33/32: Difference between revisions

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.

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 a special type of subchroma sells it short, and results in a 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 it's 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. 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.

References

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

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

[1]

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 Because of its close proximity to [[28/27]], form which it differs only by [[Pentacircle comma|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 [https://en.wikipedia.org/wiki/Genus_(music) 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 a special type of [[Diatonic, Chromatic, Enharmonic, Subchromatic|subchroma]] sells it short, as a stack of two 33/32 intervals equals [[1089/1024]], a type of chromatic semitone that has [[128/121]] as it's 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]].  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]].
+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 a special type of [[Diatonic, Chromatic, Enharmonic, Subchromatic|subchroma]] sells it short, and results in a 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 it's 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]].  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 ==

33/32: Difference between revisions

Revision as of 03:03, 18 November 2020

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