1729/1728: Difference between revisions
Put the fact in the right context: the 7, 13, and 19 are all swappable. |
elaborated on Ramanujan's tale and added the story as a footnote. I assume the efn will not form a heading. |
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* [[361/360]] and [[456/455]] | * [[361/360]] and [[456/455]] | ||
* [[385/384]] and [[495/494]] | * [[385/384]] and [[495/494]] | ||
* [[513/512]] and [[729/728]] | * [[513/512]] and [[729/728]] * | ||
* [[1001/1000]] and [[2376/2375]] | * [[1001/1000]] and [[2376/2375]] | ||
* [[1216/1215]] and [[4096/4095]] | * [[1216/1215]] and [[4096/4095]] | ||
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* [[1701/1700]] and [[104976/104975]] | * [[1701/1700]] and [[104976/104975]] | ||
* [[1716/1715]] and [[228096/228095]] | * [[1716/1715]] and [[228096/228095]] | ||
<nowiki>*</nowiki> all is within the 2.3.7.13.19 subgroup | |||
It factors into the following superparticular pairs: | It factors into the following superparticular pairs: | ||
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== Temperaments == | == 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 | 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 == | == 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''). | The name ''ramanujanisma'' was first proposed by [[User:Fredg999|Frédéric Gagné]] in reference to the anecdotal story of [[Wikipedia: Ramanujan|Srinivasa Ramanujan]] finding 1729 an interesting number<ref>Before Srinivasa Ramanujan had died of amoebic dysentery in 1920, the year prior in 1919 fellow mathematician and Ramanujan's own mentor [[wikipedia:G._H._Hardy|G. H. Hardy]] had visited him while he lay ill in Putney, London, UK. The number of the taxicab he had ridden was 1729 and Hardy would then hope its perceived dullness was not a bad omen, to that of which Ramanujan stated this fact.</ref>. This comes from its property of being the second of ''n'' 'taxicab' numbers whose cubic sum not only is restricted to two components but also to ''n'' equivalencies (the first is two and the third is 87539319). Alternative names include ''lesser massma'', proposed by [[User:Eliora|Eliora]], in reference to the number 1728 being known as the ''Maß'' in German (although this is actually an urban legend as 1728 shares in common the expression great gross with English, and is thus called ''großes Gros''), 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 == | == See also == | ||
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[[Category:Ramanujanismic]] | [[Category:Ramanujanismic]] | ||
[[Category:Commas named after mathematicians]] | |||
[[Category:Commas referencing a famous use of a number]] | |||
Latest revision as of 18:47, 1 August 2026
| Interval information |
Nothozo comma
reduced
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 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
* all is within the 2.3.7.13.19 subgroup
It factors into the following superparticular pairs:
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 chords 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 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, 13, or 19, respectively.
Terminology
The name ramanujanisma was first proposed by Frédéric Gagné in reference to the anecdotal story of Srinivasa Ramanujan finding 1729 an interesting number[1]. This comes from its property of being the second of n 'taxicab' numbers whose cubic sum not only is restricted to two components but also to n equivalencies (the first is two and the third is 87539319). Alternative names include lesser massma, proposed by Eliora, in reference to the number 1728 being known as the Maß in German (although this is actually an urban legend as 1728 shares in common the expression great gross with English, and is thus called großes Gros), and dodecentisma, proposed by Godtone, in reference to the size being close to the relative cent of 12edo (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
- ↑ Before Srinivasa Ramanujan had died of amoebic dysentery in 1920, the year prior in 1919 fellow mathematician and Ramanujan's own mentor G. H. Hardy had visited him while he lay ill in Putney, London, UK. The number of the taxicab he had ridden was 1729 and Hardy would then hope its perceived dullness was not a bad omen, to that of which Ramanujan stated this fact.