1729/1728: Difference between revisions

'''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]].  

 

 

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.  

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'').