33/32: Difference between revisions

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{{Infobox Interval

~~| Icon =~~

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

~~| Ratio = 33/32~~

| Color name = 1o1, ilo unison

~~| 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 =

~~| FJS name = P111~~

| Sound = jid_33_32_pluck_adu_dr220.mp3

| Comma = yes

}}

'''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 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]], ~~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]].

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]].

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 ~~the theory of [[User:Aura|Aura]] not only in terms of~~ [[Alpharabian tuning]]~~, but also in terms of his own ideas on how [[diatonic functional harmony]] extends to the realm of microtonality,~~ and it is from ~~these areas~~ of microtonal theory that 33/32 acquires the names '''Alpharabian parachroma''' and '''~~Alphrabian~~ ultraprime''', names that ~~are~~ at this point ~~mainly~~ used ~~by Aura himself~~.

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.

== 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 ==

[[22edo]] and [[23edo]]'s step sizes are good, albeit in[[consistent]] approximations of this interval. Since equal-step tuning of 33/32 is roughly equivalent to 22.5edo, 2 steps of [[45edo]] represent the interval with great accuracy. [[46edo]] inherits mapping from 23edo and does it consistently.

== Notation ==

This interval 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.

=== Ben Johnston's notation ===

In [[Ben Johnston's notation]], this interval is denoted with ↑, and its reciprocal as ↓. If the base note is C, then 11/8 is represented by C–F↑.

=== Sagittal notation ===

In the [[Sagittal]] system, this comma (possibly tempered) is represented by the sagittal {{sagittal | /|\ }} and is called the '''11 medium diesis''', or '''11M''' for short, because the simplest interval it notates is 11/1 (equiv. 11/8), as for example in C–F{{nbhsp}}{{sagittal | /|\ }}. The downward version is called '''1/11M''' or '''11M down''' and is represented by {{sagittal| \!/ }}.

== See also ==

* [[64/33]] – its [[octave complement]]

* [[16/11]] – its [[fifth complement]]

* [[Gallery of just intervals]]

* [[32/31]]

* [[1ed33/32]] – equal multiplication of this interval

* [[:File:Ji-33-32-csound-foscil-220hz.mp3]]

* [[64/33]] – its [[octave complement]]

* [[16/11]] – its [[fifth complement]]

* [[32/31]] – the tricesimoprimal counterpart

* [[:File:Ji-33-32-csound-foscil-220hz.mp3]] – alternative sound example

== References ==

~~[[Category:11-limit]]~~

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

~~[[Category:Superparticular]]~~

[[Category:Quartertone]]

[[Category:Alpharabian]]

[[Category:~~Medium comma~~]]

[[Category:Commas named after their color name]]

[[Category:~~Listen]]~~

[[Category:Commas named after polymaths]]

~~[[Category:Overtone~~]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Icon =
+| Name = al-Farabi quarter tone, undecimal quarter tone, io comma, Alpharabian parachroma, Alpharabian ultraprime
-| Ratio = 33/32
+| Color name = 1o1, ilo unison
-| 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 =
-| FJS name = P1<sup>11</sup>
 | Sound = jid_33_32_pluck_adu_dr220.mp3
+| Comma = yes
 }}
-'''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 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]], 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]].
+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]].
-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 the theory of [[User:Aura|Aura]] not only in terms of [[Alpharabian tuning]], but also in terms of his own ideas on how [[diatonic functional harmony]] extends to the realm of microtonality, and it is from these areas of microtonal theory that 33/32 acquires the names '''Alpharabian parachroma''' and '''Alphrabian ultraprime''', names that are at this point mainly used by Aura himself.
+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.
+== 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 ==
+[[22edo]] and [[23edo]]'s step sizes are good, albeit in[[consistent]] approximations of this interval. Since equal-step tuning of 33/32 is roughly equivalent to 22.5edo, 2 steps of [[45edo]] represent the interval with great accuracy. [[46edo]] inherits mapping from 23edo and does it consistently.
+== Notation ==
+This interval is significant in [[Functional Just System]] and [[Helmholtz&ndash;Ellis notation]] as the undecimal formal comma which translates a Pythagorean interval to a nearby undecimal interval.
+=== Ben Johnston's notation ===
+In [[Ben Johnston's notation]], this interval is denoted with &uarr;, and its reciprocal as &darr;. If the base note is C, then 11/8 is represented by C&ndash;F&uarr;.
+=== Sagittal notation ===
+In the [[Sagittal]] system, this comma (possibly tempered) is represented by the sagittal {{sagittal | /|\ }} and is called the '''11 medium diesis''', or '''11M''' for short, because the simplest interval it notates is 11/1 (equiv. 11/8), as for example in C–F{{nbhsp}}{{sagittal | /|\ }}. The downward version is called '''1/11M''' or '''11M down''' and is represented by {{sagittal| \!/ }}.
 == See also ==
-* [[64/33]] – its [[octave complement]]
-* [[16/11]] – its [[fifth complement]]
 * [[Gallery of just intervals]]
-* [[32/31]]
+* [[1ed33/32]] &ndash; equal multiplication of this interval
-* [[:File:Ji-33-32-csound-foscil-220hz.mp3]]
+* [[64/33]] &ndash; its [[octave complement]]
+* [[16/11]] &ndash; its [[fifth complement]]
+* [[32/31]] &ndash; the tricesimoprimal counterpart
+* [[:File:Ji-33-32-csound-foscil-220hz.mp3]] &ndash; alternative sound example
 == References ==
-<references />
+<references group="note" />
-[[Category:11-limit]]
-[[Category:Interval ratio]]
-[[Category:Superparticular]]
 [[Category:Quartertone]]
 [[Category:Alpharabian]]
-[[Category:Medium comma]]
+[[Category:Commas named after their color name]]
-[[Category:Listen]]
+[[Category:Commas named after polymaths]]
-[[Category:Overtone]]