5edo - Revision history

Lucius Chiaraviglio: /* Intervals */ Last formatting fix worked — do same thing for 91/60, and add its octave complement

2026-05-16T20:03:48Z

Intervals: Last formatting fix worked — do same thing for 91/60, and add its octave complement

← Older revision		Revision as of 20:03, 16 May 2026
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	\|		\|
	\| [[21/16]] (+9.219)		\| [[21/16]] (+9.219)
	\| [[33/25]] (-0.686)		\| [[33/25]] (-0.686)<br>[[120/91]] (-1.085)
	\| [[File:0-480 fourth (5-EDO).mp3\|frameless]]		\| [[File:0-480 fourth (5-EDO).mp3\|frameless]]
	\|-		\|-
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	\|		\|
	\| [[32/21]] (-9.219)		\| [[32/21]] (-9.219)
	\| [[50/33]] (+0.686)		\| [[50/33]] (+0.686)<br>[[91/60]] (+1.085)
	[[91/60]] (+1.085)
	\| [[File:0-720 fifth (5-EDO).mp3\|frameless]]		\| [[File:0-720 fifth (5-EDO).mp3\|frameless]]
	\|-		\|-

Lucius Chiaraviglio: /* Intervals */ Fix table formatting, and add 23/20 and 40/23 as even simpler intervals of lower prime limit and still decent accuracy

2026-05-16T20:01:07Z

Intervals: Fix table formatting, and add 23/20 and 40/23 as even simpler intervals of lower prime limit and still decent accuracy

← Older revision		Revision as of 20:01, 16 May 2026
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	\| [[144/125]] (-4.969)<br>[[125/108]] (-13.076)		\| [[144/125]] (-4.969)<br>[[125/108]] (-13.076)
	\| [[8/7]] (+8.826)<br>[[7/6]] (-26.871)		\| [[8/7]] (+8.826)<br>[[7/6]] (-26.871)
	\| [[31/27]] (+0.829)		\| [[23/20]] (-1.960)<br>[[31/27]] (+0.829)<br>[[224/195]] (-0.030)
	[[224/195]] (-0.030)
	\| [[File:0-240 second, third (5-EDO).mp3\|frameless]]		\| [[File:0-240 second, third (5-EDO).mp3\|frameless]]
	\|-		\|-
Line 101:		Line 100:
	\| [[216/125]] (+13.076)<br>[[125/72]] (+4.969)		\| [[216/125]] (+13.076)<br>[[125/72]] (+4.969)
	\| [[12/7]] (+26.871)<br>[[7/4]] (-8.826)		\| [[12/7]] (+26.871)<br>[[7/4]] (-8.826)
	\| [[54/31]] (-0.829)		\| [[40/23]] (+1.960)<br>[[54/31]] (-0.829)<br>[[195/112]] (+0.030)
	[[195/112]] (+0.030)
	\| [[File:0-960 sixth, seventh (5-EDO).mp3\|frameless]]		\| [[File:0-960 sixth, seventh (5-EDO).mp3\|frameless]]
	\|-		\|-

Lucius Chiaraviglio: /* Intervals */ Add 31/27 and 54/31 — higher prime limit than 224/195 and 195/112, and lower accuracy (but still pretty good), but much simpler ratios overall

2026-05-16T19:55:56Z

Intervals: Add 31/27 and 54/31 — higher prime limit than 224/195 and 195/112, and lower accuracy (but still pretty good), but much simpler ratios overall

