4096/3993
4096/3993, the Alpharabian paralimma or Alpharabian paradiatonic semilimma, is only just shy of being half of 256/243- the Pythagorean limma- being separated from the nearby 1331/1296, the interval forming the other part of the Pythagorean limma, by the nexus comma. It is also notable for being one of only two quartertone intervals in the 11-limit- specifically the 2.3.11 subgroup- needed in order to add up to a familiar 9/8 whole tone. Specifically, it is the quartertone that forms the difference between the whole tone and a stack of three 33/32 quartertones, and can thus be regarded as being some sort of second- specifically, the Alpharabian semilimmic inframinor second, not to be confused with 8192/8019, the Alpharabian inframinor second, as the two intervals are only equated when 243/242, the rastma, is tempered out.
| Interval information |
Alpharabian paradiatonic semilimma,
Alpharabian semilimmic inframinor second
reduced subharmonic
Remarkably, 4096/3993 is currently the simplest interval in terms of odd-limit that is known to result from stacking three identical quartertones with rational intervals and subtracting said stack from a 9/8 whole tone. Furthermore, although 38/37, 35/34, 32/31 and 28/27 are all simpler intervals that can be called "quarter tones" and can safely be regarded as some kind of second, subtracting any one of these intervals from 9/8 yields an interval that has a ratio lacking a cubed number in the numerator and or the denominator, and such an interval cannot be split into three equal quartertones with rational intervals.