Porcupine - Revision history

Jdfreivald: /* Music */

2026-04-30T00:22:16Z

Music

← Older revision		Revision as of 00:22, 30 April 2026
Line 478:		Line 478:

	; [[Jake Freivald]]		; [[Jake Freivald]]
	* ''[https://soundcloud.com/jdfreivald/porcupine-comma-pump Porcupine Comma Pump]''~~{{dead link}}~~		* ''[https://soundcloud.com/jdfreivald/porcupine-comma-pump Porcupine Comma Pump]''

	; [[Cody Hallenbeck]]		; [[Cody Hallenbeck]]

Jdfreivald: /* Music */ Updated link.

2026-04-30T00:21:52Z

Music: Updated link.

← Older revision		Revision as of 00:21, 30 April 2026
Line 478:		Line 478:

	; [[Jake Freivald]]		; [[Jake Freivald]]
	* [~~http~~://~~micro.soonlabel~~.com/~~gene_ward_smith/Others/Freivald~~/porcupine-comma-pump~~.mp3 ''~~Porcupine Comma Pump'']{{dead link}}		* ''[https://soundcloud.com/jdfreivald/porcupine-comma-pump Porcupine Comma Pump]''{{dead link}}

	; [[Cody Hallenbeck]]		; [[Cody Hallenbeck]]

Overthink: /* Tuning spectrum */ wording

2026-02-26T23:38:41Z

Tuning spectrum: wording

← Older revision		Revision as of 23:38, 26 February 2026
Line 324:		Line 324:
	\| [[12/11]]		\| [[12/11]]
	\| 150.637		\| 150.637
	\| Lower bound of 11- and 15-odd-limit diamond tradeoff		\| Lower bound of 11-odd-limit and 11-limit 15-odd-limit diamond tradeoff
	\|-		\|-
	\|		\|
Line 334:		Line 334:
	\|		\|
	\| '''160.000'''		\| '''160.000'''
	\| '''Lower bound of 7- to (11-limit) 15-odd-limit diamond monotone'''		\| '''Lower bound of 7-odd-limit to 11-limit 15-odd-limit diamond monotone'''
	\|-		\|-
	\|		\|
Line 379:		Line 379:
	\| [[8/5]]		\| [[8/5]]
	\| 162.737		\| 162.737
	\| 2/5-comma, 5-~~odd~~ and 7-odd minimax		\| 2/5-comma, 5- and 7-odd-limit minimax
	\|-		\|-
	\|		\|
Line 394:		Line 394:
	\|		\|
	\| '''163.636'''		\| '''163.636'''
	\| '''Upper bound of 7- to (11-limit) 15-odd-limit diamond monotone'''		\| '''Upper bound of 7-odd-limit to 11-limit 15-odd-limit diamond monotone'''
	\|-		\|-
	\|		\|
	\| [[14/9]]		\| [[14/9]]
	\| 163.743		\| 163.743
	\| 9-, 11-, and (11-limit) 15-odd-limit minimax		\| 9-, 11-, and 11-limit 15-odd-limit minimax
	\|-		\|-
	\|		\|

Overthink: Use simplest interpretation (even if less accurate)

2026-02-26T23:35:00Z

Use simplest interpretation (even if less accurate)

← Older revision		Revision as of 23:35, 26 February 2026
Line 11:		Line 11:
	\| Mapping = 1; -3 -5 6 -4		\| Mapping = 1; -3 -5 6 -4
	\| Edo join 1 = 15 \| Edo join 2 = 22		\| Edo join 1 = 15 \| Edo join 2 = 22
	\| Generators = 11/10		\| Generators = 10/9
	\| Generators tuning = 163		\| Generators tuning = 163
	\| Optimization method = CWE		\| Optimization method = CWE

