User:Zhenlige/EDO impressions - Revision history

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2026-03-14T07:01:01Z

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2026-03-13T21:01:39Z

fix typo and add link

← Older revision		Revision as of 21:01, 13 March 2026
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	*[[5edo\|5]]: Equalized [[2L 3s\|pentatonic]] scale. [[3-limit]] [[blackwood]]. Kinda familiar but everything is distorted. To me as a Chinese, it sounds like out-of-tone traditional Chinese music. A heavily streched [[slendric]] chain. The smallest EDO that roughly represents [[3-limit]] or 2.3.7 subgroup.		*[[5edo\|5]]: Equalized [[2L 3s\|pentatonic]] scale. [[3-limit]] [[blackwood]]. Kinda familiar but everything is distorted. To me as a Chinese, it sounds like out-of-tone traditional Chinese music. A heavily streched [[slendric]] chain. The smallest EDO that roughly represents [[3-limit]] or 2.3.7 subgroup.
	*[[6edo\|6]]: [[12edo]] whole tones. Incomplete 12edo. A heavily stretched [[didacus]] chain.		*[[6edo\|6]]: [[12edo]] whole tones. Incomplete 12edo. A heavily stretched [[didacus]] chain.
	*[[7edo\|7]]: Equalized [[5L 2s\|diatonic]] scale. [[3-limit]] [[whitewood]]. Similar to [[5edo]], it sounds like out-of-tone [[3L 4s\|neutral scale]] music. The smallest EDO that roughly represents [[5-limit]]. Around 3/4-comma [[meantone]].		*[[7edo\|7]]: Equalized [[5L 2s\|diatonic]] scale. [[3-limit]] [[whitewood]]. Similar to [[5edo]], it sounds like out-of-tone [[3L 4s\|neutral scale]] music. The smallest EDO that roughly represents [[5-limit]]. Around [[3/4-comma meantone\|3/4-comma]] [[meantone]].
	*[[8edo\|8]]: Incomplete [[24edo]].		*[[8edo\|8]]: Incomplete [[24edo]].
	*[[9edo\|9]]: A stack of [[7/6]]. A subset of [[ennealimmal]].		*[[9edo\|9]]: A stack of [[7/6]]. A subset of [[ennealimmal]].
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	*[[15edo\|15]]: Better than it seems to be, tho still rough. [[Blackwood]] [[augmented (temperament)\|augmented]]. A heavily stretched [[Carlos Alpha]] scale. The best you can get with blackwood. I don't know why there are “people fond of” such inaccurate temperaments.		*[[15edo\|15]]: Better than it seems to be, tho still rough. [[Blackwood]] [[augmented (temperament)\|augmented]]. A heavily stretched [[Carlos Alpha]] scale. The best you can get with blackwood. I don't know why there are “people fond of” such inaccurate temperaments.
	*...		*...
	*[[17edo\|17]]: A circle of fifths in [[34edo]]. Interesting sharp fifths. The smallest [[5L 2s\|diatonic]] EDO with neutral intervals. The boundary between neogothic and superpyth. Like [[12edo]], its diatonic thirds do not approximate any simple ratios well, and a [[well temperament]] may help. Its [[13/1\|13]] is good, and [[11/1\|11]] and [[7/1\|7]] have a similar precision to 12edo's [[5/1\|5]]. Even more suitable for making [[well temperament]]s than [[12edo]] since there are ~~mor~~ target intervals. It benefits from compression.		*[[17edo\|17]]: A circle of fifths in [[34edo]]. Interesting sharp fifths. The smallest [[5L 2s\|diatonic]] EDO with neutral intervals. The boundary between neogothic and superpyth. Like [[12edo]], its diatonic thirds do not approximate any simple ratios well, and a [[well temperament]] may help. Its [[13/1\|13]] is good, and [[11/1\|11]] and [[7/1\|7]] have a similar precision to 12edo's [[5/1\|5]]. Even more suitable for making [[well temperament]]s than [[12edo]] since there are more target intervals. It benefits from compression.
	*[[18edo\|18]]: Incomplete [[36edo]].		*[[18edo\|18]]: Incomplete [[36edo]].
	*[[19edo\|19]]: A stack of [[5/3]]. Very different tradeoffs from [[12edo]]. Usable but imperfect for many temperaments. [[Meantone]] or [[magic]], but with a too flat fifth. Strangely large minor 2nds off from [[16/15]] even more than 12edo's, kinda between a typical semitone and a neutral second, shaping its melody, making some pieces sound bad. A compressed [[Carlos Beta]] scale. For meantone [[31edo]] is more preferable, and for magic [[41edo]]. The lower bound of a good fifth. It benefits from stretching. Good as a subset of [[enneadecal]].		*[[19edo\|19]]: A stack of [[5/3]]. Very different tradeoffs from [[12edo]]. Usable but imperfect for many temperaments. [[Meantone]] or [[magic]], but with a too flat fifth. Strangely large minor 2nds off from [[16/15]] even more than 12edo's, kinda between a typical semitone and a neutral second, shaping its melody, making some pieces sound bad. A compressed [[Carlos Beta]] scale. For meantone [[31edo]] is more preferable, and for magic [[41edo]]. The lower bound of a good fifth. It benefits from stretching. Good as a subset of [[enneadecal]].

