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, undecimal 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 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, form 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: Genus (music), 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 ↓. 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.

Approximation

23edo's step size is a good albeit inconsistent approximation of this interval. 46edo contains the same approximation mapped consistently.

See also

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