Overthink: note 875/864 tempered

2026-02-26T23:33:05Z

note 875/864 tempered

← Older revision		Revision as of 23:33, 26 February 2026
Line 29:		Line 29:
	Porcupine can be thought of as a [[2.3.5.11 subgroup\|2.3.5.11-subgroup]] temperament (sometimes called ''porkypine'') without much additional damage compared to the 5-limit; the generator here represents not only 10/9, but also [[11/10]] and [[12/11]] (equivalently, [[55/54]], [[100/99]], and [[121/120]] are tempered out), with the consequence that the [[11/9]] interval, usually considered a neutral third, is in porcupine identical to the 6/5 minor third, due to the extreme flatness of 10/9. This also means that [[27/20]], the 5-limit "acute fourth", is equivalent to [[11/8]] (rather than becoming 4/3 as in meantone), found at −4 generators (tuned to about 540–560 cents). This is because as the syntonic comma has been expanded, sharpening a fourth by a comma now leads to a significantly sharp interval close to the 11th harmonic. Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup at a certain standard of accuracy.		Porcupine can be thought of as a [[2.3.5.11 subgroup\|2.3.5.11-subgroup]] temperament (sometimes called ''porkypine'') without much additional damage compared to the 5-limit; the generator here represents not only 10/9, but also [[11/10]] and [[12/11]] (equivalently, [[55/54]], [[100/99]], and [[121/120]] are tempered out), with the consequence that the [[11/9]] interval, usually considered a neutral third, is in porcupine identical to the 6/5 minor third, due to the extreme flatness of 10/9. This also means that [[27/20]], the 5-limit "acute fourth", is equivalent to [[11/8]] (rather than becoming 4/3 as in meantone), found at −4 generators (tuned to about 540–560 cents). This is because as the syntonic comma has been expanded, sharpening a fourth by a comma now leads to a significantly sharp interval close to the 11th harmonic. Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup at a certain standard of accuracy.

	It is also very easy to extend porcupine to prime 7, because the 16/9, found at +6 generators (tuned to about 960–990{{c}}), has already been flattened to merge it with (6/5)<sup>3</sup>, and therefore can be equated to [[7/4]]. This makes porcupine a weak extension of [[archy]], splitting its generator into three parts; its Pythagorean major third is mapped to [[9/7]], and its fifth is tuned sharp, ranging from around 705–720{{c}}, with the best tunings around 711–712{{c}}, which roughly splits the damage on 7/4 and 9/7.		It is also very easy to extend porcupine to prime 7, because the 16/9, found at +6 generators (tuned to about 960–990{{c}}), has already been flattened to merge it with (6/5)<sup>3</sup>, and therefore can be equated to [[7/4]]. This makes porcupine a weak extension of [[archy]], splitting its generator into three parts; its Pythagorean major third is mapped to [[9/7]], and its fifth is tuned sharp, ranging from around 705–720{{c}}, with the best tunings around 711–712{{c}}, which roughly splits the damage on 7/4 and 9/7. This extension sets [[7/6]], 6/5, 5/4, and 9/7 equidistant, thus tempering out [[875/864]], making porcupine a [[keemic temperaments\|keemic temperament]].

	See [[Porcupine family #Porcupine]] for technical data and alternative 7-limit extensions. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s.		See [[Porcupine family #Porcupine]] for technical data and alternative 7-limit extensions. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s.

FloraC: Logic. - superfluous non-breaking space.

2026-02-18T20:26:42Z

Logic. - superfluous non-breaking space.

← Older revision		Revision as of 20:26, 18 February 2026
Line 23:		Line 23:
	[[File:porcupinesymmetricminor22edo.mp3\|thumb\|Symmetric minor mode of the Porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in [[22edo]] tuning.]]		[[File:porcupinesymmetricminor22edo.mp3\|thumb\|Symmetric minor mode of the Porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in [[22edo]] tuning.]]