Zhenlige: /* Details */

2026-03-13T13:35:08Z

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← Older revision		Revision as of 13:35, 13 March 2026
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	*[[65edo\|65]]: A circle of fifths in [[130edo]].		*[[65edo\|65]]: A circle of fifths in [[130edo]].
			*...
			*[[68edo\|68]]: Good for no-11 [[23-limit]]. I may explore it some day.
	*...		*...
	*[[72edo\|72]]: The ultimate extension of [[12edo]] and [[24edo]]. Its [[11-limit]] is very accurate with a slightly flat tendency that works well with 12edo's flat [[3/1\|3]]. A real [[miracle]] (pun intended). The only reasonable way of extending [[compton]] to 11-limit. Some higher limit intervals are also usable. The only notable [[13-limit]] non-[[akea]] EDO around this size. A powerful tool for modulating quartertones, which can be difficult in 24edo itself. Suitable for octave stretching if only [[17-limit]] or below is used. Playable by using three 24edo instruments or six 12edo instruments.		*[[72edo\|72]]: The ultimate extension of [[12edo]] and [[24edo]]. Its [[11-limit]] is very accurate with a slightly flat tendency that works well with 12edo's flat [[3/1\|3]]. A real [[miracle]] (pun intended). The only reasonable way of extending [[compton]] to 11-limit. Some higher limit intervals are also usable. The only notable [[13-limit]] non-[[akea]] EDO around this size. A powerful tool for modulating quartertones, which can be difficult in 24edo itself. Suitable for octave stretching if only [[17-limit]] or below is used. Playable by using three 24edo instruments or six 12edo instruments.