← Older revision		Revision as of 19:55, 16 May 2026
Line 72:		Line 72:
	\| [[144/125]] (-4.969)<br>[[125/108]] (-13.076)		\| [[144/125]] (-4.969)<br>[[125/108]] (-13.076)
	\| [[8/7]] (+8.826)<br>[[7/6]] (-26.871)		\| [[8/7]] (+8.826)<br>[[7/6]] (-26.871)
	\| [[224/195]] (-0.030)		\| [[31/27]] (+0.829)
			[[224/195]] (-0.030)
	\| [[File:0-240 second, third (5-EDO).mp3\|frameless]]		\| [[File:0-240 second, third (5-EDO).mp3\|frameless]]
	\|-		\|-
Line 100:		Line 101:
	\| [[216/125]] (+13.076)<br>[[125/72]] (+4.969)		\| [[216/125]] (+13.076)<br>[[125/72]] (+4.969)
	\| [[12/7]] (+26.871)<br>[[7/4]] (-8.826)		\| [[12/7]] (+26.871)<br>[[7/4]] (-8.826)
	\| [[195/112]] (+0.030)		\| [[54/31]] (-0.829)
			[[195/112]] (+0.030)
	\| [[File:0-960 sixth, seventh (5-EDO).mp3\|frameless]]		\| [[File:0-960 sixth, seventh (5-EDO).mp3\|frameless]]
	\|-		\|-

MisterShafXen: fix names for 16/15

2026-05-14T19:46:00Z

fix names for 16/15

← Older revision		Revision as of 19:46, 14 May 2026
Line 296:		Line 296:
	\| 111.731		\| 111.731
	\| Gubi		\| Gubi
	\| ~~Dicot~~ comma, classic ~~chroma~~		\| Father comma, classic diatonic semitone
	\|-		\|-
	\| 5		\| 5

Overthink: /* Theory */ paragraph and link

2026-03-08T01:14:57Z

Theory: paragraph and link

← Older revision		Revision as of 01:14, 8 March 2026
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	[[File:5edo scale.mp3\|thumb\|A chromatic 5edo scale on C.]]		[[File:5edo scale.mp3\|thumb\|A chromatic 5edo scale on C.]]

	5edo is the smallest edo that contains a usable [[Perfect fifth (interval region)\|perfect fifth]] at 720{{C}}, being 18{{C}} sharp of a [[just]]ly tuned [[3/2]] ratio at 702{{C}}. As such, it is the smallest edo where elements of traditional music theory begin to make sense. The 720{{C}} fifth generates an [[equalized]] tuning of the [[pentic]] (2L 3s) scale, where every step is the same size at 240{{C}}, or one step of 5edo. It also generates a [[collapsed]] tuning of the [[diatonic]] (5L 2s) scale, where the [[diatonic semitone]] or minor second is mapped to 0 steps, meaning that E and F as well as B and C are the same note in 5edo.		5edo is the smallest edo that contains a usable [[Perfect fifth (interval region)\|perfect fifth]] at 720{{Cent}}, being 18{{C}} sharp of a [[just]]ly tuned [[3/2]] ratio at 702{{C}}. As such, it is the smallest edo where elements of traditional music theory begin to make sense.

			The 720{{C}} fifth generates an [[equalized]] tuning of the [[pentic]] (2L 3s) scale, where every step is the same size at 240{{C}}, or one step of 5edo. It also generates a [[collapsed]] tuning of the [[diatonic]] (5L 2s) scale, where the [[diatonic semitone]] or minor second is mapped to 0 steps, meaning that E and F as well as B and C are the same note in 5edo.

	5edo is the basic example of an [[equipentatonic]] scale, as in 5edo all steps are exactly the same size.		5edo is the basic example of an [[equipentatonic]] scale, as in 5edo all steps are exactly the same size.

Overthink: /* Theory */ major rewrite

2026-03-08T01:12:55Z

Theory: major rewrite

← Older revision		Revision as of 01:12, 8 March 2026
Line 14:		Line 14:
	[[File:5edo scale.mp3\|thumb\|A chromatic 5edo scale on C.]]		[[File:5edo scale.mp3\|thumb\|A chromatic 5edo scale on C.]]

