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Interval information
Ratio 33/32
Factorization 2-5 × 3 × 11
Monzo [-5 1 0 0 1
Size in cents 53.272943¢
Names al-Farabi quarter tone,
undecimal quarter tone,
undecimal comma,
Alpharabian parachroma,
Alpharabian ultraprime
Color name 1o1, ilo unison
FJS name [math]\text{P1}^{11}[/math]
Special properties superparticular,
Tenney height (log2 nd) 10.0444
Weil height (log2 max(n, d)) 10.0888
Wilson height (sopfr (nd)) 24
Harmonic entropy
(Shannon, [math]\sqrt{n\cdot d}[/math])
~4.77698 bits
Comma size medium
S-expression S9 × S10 × S11

[sound info]
open this interval in xen-calc

33/32, the al-Farabi quarter tone[1], undecimal quarter tone, or undecimal comma, is a superparticular ratio which differs by a keenanisma (385/384), from the septimal quarter tone (36/35). Raising a just perfect fourth (4/3) by the al-Farabi quarter-tone leads to the undecimal superfourth (11/8). Raising it instead by 36/35 leads to the septimal superfourth (48/35) which approximates 11/8. Apart from this, it is also the interval between 32/27 and 11/9, and between 9/8 and 12/11.

Because of its close proximity to 28/27, from which it differs only by 896/891, one could reasonably argue that 33/32 is the undecimal counterpart to 28/27 in a way, particularly if treated as an interval in its own right. However, despite this, 33/32 generally has properties more akin to a chromatic interval than to anything resembling a diatonic interval. In addition, 33/32 could arguably have been used as a melodic interval in the Greek Enharmonic Genus, and if so, there are several possibilities for the resulting tetrachord. The most obvious of these possibilities would be to include 32:33:34 within the interval of a perfect fourth, in which case this ancient Greek scale can be approximated in 22edo and 24edo, with the comma 1089/1088 being tempered out so that 33/32 and 34/33 are equated. Another possibility, however, is that the semitone was 16/15, which, according to Wikipedia, is indirectly attested to in the writings of Ptolemy, and thus, if 33/32 was in fact used, it would have been paired with 512/495.

The interval 33/32 is significant in Functional Just System and Helmholtz-Ellis notation as the undecimal formal comma which translates a Pythagorean interval to a nearby undecimal interval. Ben Johnston's notation denotes this interval with ↑, and its reciprocal as ↓. Tempering out this interval in the 2.3.11 subgroup results in the Io temperament, generated by a flat fifth (7edo and 26edo being good tunings) which represents both 3/2 and 16/11. However, it should be noted that in some significant respects, treating 33/32 as a comma rather than as an important musical interval in its own right sells it short, and results in the failure to correctly define the properties of certain intervals. Namely, a stack of two 33/32 intervals equals 1089/1024, a type of chromatic semitone that has 128/121 as its diatonic counterpart. Furthermore, 33/32 is one of two distinct 11-limit quartertone intervals required to add up to a whole tone, with 4096/3993 being the other – specifically, adding 4096/3993 to a stack of three 33/32 quartertones yields 9/8. In addition to all this, 33/32 finds a special place in Alpharabian tuning and it is from this area of microtonal theory, among a select few others, that 33/32 acquires the names "Alpharabian parachroma" and "Alpharabian ultraprime", names that at this point are only used in said theoretical contexts. While many may be accustomed to thinking of 33/32 and 729/704 as "semiaugmented primes", this analysis is only completely accurate when 243/242 is tempered out.


22edo and 23edo's step sizes are good, albeit inconsistent approximations of this interval. Since equal-step tuning of 33/32 is roughly equivalent to 22.5edo, 2 steps of 45edo represent the interval with great accuracy. 46edo inherits mapping from 23edo and does it consistently.

See also


  1. The name goes back to Abu Nasr Al-Farabi (in Western reception also Alpharabius), see Wikipedia: Al-Farabi.