	'''Porcupine''' is a [[regular temperament\|temperament]] that is [[generator\|generated]] by a minor whole tone which is tuned flat to around 160–165~~ ~~[[cent]]s~~, so that the porcupine [[comma]] ([[250/243]]) is [[tempering out\|tempered out]]~~. Two generators (stacked) represent [[6/5]], and three represent [[4/3]]; from this, the generator itself represents a (very flat~~) [[~~10/9]]. This is in stark contrast to [[meantone]] temperaments, including [[12edo]], where 10/9 is tuned sharp and equated with [[9/8]] so that two of them reach a ''major'' third of [[5/4]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.		'''Porcupine''' is a [[regular temperament\|temperament]] that is [[generator\|generated]] by a [[10/9\|minor whole tone]] which is tuned flat to around 160–165 [[cent]]s. Two generators (stacked) represent [[6/5]], and three represent [[4/3]], so that the [[250/243\|porcupine comma (250/243)]] is [[tempering out\|tempered out]]; from this, the generator itself represents a very flat 10/9. This is in stark contrast to [[meantone]] temperaments, including [[12edo]], where 10/9 is tuned sharp and equated with [[9/8]] so that two of them reach a ''major'' third of [[5/4]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.

	One may also note that in [[just intonation]], a stack of three 6/5's is flat of the classical minor seventh [[9/5]] by [[25/24]], and a stack of two 4/3's is the Pythagorean minor seventh [[16/9]], which is flat of 9/5 by [[81/80]]. Thus, it can be determined that porcupine equates the syntonic comma 81/80 with the 5-limit chromatic semitone [[25/24]], which simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to a strong extension of [[pythagorean]] (such as [[meantone]] and [[schismic]]).		One may also note that in [[just intonation]], a stack of three 6/5's is flat of the classical minor seventh [[9/5]] by [[25/24]], and a stack of two 4/3's is the Pythagorean minor seventh [[16/9]], which is flat of 9/5 by [[81/80]]. Thus, it can be determined that porcupine equates the syntonic comma 81/80 with the 5-limit chromatic semitone [[25/24]], which simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to a strong extension of [[pythagorean]] (such as [[meantone]] and [[schismic]]).

ArrowHead294 at 11:30, 18 February 2026

2026-02-18T11:30:15Z

← Older revision		Revision as of 11:30, 18 February 2026
Line 14:		Line 14:
	\| Generators tuning = 163		\| Generators tuning = 163
	\| Optimization method = CWE		\| Optimization method = CWE
	\| MOS scales = [[1L 6s]], [[7L 1s]], [[7L 8s]]		\| MOS scales = [[1L 6s]], [[7L 1s]], [[7L 8s]]
	\| Pergen = (P8, P4/3)		\| Pergen = (P8, P4/3)
	\| Color name = Triyoti		\| Color name = Triyoti
Line 23:		Line 23:
	[[File:porcupinesymmetricminor22edo.mp3\|thumb\|Symmetric minor mode of the Porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in [[22edo]] tuning.]]		[[File:porcupinesymmetricminor22edo.mp3\|thumb\|Symmetric minor mode of the Porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in [[22edo]] tuning.]]

	'''Porcupine''' is a [[regular temperament\|temperament]] that is [[generator\|generated]] by a minor whole tone which is tuned flat to around 160–165 [[cent]]s, ~~and~~ the porcupine [[comma]] ([[250/243]]) is [[tempering out\|tempered out]]. Two generators (stacked) represent [[6/5]], and three represent [[4/3]]; from this, the generator itself represents a (~~severely flattened~~) [[10/9]]. This is in stark contrast to [[meantone]] temperaments, including [[12edo]], where 10/9 is tuned sharp and equated with [[9/8]] so that two of them reach a ''major'' third of [[5/4]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.		'''Porcupine''' is a [[regular temperament\|temperament]] that is [[generator\|generated]] by a minor whole tone which is tuned flat to around 160–165 [[cent]]s, so that the porcupine [[comma]] ([[250/243]]) is [[tempering out\|tempered out]]. Two generators (stacked) represent [[6/5]], and three represent [[4/3]]; from this, the generator itself represents a (very flat) [[10/9]]. This is in stark contrast to [[meantone]] temperaments, including [[12edo]], where 10/9 is tuned sharp and equated with [[9/8]] so that two of them reach a ''major'' third of [[5/4]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.

