33/32
| Interval information |
undecimal quarter tone,
undecimal comma,
Alpharabian parachroma,
Alpharabian ultraprime
reduced,
reduced harmonic
[sound info]
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 super-fourth (11/8). Raising it instead by 36/35 leads to the septimal super-fourth (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, 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. 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 Aura's music theory not only in terms of Alpharabian tuning, but also in terms of his ideas on how diatonic functional harmony extends to the realm of microtonality, and it is from these areas of microtonal theory that 33/32 acquires the names "Alpharabian parachroma" and "Alphrabian ultraprime", names that at this point are only used in these theoretical contexts.
See also
References
- ↑ The name goes back to Abu Nasr Al-Farabi (in Western reception also Alpharabius), see Wikipedia: Al-Farabi.