33/32: Difference between revisions

Line 12:

'''33/32''', the '''al-Farabi quarter tone'''<ref>The name goes back to Abu Nasr Al-Farabi (in Western reception also Alpharabius), see [[wikipedia:Al-Farabi]] </ref>, '''undecimal quarter tone''', or '''undecimal comma''', is a [[superparticular]] [[ratio]] which differs by a [[385/384|keenanisma (385/384)]], from the [[36/35|septimal quarter tone (36/35)]]. Raising a just [[4/3|perfect fourth (4/3)]] by the al-Farabi quarter-tone leads to the [[11/8|undecimal super-fourth (11/8)]]. Raising it instead by 36/35 leads to the [[48/35|septimal super-fourth (48/35)]] which approximates 11/8.

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], ~~was~~ attested to in the writings of Ptolemy, and thus, if 33/32 was in fact used, it would have been paired with [[512/495]].

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 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]] 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]].

@@ Line 12: / Line 12: @@
 '''33/32''', the '''al-Farabi quarter tone'''<ref>The name goes back to Abu Nasr Al-Farabi (in Western reception also Alpharabius), see [[wikipedia:Al-Farabi]] </ref>, '''undecimal quarter tone''', or '''undecimal comma''', is a [[superparticular]] [[ratio]] which differs by a [[385/384|keenanisma (385/384)]], from the [[36/35|septimal quarter tone (36/35)]]. Raising a just [[4/3|perfect fourth (4/3)]] by the al-Farabi quarter-tone leads to the [[11/8|undecimal super-fourth (11/8)]]. Raising it instead by 36/35 leads to the [[48/35|septimal super-fourth (48/35)]] which approximates 11/8.
-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], was attested to in the writings of Ptolemy, and thus, if 33/32 was in fact used, it would have been paired with [[512/495]].
+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 attested to in the writings of Ptolemy, and thus, if 33/32 was in fact used, it would have been paired with [[512/495]].
 /32 is significant in [[Functional Just System]] 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]].