33/32
| Interval information |
undecimal quarter tone,
undecimal comma
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.
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.
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. Apart from the aforementioned relationship between 4/3 and 11/8, it is also the interval between 32/27 and 11/9, and between 9/8 and 12/11.
See also
References
- ↑ The name goes back to Abu Nasr Al-Farabi (in Western reception also Alpharabius), see wikipedia:Al-Farabi