33/32: Difference between revisions

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'''33/32''', the '''al-Farabi quarter tone'''<ref group="note">The name goes back to Abu Nasr Al-Farabi (in Western reception also Alpharabius), see [[Wikipedia: Al-Farabi]].</ref>, '''undecimal quarter tone''', or '''undecimal formal 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]], from 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 [[Wikipedia: 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]].

Because of its close proximity to [[28/27]], from 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 [[Wikipedia: 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]].

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.

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== Temperaments ==

If treated as a comma to be tempered out in the 2.3.11 [[subgroup]], it results in the [[no-fives subgroup temperaments #Io|io]] temperament, giving rise to the name '''io comma'''. The temperament is generated by a flat fifth ([[7edo]] and [[26edo]] being good tunings) which represents both [[3/2]] and [[16/11]].

== Interval chain ==

Intervals with excellent approximation in this interval chain are: 7/6 (5), 20/13 (14), 18/11 (16). Other intervals with good approximation are: 6/5, 7/5, 9/5, 13/7, 13/9, 11/10, 19/12, 17/16, 17/15, 16/15.

{| class="wikitable"

|+ [[Interval chain]] generated by 33/32

! #

! [[Cent]]s

! Approximated [[ratio]]s

! Associated [[comma]]s

|-

| 1

| 53.3

| [[36/35]] (-4.5{{c}}) 33/32

| [[385/384]] (keenanisma) —

|-

| 2

| 106.5

| [[1089/1024]] [[16/15]] (+5.2{{c}})

| — [[16384/16335]] (semiporwellisma)

|-

| 3

| 159.8

| [[12/11]] (-9.2{{c}}) [[35/32]] (-4.7{{c}}) [[11/10]] (+5.2{{c}})

| [[Alpharabian comma]] [[35937/35840]] [[16384/16335]] (semiporwellisma)

|-

| 4

| 213.1

| [[9/8]] (-9.2{{c}}) [[25/22]] (+8.2{{c}})

| [[Alpharabian comma]] [[13107200/13045131]]

|-

| 5

| 266.4

| [[7/6]] (-0.5{{c}})

| [[Quartisma]]

|-

| ...

|

|-

| 14

| 745.8

| [[20/13]] (+0.04{{c}})

|

|-

| 15

| 799.1

| ...

|

|-

| 16

| 852.4

| [[18/11]] (-0.2{{c}})

|

|}

== Approximation ==

@@ Line 8: / Line 8: @@
 '''33/32''', the '''al-Farabi quarter tone'''<ref group="note">The name goes back to Abu Nasr Al-Farabi (in Western reception also Alpharabius), see [[Wikipedia: Al-Farabi]].</ref>, '''undecimal quarter tone''', or '''undecimal formal 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]], from 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 [[Wikipedia: 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]].
+Because of its close proximity to [[28/27]], from 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 [[Wikipedia: 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]].
 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&mdash;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.
@@ Line 14: / Line 14: @@
 == Temperaments ==
 If treated as a comma to be tempered out in the 2.3.11 [[subgroup]], it results in the [[no-fives subgroup temperaments #Io|io]] temperament, giving rise to the name '''io comma'''. The temperament is generated by a flat fifth ([[7edo]] and [[26edo]] being good tunings) which represents both [[3/2]] and [[16/11]].
+== Interval chain ==
+Intervals with excellent approximation in this interval chain are: 7/6 (5), 20/13 (14), 18/11 (16). Other intervals with good approximation are: 6/5, 7/5, 9/5, 13/7, 13/9, 11/10, 19/12, 17/16, 17/15, 16/15.
+{| class="wikitable"
+|+ [[Interval chain]] generated by 33/32
+! #
+! [[Cent]]s
+! Approximated [[ratio]]s
+! Associated [[comma]]s
+|-
+| 1
+| 53.3
+| [[36/35]] (-4.5{{c}})<br>33/32
+| [[385/384]] (keenanisma)<br>&mdash;
+|-
+| 2
+| 106.5
+| [[1089/1024]]<br>[[16/15]] (+5.2{{c}})
+| &mdash;<br>[[16384/16335]] (semiporwellisma)
+|-
+| 3
+| 159.8
+| [[12/11]] (-9.2{{c}})<br>[[35/32]] (-4.7{{c}})<br>[[11/10]] (+5.2{{c}})
+| [[Alpharabian comma]]<br>[[35937/35840]]<br>[[16384/16335]] (semiporwellisma)
+|-
+| 4
+| 213.1
+| [[9/8]] (-9.2{{c}})<br>[[25/22]] (+8.2{{c}})
+| [[Alpharabian comma]]<br>[[13107200/13045131]]
+|-
+| 5
+| 266.4
+| [[7/6]] (-0.5{{c}})
+| [[Quartisma]]
+|-
+| ...
+|
+|
+|
+|-
+| 14
+| 745.8
+| [[20/13]] (+0.04{{c}})
+|
+|-
+| 15
+| 799.1
+| ...
+|
+|-
+| 16
+| 852.4
+| [[18/11]] (-0.2{{c}})
+|
+|}
+{{todo|complete table|inline=1}}
 == Approximation ==