	One may also note that in [[just intonation]], a stack of three 6/5's is flat of the classical minor seventh [[9/5]] by [[25/24]], and a stack of two 4/3's is the Pythagorean minor seventh [[16/9]], which is flat of 9/5 by [[81/80]]. Thus, it can be determined that porcupine equates the syntonic comma 81/80 with the 5-limit chromatic semitone [[25/24]], which simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to a strong extension of [[pythagorean]] (such as [[meantone]] and [[schismic]]).		One may also note that in [[just intonation]], a stack of three 6/5's is flat of the classical minor seventh [[9/5]] by [[25/24]], and a stack of two 4/3's is the Pythagorean minor seventh [[16/9]], which is flat of 9/5 by [[81/80]]. Thus, it can be determined that porcupine equates the syntonic comma 81/80 with the 5-limit chromatic semitone [[25/24]], which simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to a strong extension of [[pythagorean]] (such as [[meantone]] and [[schismic]]).
Line 29:		Line 29:
	Porcupine can be thought of as a [[2.3.5.11 subgroup\|2.3.5.11-subgroup]] temperament (sometimes called ''porkypine'') without much additional damage compared to the 5-limit; the generator here represents not only 10/9, but also [[11/10]] and [[12/11]] (equivalently, [[55/54]], [[100/99]], and [[121/120]] are tempered out), with the consequence that the [[11/9]] interval, usually considered a neutral third, is in porcupine identical to the 6/5 minor third, due to the extreme flatness of 10/9. This also means that [[27/20]], the 5-limit "acute fourth", is equivalent to [[11/8]] (rather than becoming 4/3 as in meantone), found at −4 generators (tuned to about 540–560 cents). This is because as the syntonic comma has been expanded, sharpening a fourth by a comma now leads to a significantly sharp interval close to the 11th harmonic. Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup at a certain standard of accuracy.		Porcupine can be thought of as a [[2.3.5.11 subgroup\|2.3.5.11-subgroup]] temperament (sometimes called ''porkypine'') without much additional damage compared to the 5-limit; the generator here represents not only 10/9, but also [[11/10]] and [[12/11]] (equivalently, [[55/54]], [[100/99]], and [[121/120]] are tempered out), with the consequence that the [[11/9]] interval, usually considered a neutral third, is in porcupine identical to the 6/5 minor third, due to the extreme flatness of 10/9. This also means that [[27/20]], the 5-limit "acute fourth", is equivalent to [[11/8]] (rather than becoming 4/3 as in meantone), found at −4 generators (tuned to about 540–560 cents). This is because as the syntonic comma has been expanded, sharpening a fourth by a comma now leads to a significantly sharp interval close to the 11th harmonic. Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup at a certain standard of accuracy.

	It is also very easy to extend porcupine to prime 7, because the 16/9, found at +6 generators (tuned to about 960–990 ~~cents~~), has already been flattened to merge it with (6/5)<sup>3</sup>, and therefore can be equated to [[7/4]]. This makes porcupine a weak extension of [[archy]], splitting its generator into three parts; its Pythagorean major third is mapped to [[9/7]], and its fifth is tuned sharp, ranging from around 705–720 ~~cents~~, with the best tunings around 711–712 ~~cents~~, which roughly splits the damage on 7/4 and 9/7.		It is also very easy to extend porcupine to prime 7, because the 16/9, found at +6 generators (tuned to about 960–990{{c}}), has already been flattened to merge it with (6/5)<sup>3</sup>, and therefore can be equated to [[7/4]]. This makes porcupine a weak extension of [[archy]], splitting its generator into three parts; its Pythagorean major third is mapped to [[9/7]], and its fifth is tuned sharp, ranging from around 705–720{{c}}, with the best tunings around 711–712{{c}}, which roughly splits the damage on 7/4 and 9/7.

