33/32 - Revision history

Lucius Chiaraviglio: Link to 34/33

2026-05-16T23:19:19Z

Link to 34/33

← Older revision		Revision as of 23:19, 16 May 2026
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]].		'''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.		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.

Lucius Chiaraviglio: Add link to 1089/1088

2025-03-28T08:55:47Z

Add link to 1089/1088

← Older revision		Revision as of 08:55, 28 March 2025
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]].		'''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.		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.

BudjarnLambeth: Links

2024-12-24T05:54:31Z

Links

← Older revision		Revision as of 05:54, 24 December 2024
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]].		'''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.		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 ==		== 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]].		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]].

	== Approximation ==		== Approximation ==

ArrowHead294 at 16:10, 18 November 2024

2024-11-18T16:10:31Z

← Older revision		Revision as of 16:10, 18 November 2024
Line 6:		Line 6:
	}}		}}

	'''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 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]].		'''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.		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 ==		== Temperaments ==
Line 19:		Line 19:

	== Notation ==		== 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.		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 ===		=== 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↑~~.		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 ===		=== Sagittal notation ===
Line 28:		Line 28:

	== See also ==		== See also ==
	* [[1ed33/32]] – equal multiplication of this interval
	* [[64/33]] – its [[octave complement]]
	* [[16/11]] – its [[fifth complement]]
	* [[Gallery of just intervals]]		* [[Gallery of just intervals]]
	* [[32/31]] – the tricesimoprimal counterpart		* [[1ed33/32]] – equal multiplication of this interval
	* [[:File:Ji-33-32-csound-foscil-220hz.mp3]] – alternative sound example		* [[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 ==		== References ==
	<references />		<references group="note" />

	[[Category:Quartertone]]		[[Category:Quartertone]]

FloraC: +temperaments section

2024-11-13T11:55:36Z

+temperaments section

← Older revision		Revision as of 11:55, 13 November 2024
Line 8:		Line 8:
	'''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 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]].		'''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 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]].

	Tempering out this interval in the 2.3.11 subgroup 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]]. 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.		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]].

	== Approximation ==		== Approximation ==

FloraC: /* Notation */ expand on Johnston notation

2024-11-13T11:52:59Z

Notation: expand on Johnston notation

← Older revision		Revision as of 11:52, 13 November 2024
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	== Notation ==		== Notation ==
	~~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. [[Ben Johnston's notation]] ~~denotes~~ this interval with ↑, and its reciprocal as ↓.		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 ===		=== 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\| \!/ }}.		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 ==		== See also ==

BudjarnLambeth at 20:39, 6 November 2024

2024-11-06T20:39:36Z

← Older revision		Revision as of 20:39, 6 November 2024
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	[[Category:Alpharabian]]		[[Category:Alpharabian]]
	[[Category:Commas named after their color name]]		[[Category:Commas named after their color name]]
			[[Category:Commas named after polymaths]]

BudjarnLambeth at 07:25, 3 November 2024

2024-11-03T07:25:24Z

← Older revision		Revision as of 07:25, 3 November 2024
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	[[Category:Quartertone]]		[[Category:Quartertone]]
	[[Category:Alpharabian]]		[[Category:Alpharabian]]
			[[Category:Commas named after their color name]]

Lériendil at 16:06, 22 October 2024

2024-10-22T16:06:41Z

← Older revision		Revision as of 16:06, 22 October 2024
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	}}		}}

	'''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>, or '''undecimal quarter tone''', 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]].		'''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 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]].

FloraC: Withdraw "undecimal comma"; replace with "io comma" after the correponding temp. Collect notation stuff into the notation section. Misc. cleanup

2024-10-22T15:17:02Z

Withdraw "undecimal comma"; replace with "io comma" after the correponding temp. Collect notation stuff into the notation section. Misc. cleanup

← Older revision		Revision as of 15:17, 22 October 2024
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	{{Infobox Interval		{{Infobox Interval
	\| Name = al-Farabi quarter tone, undecimal quarter tone, ~~undecimal~~ comma, Alpharabian parachroma, Alpharabian ultraprime		\| Name = al-Farabi quarter tone, undecimal quarter tone, io comma, Alpharabian parachroma, Alpharabian ultraprime
	\| Color name = 1o1, ilo unison		\| Color name = 1o1, ilo unison
	\| Sound = jid_33_32_pluck_adu_dr220.mp3		\| Sound = jid_33_32_pluck_adu_dr220.mp3
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	}}		}}

	'''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]].		'''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>, or '''undecimal quarter tone''', 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]].

	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. [[Ben Johnston's notation]] denotes this interval with ↑, and its reciprocal as ↓. Tempering out this interval in the 2.3.11 subgroup results in the [[No-~~fives_subgroup_temperaments~~#~~Io_aka_undecimal~~\|Io]] temperament, generated by a flat fifth ([[7edo]] and [[26edo]] being good tunings) which represents both [[3/2]] and [[16/11]]. 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.		Tempering out this interval in the 2.3.11 subgroup 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]]. 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.

	== Approximation ==		== 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.		[[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.

	== Sagittal notation ==		== Notation ==
			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. [[Ben Johnston's notation]] denotes this interval with ↑, and its reciprocal as ↓.

			=== 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\| \!/ }}.		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 ==		== See also ==
	* [[33/32 ~~equal step tuning~~]] – equal multiplication of this interval		* [[1ed33/32]] – equal multiplication of this interval
	* [[64/33]] – its [[octave complement]]		* [[64/33]] – its [[octave complement]]
	* [[16/11]] – its [[fifth complement]]		* [[16/11]] – its [[fifth complement]]