	5edo is the smallest edo that contains a usable ~~(albeit very sharp)~~ [[Perfect fifth (interval region)\|perfect fifth]]~~. 5edo can be seen as~~ a ~~simplified form of the familiar pentatonic scale, which has the~~ [[~~mos~~]] ~~pattern~~ [[~~2L 3s~~]] ~~(known as ''pentic'')~~, ~~with 5edo specifically being~~ the [[equalized]] tuning of ~~2L 3s. 5edo is~~ the ~~basic example of an~~ [[~~equipentatonic~~]] scale, ~~as in 5edo all steps are exactly~~ the same size. {{~~W\|Tertian harmony~~}} is also ~~possible in 5edo, but barely:~~ the ~~only chords available are suspended chords, which may also be seen as inframinor~~ (~~very flat minor~~) ~~and ultramajor (very sharp major) chords~~, ~~also known as~~ [[~~Extraclassical tonality\|arto and tendo~~]] ~~chords, due~~ to ~~how sharp the fifth is. As a result~~, ~~many triads will share~~ the same ~~three notes, so rootedness is much more important to explicitly establish~~.		5edo is the smallest edo that contains a usable [[Perfect fifth (interval region)\|perfect fifth]] at 720{{C}}, being 18{{C}} sharp of a [[just]]ly tuned [[3/2]] ratio at 702{{C}}. As such, it is the smallest edo where elements of traditional music theory begin to make sense. The 720{{C}} fifth generates an [[equalized]] tuning of the [[pentic]] (2L 3s) scale, where every step is the same size at 240{{C}}, or one step of 5edo. It also generates a [[collapsed]] tuning of the [[diatonic]] (5L 2s) scale, where the [[diatonic semitone]] or minor second is mapped to 0 steps, meaning that E and F as well as B and C are the same note in 5edo.

	~~In terms of just intonation,~~ 5edo represents the perfect fifth 3/2 and harmonic seventh 7/4 rather accurately for how wide the steps are, with 3 being about 18 [[cent]]s sharp, and 7 being about 9 cents flat. In 5edo, the perfect fifth is ~~3 steps, meaning it can be divided into 3 equal parts, each representing~~ the ~~supermajor second 8/7. This is [[slendric]] temperament. Two~~ of ~~these parts make the perfect fourth~~ [[~~4/3~~]], ~~which is [[semaphore]] temperament, and finally the harmonic seventh may be found by going up two perfect fourths, which is [[superpyth]] or "archy" temperament. This~~ all ~~means that 5edo contains a very simplified form of~~ the ~~[[2.3.7 subgroup]], and many scales in 2.3.7 take a pentatonic form~~.		5edo is the basic example of an [[equipentatonic]] scale, as in 5edo all steps are exactly the same size.

	~~With more complex intervals, however,~~ 5edo ~~becomes increasingly inaccurate. For example~~, the ~~supermajor third 9/7 is mapped to the same interval as the perfect fourth~~, which ~~is a rather inaccurate equivalence~~ (~~specifically~~, [[~~Trienstonic clan~~\|~~trienstonic~~]] ~~temperament)~~. ~~However, this can still be used as~~ a ~~third~~, ~~as referenced in~~ the ~~top paragraph~~.		{{W\|Tertian harmony}} is also possible in 5edo, but barely: the only chords available are suspended chords, which may also be seen as inframinor (very flat minor) and ultramajor (very sharp major) chords, also known as [[Extraclassical tonality\|arto and tendo]] chords, due to how sharp the fifth is. As a result, many triads will share the same three notes, so rootedness is much more important to explicitly establish.