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2026-03-13T12:45:45Z

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2026-03-13T02:07:55Z

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Zhenlige: /* Details */

2026-03-12T05:34:48Z

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← Older revision		Revision as of 05:34, 12 March 2026
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	*[[22edo\|22]]: The smallest EDO with decent [[11-limit]] and the smallest non-meantone EDO with decent [[5-limit]]. [[Superpyth]] and [[porcupine]]. Close to optimal 2.3.7 [[archy]] with compression. With pure octaves it is almost the best archy can give, since archy highly relies on octave compression. The upper bound of a good fifth. The best you can get with [[50/49]] tempered out.		*[[22edo\|22]]: The smallest EDO with decent [[11-limit]] and the smallest non-meantone EDO with decent [[5-limit]]. [[Superpyth]] and [[porcupine]]. Close to optimal 2.3.7 [[archy]] with compression. With pure octaves it is almost the best archy can give, since archy highly relies on octave compression. The upper bound of a good fifth. The best you can get with [[50/49]] tempered out.
	*[[23edo\|23]]: Incomplete [[46edo]]. The largest EDO without a [[5L 2s\|diatonic]], [[5edo\|blackwood]] or [[7edo\|whitewood]] fifth.		*[[23edo\|23]]: Incomplete [[46edo]]. The largest EDO without a [[5L 2s\|diatonic]], [[5edo\|blackwood]] or [[7edo\|whitewood]] fifth.
	*[[24edo\|24]]: What some non-microtonalists think microtonality is. 12edo with neutrals. Good for prime [[11/1\|11]]. Accurate in subgroup 2.3.11.17.19.		*[[24edo\|24]]: What some non-microtonalists think microtonality is. [[12edo]] with neutrals. Good for prime [[11/1\|11]]. Accurate in subgroup 2.3.11.17.19. Nearly optimal for 2.3.11.19 tempering out [[243/242]] and [[513/512]], where the next EDO that significanly improves is [[89edo]].
	*...		*...
	*[[26edo\|26]]: The fact that [[53edo]] is good indicates that 26- and 27edo are probably bad. A stack of [[7/4]]. Good for 2.7.11 subgroup. Other intervals suck. Since it is relatively small, consistency does not implies high accuracy. [[Meantone]] in [[5-limit]] but tuned terribly. Incomplete [[130edo]].		*[[26edo\|26]]: The fact that [[53edo]] is good indicates that 26- and 27edo are probably bad. A stack of [[7/4]]. Good for 2.7.11 subgroup. Other intervals suck. Since it is relatively small, consistency does not implies high accuracy. [[Meantone]] in [[5-limit]] but tuned terribly. Incomplete [[130edo]].
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	*[[72edo\|72]]: The ultimate extension of [[12edo]] and [[24edo]]. Its [[11-limit]] is very accurate with a slightly flat tendency that works well with 12edo's flat [[3/1\|3]]. A real [[miracle]] (pun intended). The only reasonable way of extending [[compton]] to 11-limit. Some higher limit intervals are also usable. The only notable [[13-limit]] non-[[akea]] EDO around this size. A powerful tool for modulating quartertones, which can be difficult in 24edo itself. Suitable for octave stretching if only [[17-limit]] or below is used. Playable by using three 24edo instruments or six 12edo instruments.		*[[72edo\|72]]: The ultimate extension of [[12edo]] and [[24edo]]. Its [[11-limit]] is very accurate with a slightly flat tendency that works well with 12edo's flat [[3/1\|3]]. A real [[miracle]] (pun intended). The only reasonable way of extending [[compton]] to 11-limit. Some higher limit intervals are also usable. The only notable [[13-limit]] non-[[akea]] EDO around this size. A powerful tool for modulating quartertones, which can be difficult in 24edo itself. Suitable for octave stretching if only [[17-limit]] or below is used. Playable by using three 24edo instruments or six 12edo instruments.
	*...		*...
	*[[74edo\|74]]: [[37edo]] with meantone fifths. Close to [[tungsten meantone]]. The intrinsic error of meantone becomes a problem at this size, making [[9/1\|9]] inconsistent.		*[[74edo\|74]]: [[37edo]] with [[meantone]] fifths. Close to [[tungsten meantone]]. The intrinsic error of meantone becomes a problem at this size, making [[9/1\|9]] inconsistent.
	*...		*...
	*[[77edo\|77]]: Good for [[valentine]]. Its slightly flat [[3/2]] gives a good [[19/1\|19]] via [[boethius~~]]. Containing [[Carlos Alpha~~]]. [[40/27]] as 4\7 aka [[absurdity]]. Usable for high limit JI. At this size even some inconsistent intervals are usable via val mapping since its step size is only ~16 cents.		*[[77edo\|77]]: Good for [[valentine]] therefore containing [[Carlos Alpha]]. Its slightly flat [[3/2]] gives a good [[19/1\|19]] via [[boethius]]. [[40/27]] as 4\[[7edo\|7]] aka [[absurdity]]. Usable for high limit JI. At this size even some inconsistent intervals are usable via val mapping since its step size is only ~16 cents.
	*...		*...
	*[[81edo\|81]]: The [[optimal patent val]] for [[meantone]] and some of its higher-limit extentions, but I won't use such a large EDO for a temperament with relatively low accuracy, and rather use [[golden meantone]] instead, which is simpler and more elegant mathematically.		*[[81edo\|81]]: The [[optimal patent val]] for [[meantone]] and some of its higher-limit extentions, but I won't use such a large EDO for a temperament with relatively low accuracy, and rather use [[golden meantone]] instead, which is simpler and more elegant mathematically.

