1729/1728
Ratio | 1729/1728 |
Factorization | 2^{-6} × 3^{-3} × 7 × 13 × 19 |
Monzo | [-6 -3 0 1 0 1 0 1⟩ |
Size in cents | 1.0015818¢ |
Name | ramanujanisma |
Color name | 19o3oz2, nothozo 2nd, Nothozo comma |
FJS name | [math]\text{d2}^{7,13,19}[/math] |
Special properties | superparticular, reduced |
Tenney height (log_{2} nd) | 21.5106 |
Weil height (log_{2} max(n, d)) | 21.5114 |
Wilson height (sopfr (nd)) | 60 |
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math]) |
~1.20759 bits |
Comma size | unnoticeable |
open this interval in xen-calc |
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 = 1^{3} + 12^{3} = 9^{3} + 10^{3}).
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 Ramanujan finding 1729 an interesting number. Alternative names include lesser massma, proposed by Eliora, in reference to the number 1728 being known as the Maß in German, 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).