	~~If we extend our scope to~~ the ~~full~~ [[7~~-limit~~]] (~~including prime~~ [[~~5/1\|5~~]], and ~~thus conventional major and minor thirds)~~, ~~then~~ the ~~most salient fact~~ is ~~that~~ the ~~best approximation of the major third 5~~/4 is ~~extremely inaccurate~~, ~~almost a full semitone sharper than just. This results in 5edo supporting several~~ [[~~Exotemperament\|exotemperaments~~]] ~~when~~ intervals of ~~5 are introduced. For example, the best 5~~/~~4 of 480 cents is in fact~~ the ~~same interval as~~ 4/3, ~~meaning that the semitone that usually separates them,~~ [[16/15]], is [[~~tempered~~ out]] (which is ~~the very inaccurate~~ [[~~father~~]] temperament).		In terms of just intonation, besides the perfect fifth, 5edo also contains a relatively accurate approximation the harmonic seventh [[7/4]] at 4 steps (960{{C}}), being 8.8{{C}} flat of just. 5edo can thus be used as a simplified version of the [[2.3.7 subgroup]], and defines much of its underlying structure. For example, in 5edo, the perfect fifth is 3 steps, meaning it can be divided into 3 equal parts, each representing the supermajor second [[8/7]]. This is known as [[slendric]] temperament, where [[1029/1024]], the gamelisma, is tempered out. Two intervals of [[7/6]] or 8/7 make the perfect fourth [[4/3]], tempering out [[49/48]], known as [[semaphore]] temperament. Finally, the harmonic seventh may be found by going up two perfect fourths, tempering out [[64/63]], which is [[superpyth]] temperament (sometimes known as ''archy'' in the 2.3.7 subgroup).

	~~Exploring~~ more complex intervals, ~~we find that~~ the ~~minor tone~~ [[~~10/~~9~~]] and the minor third [[6~~/5]] ~~are best~~ mapped to the same ~~step of 240 cents, meaning that~~ the ~~semitone separating them,~~ [[27~~/25~~]], is tempered out ~~as well - this is~~ [[~~bug~~]] temperament, ~~which is~~ a ~~little more perverse even than father~~.		With more complex intervals, however, 5edo becomes increasingly inaccurate. For example, the supermajor third [[9/7]] is mapped very sharply to 480{{C}}, which is the same interval as the perfect fourth. Thus [[28/27]] is tempered out, leading to the rather inaccurate [[Trienstonic clan\|trienstonic]] temperament. However, this interval can still be used as a third, as referenced above.

	~~Because these intervals are so large~~, ~~this sort of analysis~~ is ~~less significant with 5edo than it becomes with larger and more accurate divisions~~, ~~but it still plays~~ a ~~role~~. For example, ~~if we attempt to analyze 5edo~~ as ~~supporting standard~~ [[~~Diatonic functional harmony\|diatonic harmony~~]], ~~I–IV–V–I~~ is the same as ~~I–III–V–I and involves triads with common intervals because of fourth-thirds equivalence~~.		If we attempt to add prime [[5/1\|5]] to the mix and extend 5et to the full [[7-limit]], then the major third [[5/4]] is mapped very sharply to 2 steps (480{{C}}), almost a full semitone sharper than the just 5/4 at 386.3{{C}}. This results in 5edo supporting several [[exotemperament]]s when intervals of 5 are introduced. For example, the best 5/4 is the same interval as 4/3, meaning that the semitone that separates them in JI, [[16/15]], is tempered out, leading to the very inaccurate [[father]] temperament. Exploring more complex intervals, we find that the minor tone [[10/9]] and the minor third [[6/5]] are best mapped to the same step of 240 cents, meaning that the semitone separating them, [[27/25]], is tempered out as well—this is [[bug]] temperament, which is a little more perverse even than father.

	If 5edo is ~~taken as only a tuning of the 3-limit~~, ~~we find that the circle of fifths returns to the unison after only 5 steps, rather~~ than ~~12. This is called [[blackwood]] temperament,~~ and ~~in 5edo, this is a "good" tuning of a circle of fifths -~~ more ~~formally, since the comma being tempered out, the semitone 256/243~~, ~~is smaller than half~~ a ~~step (120 cents)~~, 5edo ~~demonstrates~~ [[~~Telicity~~\|~~3-to-2 telicity~~]] ~~(and~~ is the ~~third EDO to do so after [[1edo]]~~ and ~~[[2edo]])~~.		Because 5edo's step is so large, such analysis is less significant with 5edo than it becomes with larger and more accurate divisions, but it still plays a role. For example, if we attempt to analyze 5edo as supporting standard [[Diatonic functional harmony\|diatonic harmony]], I–IV–V–I is the same as I–III–V–I and involves triads with common intervals because major thirds and fourths are equivalent.