	See [[Porcupine family #Porcupine]] for technical data and alternative 7-limit extensions. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s.		See [[Porcupine family #Porcupine]] for technical data and alternative 7-limit extensions. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s.
Line 197:		Line 197:
	\| M3		\| M3
	\|}		\|}
	<nowiki>*~~</nowiki>~~ In 11-limit [[CWE tuning]], octave reduced		<nowiki/>* In 11-limit [[CWE tuning]], octave reduced

	In the ups and downs notation, the [[enharmonic unison]] is the trudsharp, the triple-down augmented unison. The porcupine notation does not have an enharmonic unison.		In the ups and downs notation, the [[enharmonic unison]] is the trudsharp, the triple-down augmented unison. The porcupine notation does not have an enharmonic unison.

	Besides the specific tuning shown here, there is a range of acceptable porcupine tunings that includes generators as small as 160 ~~cents~~ ([[15edo]]) and as large as 165.5 ~~cents~~ ([[29edo]]). However, the 29edo patent val does not support 11-limit porcupine proper, since it does not temper out [[64/63]].		Besides the specific tuning shown here, there is a range of acceptable porcupine tunings that includes generators as small as 160{{c}} ([[15edo]]) and as large as 165.5{{c}} ([[29edo]]). However, the 29edo patent val does not support full 11-limit porcupine proper, since it does not temper out [[64/63]].

	== Chords and harmony ==		== Chords and harmony ==
Line 451:		Line 451:
	\| Untempered generator; upper bound of 9- to 15-odd-limit diamond tradeoff		\| Untempered generator; upper bound of 9- to 15-odd-limit diamond tradeoff
	\|}		\|}
	<nowiki>*~~</nowiki> besides~~ the octave		<nowiki/>* Besides the octave

	== History ==		== History ==

FloraC: Cleanup on infobox

2026-02-09T09:36:18Z

Cleanup on infobox

← Older revision		Revision as of 09:36, 9 February 2026
Line 1:		Line 1:
	{{~~interwiki~~		{{Interwiki
			\| en = Porcupine
	\| de = Porcupine		\| de = Porcupine
	~~\| en = Porcupine~~
	\| es =		\| es =
	\| ja =		\| ja =
Line 11:		Line 11:
	\| Mapping = 1; -3 -5 6 -4		\| Mapping = 1; -3 -5 6 -4
	\| Edo join 1 = 15 \| Edo join 2 = 22		\| Edo join 1 = 15 \| Edo join 2 = 22
	\| ~~Generator~~ = 11/10		\| Generators = 11/10
	\| ~~Generator~~ tuning = 163		\| Generators tuning = 163
	\| Optimization method = CWE		\| Optimization method = CWE
	\| MOS scales = [[1L 6s]], [[7L 1s]], [[7L 8s]]		\| MOS scales = [[1L 6s]], [[7L 1s]], [[7L 8s]]
Line 18:		Line 18:
	\| Color name = Triyoti		\| Color name = Triyoti
	\| Odd limit 1 = 5 \| Mistuning 1 = 9.8 \| Complexity 1 = 7		\| Odd limit 1 = 5 \| Mistuning 1 = 9.8 \| Complexity 1 = 7
	\| Odd limit 2 = (11-limit) 15 \| Mistuning 2 = 19.9 \| Complexity 2 = 15		\| Odd limit 2 = 11-limit 15 \| Mistuning 2 = 19.9 \| Complexity 2 = 15
	}}		}}
	[[File:porcupine.png\|thumb\|Porcupine equates three minor thirds (6/5, in red) with two perfect fourths (4/3, in green). To do so, it tempers out 250/243, which implies a generator of a flat 10/9.\|600x600px]]		[[File:porcupine.png\|thumb\|Porcupine equates three minor thirds (6/5, in red) with two perfect fourths (4/3, in green). To do so, it tempers out 250/243, which implies a generator of a flat 10/9.\|600x600px]]

Inthar: /* History */

2026-01-25T12:23:29Z

History

← Older revision		Revision as of 12:23, 25 January 2026
Line 454:		Line 454:

