4096/3993

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Interval information
Ratio 4096/3993
Factorization 212 × 3-1 × 11-3
Monzo [12 -1 0 0 -3
Size in cents 44.091172¢
Names Alpharabian paralimma,
Alpharabian paradiatonic semilimma,
Alpharabian semilimmic inframinor second
Color name s1u32, satrilu 2nd
FJS name [math]\text{M2}_{11,11,11}[/math]
Special properties reduced
Tenney height (log2 nd) 23.9633
Weil height (log2 max(n, d)) 24
Wilson height (sopfr (nd)) 60
Harmonic entropy
(Shannon, [math]\sqrt{nd}[/math])
~4.50767 bits
Comma size medium
open this interval in xen-calc

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.

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.

Temperaments

Tempering out the paralimma in the 2.3.11 subgroup results in paralimmic temperament, where 3/1 is divided into 3 flat 16/11 generators.

See also