	5edo is the smallest edo representing the [[9-odd-limit]] [[consistent]]ly, giving a distinct [[octave-reduced]] step to harmonics 1, 3, 5, 7 and ~~9 - specifically~~, 3 is mapped to 3 steps (720 cents), 5 is very inaccurately mapped to 2 steps (480 cents), 7 is mapped to 4 steps (960 cents), and 9 is mapped to 1 step (240 cents). However, while 2edo represents the [[3-odd-limit]] consistently, 3edo the [[5-odd-limit]], 4edo the [[7-odd-limit]] and 5edo the 9-odd-limit, to represent the [[11-odd-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].		If 5edo is taken as only a tuning of the [[3-limit]], we find that the circle of fifths closes after only 5 steps, rather than 12, meaning [[256/243]] is tempered out. This is called [[blackwood]] temperament, and in 5edo, this is a "good" tuning of a circle of fifths—more formally, since the comma being tempered out, the 256/243 semitone at 90.2{{C}}, is smaller than half a step at 120{{C}}, 5edo demonstrates [[Telicity\|3-to-2 telicity]], and is in fact the third edo to do so after [[1edo]] and [[2edo]].

			5edo is the smallest edo representing the [[9-odd-limit]] [[consistent]]ly, giving a distinct [[octave-reduced]] step to harmonics 1, 3, 5, 7 and 9—specifically, 3 is mapped to 3 steps (720 cents), 5 is very inaccurately mapped to 2 steps (480 cents), 7 is mapped to 4 steps (960 cents), and 9 is mapped to 1 step (240 cents). However, while [[2edo]] represents the [[3-odd-limit]] consistently, [[3edo]] the [[5-odd-limit]], [[4edo]] the [[7-odd-limit]] and 5edo the 9-odd-limit, to represent the [[11-odd-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].

	Despite its lack of accuracy in the 5-limit, 5edo is the second [[zeta integral edo]], after [[2edo]].		Despite its lack of accuracy in the 5-limit, 5edo is the second [[zeta integral edo]], after [[2edo]].
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	5edo is the 3rd [[prime edo]], after [[2edo]] and [[3edo]] and before [[7edo]]. It does not contain any nontrivial subset edos, though it contains [[5ed4]]. Multiples such as [[10edo]], [[15edo]], … up to [[35edo]], share the same tuning of the perfect fifth as 5edo, while improving on other intervals.		5edo is the 3rd [[prime edo]], after [[2edo]] and [[3edo]] and before [[7edo]]. It does not contain any nontrivial subset edos, though it contains 5 equal divisions of the double octave [[4/1]], or [[5ed4]]. Multiples of 5edo, such as [[10edo]], [[15edo]], …, up to [[35edo]], share the same tuning of the perfect fifth as 5edo, while improving on other intervals.

	== Intervals ==		== Intervals ==

Overthink: /* Theory */ Improve writing (mostly at the start, further improvement still seems needed)

2026-03-06T23:56:17Z

Theory: Improve writing (mostly at the start, further improvement still seems needed)

← Older revision		Revision as of 23:56, 6 March 2026
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	[[File:5edo scale.mp3\|thumb\|A chromatic 5edo scale on C.]]		[[File:5edo scale.mp3\|thumb\|A chromatic 5edo scale on C.]]

	5edo is the ~~basic example of an [[equipentatonic]] scale, containing~~ a sharp ~~but usable~~ [[Perfect fifth (interval region)\|perfect fifth]]~~, and~~ can be seen as a simplified form of the familiar [[pentic]] scale. Tertian harmony is possible in 5edo, but barely: the only chords available are suspended chords, which ~~[[Extraclassical tonality\|~~may also be seen as]] inframinor (very flat minor) and ultramajor (very sharp major) chords, due to how sharp the fifth is. As a result, many triads will share the same three notes, so rootedness is much more important to explicitly establish.		5edo is the smallest edo that contains a usable (albeit very sharp) [[Perfect fifth (interval region)\|perfect fifth]]. 5edo can be seen as a simplified form of the familiar pentatonic scale, which has the [[mos]] pattern [[2L 3s]] (known as ''pentic''), with 5edo specifically being the [[equalized]] tuning of 2L 3s. 5edo is the basic example of an [[equipentatonic]] scale, as in 5edo all steps are exactly the same size. {{W\|Tertian harmony}} is also possible in 5edo, but barely: the only chords available are suspended chords, which may also be seen as inframinor (very flat minor) and ultramajor (very sharp major) chords, also known as [[Extraclassical tonality\|arto and tendo]] chords, due to how sharp the fifth is. As a result, many triads will share the same three notes, so rootedness is much more important to explicitly establish.

