1729/1728: Difference between revisions

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{{Infobox Interval

| ~~Icon~~ =

| Name = ramanujanisma

| ~~Ratio = 1729/1728~~

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

~~| Monzo = -6 -3 0 1 0 1 0 1~~

| Comma = yes

~~| Cents~~ = ~~1.00158~~

~~| Name = lesser massma~~, ~~ ramanujanisma~~, ~~dodecentisma~~

~~| Color name =~~

~~| FJS name =~~

| ~~Sound~~ =

}}

'''1729/1728''' is a [[19-limit]] (more accurately, 2.3.7.13.19 subgroup) [[superparticular]] interval and an [[unnoticeable comma]]~~. '''Lesser massma'''<ref>referring~~ to ~~number 1728 being known as the ''Maß''~~ in ~~German. </ref>, '''ramanujanisma'''<ref>referring to the anecdotal story of [[Wikipedia: Ramanujan|Ramanujan]] finding 1729 an interesting number. </ref>, and '''dodecentisma'''<ref>referring to the~~ size ~~being close to the cent relative to 12edo~~. ~~</ref> have been proposed as the name. The comma~~ 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]].

'''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. [[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).

~~Remarkably, this~~ comma is ~~very close to one cent~~.

== 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

Tempering out this comma enables the ~~related~~ essentially tempered ~~chords~~ in the 19-odd-limit.

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 ==

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* [[Unnoticeable comma]]

~~== Notes ==~~

[[Category:Ramanujanismic]]

[[Category:Commas named after mathematicians]]

[[Category:~~19-limit~~]]

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

[[Category:~~Unnoticeable comma~~]]

[[Category:~~Superparticular~~]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Icon =
+| Name = ramanujanisma
-| Ratio = 1729/1728
+| Color name = 19o3oz2, nothozo 2nd,<br>Nothozo comma
-| Monzo = -6 -3 0 1 0 1 0 1
+| Comma = yes
-| Cents = 1.00158
-| Name = lesser massma, <br>ramanujanisma, <br>dodecentisma
-| Color name =
-| FJS name =
-| Sound =
 }}
-'''1729/1728''' is a [[19-limit]] (more accurately, 2.3.7.13.19 subgroup) [[superparticular]] interval and an [[unnoticeable comma]]. '''Lesser massma'''<ref>referring to number 1728 being known as the ''Maß'' in German. </ref>, '''ramanujanisma'''<ref>referring to the anecdotal story of [[Wikipedia: Ramanujan|Ramanujan]] finding 1729 an interesting number. </ref>, and '''dodecentisma'''<ref>referring to the size being close to the cent relative to 12edo. </ref> have been proposed as the name. The comma 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]].
+'''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. [[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>).
-Remarkably, this comma is very close to one cent.
+== 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
-Tempering out this comma enables the related essentially tempered chords in the 19-odd-limit.
+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 ==
@@ Line 21: / Line 42: @@
 * [[Unnoticeable comma]]
-== Notes ==
+[[Category:Ramanujanismic]]
+[[Category:Commas named after mathematicians]]
-[[Category:19-limit]]
+[[Category:Commas referencing a famous use of a number]]
-[[Category:Unnoticeable comma]]
-[[Category:Superparticular]]