33/32: Difference between revisions

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{{Infobox Interval

~~| Icon =~~

| Ratio = 33/32

| Monzo = -5 1 0 0 1

| Cents = 53.27294

| Name = al-Farabi quarter tone, undecimal quarter tone, undecimal comma, Alpharabian parachroma, Alpharabian ultraprime

| Color name =

| Color name = 1o1, ilo unison

| FJS name = P111

| Sound = jid_33_32_pluck_adu_dr220.mp3

}}

'''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. Apart from this, it is also the interval between [[32/27]] and [[11/9]], and between [[9/8]] and [[12/11]].

'''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 superfourth (11/8)]]. Raising it instead by 36/35 leads to the [[48/35|septimal superfourth (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 [[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 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.

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 "'''Alpharabian 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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[[Category:11-limit]]

~~[[Category:Interval ratio]]~~

[[Category:Superparticular]]

[[Category:Quartertone]]

[[Category:Alpharabian]]

[[Category:Medium comma]]

[[Category:~~Listen~~]]

[[Category:Octave-reduced harmonics]]

[[Category:~~Overtone~~]]

[[Category:Pages with internal sound examples]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Icon =
 | Ratio = 33/32
 | Monzo = -5 1 0 0 1
 | Cents = 53.27294
 | Name = al-Farabi quarter tone, <br>undecimal quarter tone, <br>undecimal comma, <br>Alpharabian parachroma, <br>Alpharabian ultraprime
-| Color name =
+| Color name = 1o1, ilo unison
 | FJS name = P1<sup>11</sup>
 | Sound = jid_33_32_pluck_adu_dr220.mp3
 }}
-'''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.  Apart from this, it is also the interval between [[32/27]] and [[11/9]], and between [[9/8]] and [[12/11]].
+'''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 superfourth (11/8)]]. Raising it instead by 36/35 leads to the [[48/35|septimal superfourth (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 [[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 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.
+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 "'''Alpharabian 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 ==
@@ Line 28: / Line 27: @@
 [[Category:11-limit]]
-[[Category:Interval ratio]]
 [[Category:Superparticular]]
 [[Category:Quartertone]]
 [[Category:Alpharabian]]
 [[Category:Medium comma]]
-[[Category:Listen]]
+[[Category:Octave-reduced harmonics]]
-[[Category:Overtone]]
+[[Category:Pages with internal sound examples]]