Zhenlige: /* Details */

2026-03-11T17:46:09Z

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← Older revision		Revision as of 17:46, 11 March 2026
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	*[[10edo\|10]]: A stack of [[13/8]]. A subset of [[130edo]] and [[270edo]].		*[[10edo\|10]]: A stack of [[13/8]]. A subset of [[130edo]] and [[270edo]].
	*[[11edo\|11]]: Incomplete [[22edo]].		*[[11edo\|11]]: Incomplete [[22edo]].
	*[[12edo\|12]]: Equalized [[5L 7s\|chromatic]] scale. Both [[augmented (temperament)\|augmented]] and [[diminished (temperament)\|diminished]]. The smallest EDO with decent [[3-limit\|3-]], [[5-limit\|5-]] and [[7-limit]]. The only reasonable tuning for the [[dominant (temperament)\|dominant]] temperament, ignoring the difference of overall streching. A sharper fifth makes [[garibaldi]] better and a flatter fifth makes [[septimal meantone]] better. The boundary between [[meantone]] and [[schismatic]]. The smallest [[5L 2s\|diatonic]] EDO. Efficient at its size. Very excellent [[3/1\|3]] as well as good [[17/1\|17]] and [[19/1\|19]] for its size, but inaccurate [[5/1\|5]] and worse [[7/1\|7]]. Since its 7 is around twice as off as 5, adding a comma-sized generator gives [[7-limit]] [[compton]], a simple and accurate rank-2 temperament. Suitable for symmetric scales. Easy to make [[../12neji\|accurate NEJIs]]. Its 2.3.17.19 subgroup deserves more exploration (something “xenharmonic” but not “microtonal”). Its thirds do not accurately approximate common JI intervals. A [[well temperament]] can make some of them do better. AFAIK 12edo and [[17edo]] are the most suitable EDOs for making well temperaments.		*[[12edo\|12]]: Equalized [[5L 7s\|chromatic]] scale. Both [[augmented (temperament)\|augmented]] and [[diminished (temperament)\|diminished]]. The smallest EDO with decent [[3-limit\|3-]], [[5-limit\|5-]] and [[7-limit]]. The only reasonable tuning for the [[dominant (temperament)\|dominant]] temperament, ignoring the difference of overall streching. A sharper fifth makes [[garibaldi]] better and a flatter fifth makes [[septimal meantone]] better. The boundary between [[meantone]] and [[schismatic]]. The smallest [[5L 2s\|diatonic]] EDO. Efficient at its size. Very excellent [[3/1\|3]] as well as good [[17/1\|17]] and [[19/1\|19]] for its size, but inaccurate [[5/1\|5]] and worse [[7/1\|7]]. Since its 7 is around twice as off as 5, adding a comma-sized generator gives [[7-limit]] [[compton]], a simple and accurate rank-2 temperament which is friendly to 12edo-based softwares. Suitable for symmetric scales. Easy to make [[../12neji\|accurate NEJIs]]. Its 2.3.17.19 subgroup deserves more exploration (something “xenharmonic” but not “microtonal”). Its thirds do not accurately approximate common JI intervals. A [[well temperament]] can make some of them do better. AFAIK 12edo and [[17edo]] are the most suitable EDOs for making well temperaments.
	*...		*...
	*[[15edo\|15]]: Better than it seems to be, tho still rough. [[Blackwood]] [[augmented (temperament)\|augmented]]. A heavily stretched [[Carlos Alpha]] scale. The best you can get with blackwood. I don't know why there are “people fond of” such inaccurate temperaments.		*[[15edo\|15]]: Better than it seems to be, tho still rough. [[Blackwood]] [[augmented (temperament)\|augmented]]. A heavily stretched [[Carlos Alpha]] scale. The best you can get with blackwood. I don't know why there are “people fond of” such inaccurate temperaments.
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	*[[65edo\|65]]: A circle of fifths in [[130edo]].		*[[65edo\|65]]: A circle of fifths in [[130edo]].
	*...		*...
	*[[72edo\|72]]: The ultimate extension of [[12edo]] and [[24edo]]. Its [[11-limit]] is very accurate with a slightly flat tendency that works well with 12edo's flat [[3/1\|3]]~~, and some~~ higher limit intervals are also usable~~. [[Compton]] which is useful in 12edo-based programs~~. The only notable [[13-limit]] non-[[akea]] EDO around this size. A powerful tool for modulating quartertones, which can be difficult in 24edo itself~~. Good for [[miracle]]~~. Suitable for octave stretching if only [[17-limit]] or below is used. Playable by using three 24edo instruments or six 12edo instruments.		*[[72edo\|72]]: The ultimate extension of [[12edo]] and [[24edo]]. Its [[11-limit]] is very accurate with a slightly flat tendency that works well with 12edo's flat [[3/1\|3]]. A real [[miracle]] (pun intended). The only reasonable way of extending [[compton]] to 11-limit. Some higher limit intervals are also usable. The only notable [[13-limit]] non-[[akea]] EDO around this size. A powerful tool for modulating quartertones, which can be difficult in 24edo itself. Suitable for octave stretching if only [[17-limit]] or below is used. Playable by using three 24edo instruments or six 12edo instruments.
	*...		*...
	*[[74edo\|74]]: [[37edo]] with meantone fifths. Close to [[tungsten meantone]]. The intrinsic error of meantone becomes a problem at this size, making [[9/1\|9]] inconsistent.		*[[74edo\|74]]: [[37edo]] with meantone fifths. Close to [[tungsten meantone]]. The intrinsic error of meantone becomes a problem at this size, making [[9/1\|9]] inconsistent.