	In terms of just intonation, 5edo represents the perfect fifth 3/2 and harmonic seventh 7/4 rather accurately for how wide the steps are, with 3 being about ~~20 cents~~ sharp, and 7 being about 10 cents flat. In 5edo, the perfect fifth is 3 steps, meaning it can be divided into 3 equal parts, each representing the supermajor second 8/7. This is [[slendric]] temperament. Two of these parts make the perfect fourth [[4/3]], which is [[semaphore]] temperament, and finally the harmonic seventh may be found by going up two perfect fourths, which is [[superpyth]] or "archy" temperament. This all means that 5edo contains a very simplified form of the [[2.3.7 subgroup]], and many scales in 2.3.7 take a pentatonic form.		In terms of just intonation, 5edo represents the perfect fifth 3/2 and harmonic seventh 7/4 rather accurately for how wide the steps are, with 3 being about 18 [[cent]]s sharp, and 7 being about 9 cents flat. In 5edo, the perfect fifth is 3 steps, meaning it can be divided into 3 equal parts, each representing the supermajor second 8/7. This is [[slendric]] temperament. Two of these parts make the perfect fourth [[4/3]], which is [[semaphore]] temperament, and finally the harmonic seventh may be found by going up two perfect fourths, which is [[superpyth]] or "archy" temperament. This all means that 5edo contains a very simplified form of the [[2.3.7 subgroup]], and many scales in 2.3.7 take a pentatonic form.

	With more complex intervals, however, 5edo becomes increasingly inaccurate. For example, the supermajor third 9/7 is the same interval as the perfect fourth, which is a rather inaccurate equivalence (specifically, [[Trienstonic clan\|trienstonic]] temperament). However, this can still be used as a third, as referenced in the top paragraph.		With more complex intervals, however, 5edo becomes increasingly inaccurate. For example, the supermajor third 9/7 is mapped to the same interval as the perfect fourth, which is a rather inaccurate equivalence (specifically, [[Trienstonic clan\|trienstonic]] temperament). However, this can still be used as a third, as referenced in the top paragraph.

	If we extend our scope to the full 7-limit (including 5, and thus conventional major and minor thirds), then the most salient fact is that the best approximation of the major third 5/4 is extremely inaccurate, almost a full semitone sharper than just. This results in 5edo supporting several [[Exotemperament\|exotemperaments]] when intervals of 5 are introduced. For example, the best 5/4 of 480 cents is in fact the same interval as 4/3, meaning that the semitone that usually separates them, [[16/15]], is [[tempered out]] (which is the very inaccurate [[father]] temperament).		If we extend our scope to the full [[7-limit]] (including prime [[5/1\|5]], and thus conventional major and minor thirds), then the most salient fact is that the best approximation of the major third 5/4 is extremely inaccurate, almost a full semitone sharper than just. This results in 5edo supporting several [[Exotemperament\|exotemperaments]] when intervals of 5 are introduced. For example, the best 5/4 of 480 cents is in fact the same interval as 4/3, meaning that the semitone that usually separates them, [[16/15]], is [[tempered out]] (which is the very inaccurate [[father]] temperament).

