is also known as mass. [[Wikipedia:1729 (number)|1729]] is known for being Ramanujan's number and the first number that can be expressed as the sum of two cubes in two different ways (1729 = 1<sup>3</sup> + 12<sup>3</sup> = 9<sup>3</sup> + 10<sup>3</sup>).
+== Commatic relations ==
+This comma is the difference between the following superparticular pairs:
+* [[91/90]] and [[96/95]]
+* [[133/132]] and [[144/143]]
+* [[273/272]] and [[324/323]]
+* [[325/324]] and [[400/399]]
+* [[361/360]] and [[456/455]]
+* [[385/384]] and [[495/494]]
+* [[513/512]] and [[729/728]] *
+* [[1001/1000]] and [[2376/2375]]
+* [[1216/1215]] and [[4096/4095]]
+* [[1225/1224]] and [[4200/4199]]
+* [[1521/1520]] and [[12636/12635]]
+* [[1540/1539]] and [[14080/14079]]
+* [[1701/1700]] and [[104976/104975]]
+* [[1716/1715]] and [[228096/228095]]
+<nowiki>*</nowiki> all is within the 2.3.7.13.19 subgroup
+It factors into the following superparticular pairs:
+* [[2926/2925]] and [[4225/4224]]
+* [[2431/2430]] and [[5985/5984]]
+* [[2401/2400]] and [[6175/6174]]
+* [[2080/2079]] and [[10241/10240]]
+== Temperaments ==
+Tempering out this comma in the 19-limit leads to the rank-7 '''ramanujanismic temperament''', or in the 2.3.7.13.19 subgroup, the rank-4 '''ramanujanic temperament'''. In either case it enables the [[ramanujanismic chords]], the [[essentially tempered chord]]s in the 19- or 21-odd-limit. The basic equivalence related to all these chords can be expressed as (7/6)(13/12)(19/12)~2/1, similar to (7/5)(11/10)(13/10)~2/1 as is enabled by the [[1001/1000|sinbadma (1001/1000)]]. Futhermore, [[8/7]] is short of a stack consisting of 19/18 and 13/12, [[16/13]] short of a stack consisting of 19/18 and 7/6, and [[32/19]] short of a stack consisting of 7/6 and 13/9, all by the ramanujanisma, so that any accurate tuning of the 2.3.13.19, 2.3.7.19, or 2.3.7.13 subgroup will naturally have an accurate approximation to [[7/1|7]], [[13/1|13]], or [[19/1|19]], respectively.
+== Terminology ==
+The name ''ramanujanisma'' was first proposed by [[User:Fredg999|Frédéric Gagné]] in reference to the anecdotal story of [[Wikipedia: Ramanujan|Ramanujan]] finding 1729 an interesting number. Alternative names include ''lesser massma'', proposed by [[User:Eliora|Eliora]], in reference to the number 1728 being known as the ''Maß'' in German, and ''dodecentisma'', proposed by [[User:Godtone|Godtone]], in reference to the size being close to the relative ''cent'' of ''12''edo (''dodeca'') (12 × 100 = 1200 and this comma is a low [[prime limit]] superparticular approximating 1/1200 of an octave) and in reference to 1728 being a power of 12 (''dodeca'').
 == See also ==
-[[Wikipedia:1728 (number)]]
+* [[List of superparticular intervals]]
-[[Wikipedia:1729 (number)]]
+* [[Unnoticeable comma]]
+[[Category:Ramanujanismic]]
+[[Category:Commas named after mathematicians]]
+[[Category:Commas referencing a famous use of a number]]