33/32: Difference between revisions

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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 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 ~~areas~~ of microtonal theory, among a select few others, that 33/32 acquires the names "'''Alpharabian parachroma'''" and "'''Alphrabian 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.

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 [[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 "'''Alphrabian 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.

== See also ==

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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 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 areas of microtonal theory, among a select few others, that 33/32 acquires the names "'''Alpharabian parachroma'''" and "'''Alphrabian 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.
+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 [[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 "'''Alphrabian 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.
 == See also ==