Zhenlige: /* Details */

2026-03-11T17:33:31Z

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← Older revision		Revision as of 17:33, 11 March 2026
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	*[[87edo\|87]]: Good [[mystery]] EDO. Useful for high limit JI. Playable by using three [[29edo]] instruments.		*[[87edo\|87]]: Good [[mystery]] EDO. Useful for high limit JI. Playable by using three [[29edo]] instruments.
			*...
			*[[89edo\|89]]: Like [[77edo]], the slightly flat fifth gives good [[19/1\|19]]. In addition, since 89edo has neutrals, it gives good [[11/1\|11]] too, together making a strong 2.3.11.19 system after [[24edo]]. Other intervals are just meh.
	*...		*...
	*[[94edo\|94]]: Good for high-limit JI with the [[garibaldi]] structure similar to [[41edo]] and [[53edo]]. Containing [[Carlos Beta]].		*[[94edo\|94]]: Good for high-limit JI with the [[garibaldi]] structure similar to [[41edo]] and [[53edo]]. Containing [[Carlos Beta]].

Zhenlige: /* Details */

2026-03-11T17:16:16Z

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← Older revision		Revision as of 17:16, 11 March 2026
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	*[[10edo\|10]]: A stack of [[13/8]]. A subset of [[130edo]] and [[270edo]].		*[[10edo\|10]]: A stack of [[13/8]]. A subset of [[130edo]] and [[270edo]].
	*[[11edo\|11]]: Incomplete [[22edo]].		*[[11edo\|11]]: Incomplete [[22edo]].
	*[[12edo\|12]]: Equalized [[5L 7s\|chromatic]] scale. Both [[augmented (temperament)\|augmented]] and [[diminished (temperament)\|diminished]]. The smallest EDO with decent [[3-limit\|3-]], [[5-limit\|5-]] and [[7-limit]]. The only reasonable tuning for the [[dominant (temperament)\|dominant]] temperament, ignoring the difference of overall streching. A sharper fifth makes [[garibaldi]] better and a flatter fifth makes [[septimal meantone]] better. The boundary between [[meantone]] and [[schismatic]]. The smallest [[5L 2s\|diatonic]] EDO. Efficient at its size. Very excellent [[3/1\|3]] as well as good [[17/1\|17]] and [[19/1\|19]] for its size, but inaccurate [[5/1\|5]] and worse [[7/1\|7]]. Since its 7 is around twice as off as 5, adding a comma-sized generator gives [[7-limit]] [[compton]], a simple and accurate rank-2 temperament. Suitable for symmetric scales. Easy to make [[../12neji\|accurate NEJIs]]. Its 2.3.17.19 subgroup deserves more exploration (something “xenharmonic” but not “microtonal”). Its thirds do not accurately approximate common JI intervals. A [[well temperament]] can make some of them do better.		*[[12edo\|12]]: Equalized [[5L 7s\|chromatic]] scale. Both [[augmented (temperament)\|augmented]] and [[diminished (temperament)\|diminished]]. The smallest EDO with decent [[3-limit\|3-]], [[5-limit\|5-]] and [[7-limit]]. The only reasonable tuning for the [[dominant (temperament)\|dominant]] temperament, ignoring the difference of overall streching. A sharper fifth makes [[garibaldi]] better and a flatter fifth makes [[septimal meantone]] better. The boundary between [[meantone]] and [[schismatic]]. The smallest [[5L 2s\|diatonic]] EDO. Efficient at its size. Very