4096/3993

Interval information

Ratio	4096/3993
Factorization	212 × 3-1 × 11-3
Monzo	[12 -1 0 0 -3⟩
Size in cents	44.09117¢
Names	Alpharabian paralimma, Alpharabian paradiatonic semilimma
Color name	Satrilu 2nd
FJS name	[math]\displaystyle{ \text{M2}_{11,11,11} }[/math]
Special properties	reduced, reduced subharmonic
Tenney norm (log2 nd)	23.9633
Weil norm (log2 max(n, d))	24
Wilson norm (sopfr(nd))	60
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4096/3993, the Alpharabian paralimma or Alpharabian paradiatonic semilimma, is 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. It is of further note that this interval 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.

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.

See also

• 3993/2048 – its octave complement
• 1331/1024 – its fourth complement
• Gallery of just intervals