33/32

Interval information

Ratio	33/32
Factorization	2-5 × 3 × 11
Monzo	[-5 1 0 0 1⟩
Size in cents	53.27294¢
Names	al-Farabi quarter tone, undecimal quarter tone, io comma, Alpharabian parachroma, Alpharabian ultraprime
Color name	1o1, ilo unison
FJS name	[math]\displaystyle{ \text{P1}^{11} }[/math]
Special properties	superparticular, reduced, reduced harmonic
Tenney height (log2 nd)	10.0444
Weil height (log2 max(n, d))	10.0888
Wilson height (sopfr(nd))	24
Comma size	medium
S-expression	S9 × S10 × S11
[sound info]
Open this interval in xen-calc

33/32, the al-Farabi quarter tone, undecimal quarter tone, or undecimal formal 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.

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.

Temperaments

If treated as a comma to be tempered out in the 2.3.11 subgroup, it results in the io temperament, giving rise to the name io comma. The temperament is generated by a flat fifth (7edo and 26edo being good tunings) which represents both 3/2 and 16/11.

Approximation

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.

Notation

This interval 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

In Ben Johnston's notation, this interval is denoted with ↑, and its reciprocal as ↓. If the base note is C, then 11/8 is represented by C–F↑.

Sagittal notation

In the Sagittal system, this comma (possibly tempered) is represented by the sagittal ⁠ ⁠ and is called the 11 medium diesis, or 11M for short, because the simplest interval it notates is 11/1 (equiv. 11/8), as for example in C–F⁠ ⁠⁠ ⁠. The downward version is called 1/11M or 11M down and is represented by ⁠ ⁠.

See also

• Gallery of just intervals
• 1ed33/32 – equal multiplication of this interval
• 64/33 – its octave complement
• 16/11 – its fifth complement
• 32/31 – the tricesimoprimal counterpart
• File:Ji-33-32-csound-foscil-220hz.mp3 – alternative sound example