excellent [[3/1\|3]] as well as good [[17/1\|17]] and [[19/1\|19]] for its size, but inaccurate [[5/1\|5]] and worse [[7/1\|7]]. Since its 7 is around twice as off as 5, adding a comma-sized generator gives [[7-limit]] [[compton]], a simple and accurate rank-2 temperament. Suitable for symmetric scales. Easy to make [[../12neji\|accurate NEJIs]]. Its 2.3.17.19 subgroup deserves more exploration (something “xenharmonic” but not “microtonal”). Its thirds do not accurately approximate common JI intervals. A [[well temperament]] can make some of them do better. AFAIK 12edo and [[17edo]] are the most suitable EDOs for making well temperaments.
	*...		*...
	*[[15edo\|15]]: Better than it seems to be, tho still rough. [[Blackwood]] [[augmented (temperament)\|augmented]]. A heavily stretched [[Carlos Alpha]] scale. The best you can get with blackwood. I don't know why there are “people fond of” such inaccurate temperaments.		*[[15edo\|15]]: Better than it seems to be, tho still rough. [[Blackwood]] [[augmented (temperament)\|augmented]]. A heavily stretched [[Carlos Alpha]] scale. The best you can get with blackwood. I don't know why there are “people fond of” such inaccurate temperaments.
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	*[[29edo\|29]]: A circle of fifths in [[mystery]], which supports [[pele]] (and is close to its optimal tuning), a convenient temperament with [[5/1\|5]], [[7/1\|7]], [[11/1\|11]] and [[13/1\|13]] on the same chain of fifths. Near pure [[13/11]]. The smallest EDO with a better [[3/2]] than [[12edo]].		*[[29edo\|29]]: A circle of fifths in [[mystery]], which supports [[pele]] (and is close to its optimal tuning), a convenient temperament with [[5/1\|5]], [[7/1\|7]], [[11/1\|11]] and [[13/1\|13]] on the same chain of fifths. Near pure [[13/11]]. The smallest EDO with a better [[3/2]] than [[12edo]].
	*...		*...
	*[[31edo\|31]]: Ideal for pure-octave [[meantone]], combining lots of [[11-limit]] extensions in a single tuning. The meantone flat fifth makes its neutral thirds close to [[11/9]] (tho there is the JI subgroup problem since [[11/1\|11]] and [[9/1\|9]] themselves are not so good). ~~Also~~ [[valentine]] and [[miracle]]. IMO the best meantone EDO. Nearly the best meantone can give for high limit. For other temperaments its flat fifth may be a drawback, making 9 bad.		*[[31edo\|31]]: Ideal for pure-octave [[meantone]], combining lots of [[11-limit]] extensions in a single tuning. The meantone flat fifth makes its neutral thirds close to [[11/9]] (tho there is the JI subgroup problem since [[11/1\|11]] and [[9/1\|9]] themselves are not so good). The intersection of [[valentine]] and [[miracle]]. IMO the best meantone EDO. Nearly the best meantone can give for high limit. For other temperaments its flat fifth may be a drawback, making 9 bad.
	*...		*...
	*[[34edo\|34]]: [[17edo]] with [[5/1\|5]] and [[17/1\|17]] added, making a good 2.3.5.13.17 system. A slightly stretched [[Carlos Gamma]] scale.		*[[34edo\|34]]: [[17edo]] with [[5/1\|5]] and [[17/1\|17]] added, making a good 2.3.5.13.17 system. A slightly stretched [[Carlos Gamma]] scale.