	== History ==		== History ==
	Porcupine temperament/scales were discovered by [[Dave Keenan]], but did not have a name until [[Herman Miller]] mentioned that his ''Mizarian Porcupine Overture'' in 15et had a section that pumps the 250/243 comma. Although this music did not use a porcupine mos or [[modmos]] (which would have 7 or 8 notes), the name was adopted for such scales as well, once the essentially one-to-one relationship between vanishing commas and sequences of [[DE]] scales was fully evident. It was clear that even though Herman's piece was in 15edo, 22edo was a porcupine tuning par excellence, and that was an interesting development in itself.		Porcupine temperament/scales were discovered by [[Dave Keenan]], but did not have a name until [[Herman Miller]] mentioned that his ''Mizarian Porcupine Overture'' in 15et had a section that pumps the 250/243 comma. Although this music did not use a porcupine mos or [[modmos]] (which would have 7 or 8 notes), the name was adopted for such scales as well, once the essentially one-to-one relationship between vanishing commas and sequences of [[MOS]] scales was fully evident. It was clear that even though Herman's piece was in 15edo, 22edo was a porcupine tuning par excellence, and that was an interesting development in itself.

	== See also ==		== See also ==

Fynn Cerulean: Removed the belittling 'obviously' in 'This is obviously in stark contrast to meantone temperaments...' in the first paragraph

2025-12-26T05:33:08Z

Removed the belittling 'obviously' in 'This is obviously in stark contrast to meantone temperaments...' in the first paragraph

← Older revision		Revision as of 05:33, 26 December 2025
Line 23:		Line 23:
	[[File:porcupinesymmetricminor22edo.mp3\|thumb\|Symmetric minor mode of the Porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in [[22edo]] tuning.]]		[[File:porcupinesymmetricminor22edo.mp3\|thumb\|Symmetric minor mode of the Porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in [[22edo]] tuning.]]

	'''Porcupine''' is a [[regular temperament\|temperament]] that is [[generator\|generated]] by a minor whole tone which is tuned flat to around 160–165 [[cent]]s, and the porcupine [[comma]] ([[250/243]]) is [[tempering out\|tempered out]]. Two generators (stacked) represent [[6/5]], and three represent [[4/3]]; from this, the generator itself represents a (severely flattened) [[10/9]]. This is ~~obviously~~ in stark contrast to [[meantone]] temperaments, including [[12edo]], where 10/9 is tuned sharp and equated with [[9/8]] so that two of them reach a ''major'' third of [[5/4]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.		'''Porcupine''' is a [[regular temperament\|temperament]] that is [[generator\|generated]] by a minor whole tone which is tuned flat to around 160–165 [[cent]]s, and the porcupine [[comma]] ([[250/243]]) is [[tempering out\|tempered out]]. Two generators (stacked) represent [[6/5]], and three represent [[4/3]]; from this, the generator itself represents a (severely flattened) [[10/9]]. This is in stark contrast to [[meantone]] temperaments, including [[12edo]], where 10/9 is tuned sharp and equated with [[9/8]] so that two of them reach a ''major'' third of [[5/4]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.

	One may also note that in [[just intonation]], a stack of three 6/5's is flat of the classical minor seventh [[9/5]] by [[25/24]], and a stack of two 4/3's is the Pythagorean minor seventh [[16/9]], which is flat of 9/5 by [[81/80]]. Thus, it can be determined that porcupine equates the syntonic comma 81/80 with the 5-limit chromatic semitone [[25/24]], which simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to a strong extension of [[pythagorean]] (such as [[meantone]] and [[schismic]]).		One may also note that in [[just intonation]], a stack of three 6/5's is flat of the classical minor seventh [[9/5]] by [[25/24]], and a stack of two 4/3's is the Pythagorean minor seventh [[16/9]], which is flat of 9/5 by [[81/80]]. Thus, it can be determined that porcupine equates the syntonic comma 81/80 with the 5-limit chromatic semitone [[25/24]], which simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to a strong extension of [[pythagorean]] (such as [[meantone]] and [[schismic]]).