1729/1728: Difference between revisions
Put the fact in the right context: the 7, 13, and 19 are all swappable. |
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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 == | ||
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[[Category:Ramanujanismic]] | [[Category:Ramanujanismic]] | ||
[[Category:Commas named after mathematicians]] | |||
[[Category:Commas referencing a famous use of a number]] | |||