	Exploring more complex intervals, we find that the minor tone [[10/9]] and the minor third [[6/5]] are best mapped to the same step of 240 cents, meaning that the semitone separating them, [[27/25]], is tempered out as well - this is [[bug]] temperament, which is a little more perverse even than father.		Exploring more complex intervals, we find that the minor tone [[10/9]] and the minor third [[6/5]] are best mapped to the same step of 240 cents, meaning that the semitone separating them, [[27/25]], is tempered out as well - this is [[bug]] temperament, which is a little more perverse even than father.

Overthink: /* Theory */ explain better

2026-03-06T23:30:16Z

Theory: explain better

← Older revision		Revision as of 23:30, 6 March 2026
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	If 5edo is taken as only a tuning of the 3-limit, we find that the circle of fifths returns to the unison after only 5 steps, rather than 12. This is called [[blackwood]] temperament, and in 5edo, this is a "good" tuning of a circle of fifths - more formally, since the comma being tempered out, the semitone 256/243, is smaller than half a step (120 cents), 5edo demonstrates [[Telicity\|3-to-2 telicity]] (and is the third EDO to do so after [[1edo]] and [[2edo]]).		If 5edo is taken as only a tuning of the 3-limit, we find that the circle of fifths returns to the unison after only 5 steps, rather than 12. This is called [[blackwood]] temperament, and in 5edo, this is a "good" tuning of a circle of fifths - more formally, since the comma being tempered out, the semitone 256/243, is smaller than half a step (120 cents), 5edo demonstrates [[Telicity\|3-to-2 telicity]] (and is the third EDO to do so after [[1edo]] and [[2edo]]).

	5edo is the smallest edo representing the [[9-odd-limit]] [[consistent]]ly, giving a distinct ~~value modulo 5~~ to 1, 3, 5, 7 and 9 - specifically, 3 is mapped to 3 steps (720 cents), 5 is very inaccurately mapped to 2 steps (480 cents), 7 is mapped to 4 steps (960 cents), and 9 is mapped to 1 step (240 cents). However, while 2edo represents the [[3-odd-limit]] consistently, 3edo the [[5-odd-limit]], 4edo the [[7-odd-limit]] and 5edo the 9-odd-limit, to represent the [[11-odd-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].		5edo is the smallest edo representing the [[9-odd-limit]] [[consistent]]ly, giving a distinct [[octave-reduced]] step to harmonics 1, 3, 5, 7 and 9 - specifically, 3 is mapped to 3 steps (720 cents), 5 is very inaccurately mapped to 2 steps (480 cents), 7 is mapped to 4 steps (960 cents), and 9 is mapped to 1 step (240 cents). However, while 2edo represents the [[3-odd-limit]] consistently, 3edo the [[5-odd-limit]], 4edo the [[7-odd-limit]] and 5edo the 9-odd-limit, to represent the [[11-odd-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].

	Despite its lack of accuracy in the 5-limit, 5edo is the second [[zeta integral edo]], after [[2edo]].		Despite its lack of accuracy in the 5-limit, 5edo is the second [[zeta integral edo]], after [[2edo]].

Xenllium at 09:06, 26 February 2026

2026-02-26T09:06:44Z

← Older revision		Revision as of 09:06, 26 February 2026
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	== See also ==		== See also ==
	* [~~https://en.xen.wiki/w/Equal-step_tuning#~~Alpha-beta~~-gamma_family_of_equal_divisions Alpha-beta-gamma_family_of_equal_divisions~~]		* [[Alpha, beta, and gamma family of equal divisions]]

	== Notes ==		== Notes ==

Contribution at 01:44, 28 October 2025

2025-10-28T01:44:29Z

← Older revision		Revision as of 01:44, 28 October 2025
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	There is also much 5edo-like world music, just search for "[[gyil]]" or "[[amadinda]]" or "[[slendro]]".		There is also much 5edo-like world music, just search for "[[gyil]]" or "[[amadinda]]" or "[[slendro]]".

			== See also ==
			* [https://en.xen.wiki/w/Equal-step_tuning#Alpha-beta-gamma_family_of_equal_divisions Alpha-beta-gamma_family_of_equal_divisions]

	== Notes ==		== Notes ==