Zhenlige: /* Details */

2026-03-11T12:24:22Z

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← Older revision		Revision as of 12:24, 11 March 2026
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	*[[38edo\|38]]: [[19edo]] with neutrals. Near pure [[11/9]]. Doubling such a coarse EDO won't give anything very notable, and the acceptable error of 19edo really becomes a problem at this size.		*[[38edo\|38]]: [[19edo]] with neutrals. Near pure [[11/9]]. Doubling such a coarse EDO won't give anything very notable, and the acceptable error of 19edo really becomes a problem at this size.
	*...		*...
	*[[41edo\|41]]: Prime steps in an octave and highly composite steps in a fifth, opposite from [[12edo]], thus good for fifth-dividing temperaments. Containing [[Bohlen-Pierce scale]]. ~~Optimal~~ [[magic]] ~~EDO~~. The [[Kite guitar]] shows its elegance, with many simple intervals evenly spaced. Also [[garibaldi]] and [[miracle]]. The largest problem is its relatively inaccurate [[5/1\|5]]. From here on, most EDOs with good [[13-limit]] support [[akea]].		*[[41edo\|41]]: Prime steps in an octave and highly composite steps in a fifth, opposite from [[12edo]], thus good for fifth-dividing temperaments. Containing [[Bohlen-Pierce scale]]. Good for [[magic]]. The [[Kite guitar]] shows its elegance, with many simple intervals evenly spaced. Also [[garibaldi]] and [[miracle]]. The largest problem is its relatively inaccurate [[5/1\|5]]. From here on, most EDOs with good [[13-limit]] support [[akea]].
	*[[42edo\|42]]: Incomplete [[84edo]].		*[[42edo\|42]]: Incomplete [[84edo]].
	*[[43edo\|43]]: Close to 1/5-comma [[meantone]] which gives pure [[15/8]]. Not very notable besides that. Its fifth is too sharp for [[septimal meantone]].		*[[43edo\|43]]: Close to 1/5-comma [[meantone]] which gives pure [[15/8]]. Not very notable besides that. Its fifth is too sharp for [[septimal meantone]].

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	*[[5edo\|5]]: Equalized [[2L 3s\|pentatonic]] scale. [[3-limit]] [[blackwood]]. Kinda familiar but everything is distorted. To me as a Chinese, it sounds like out-of-tone traditional Chinese music. A heavily streched [[slendric]] chain. The smallest EDO that roughly represents [[3-limit]] or 2.3.7 subgroup.		*[[5edo\|5]]: Equalized [[2L 3s\|pentatonic]] scale. [[3-limit]] [[blackwood]]. Kinda familiar but everything is distorted. To me as a Chinese, it sounds like out-of-tone traditional Chinese music. A heavily streched [[slendric]] chain. The smallest EDO that roughly represents [[3-limit]] or 2.3.7 subgroup.
	*[[6edo\|6]]: [[12edo]] whole tones. Incomplete 12edo. A heavily stretched [[didacus]] chain.		*[[6edo\|6]]: [[12edo]] whole tones. Incomplete 12edo. A heavily stretched [[didacus]] chain.
	*[[7edo\|7]]: Equalized [[5L 2s\|diatonic]] scale. [[3-limit]] [[whitewood]]. Similar to [[5edo]], it sounds like out-of-tone [[3L 4s\|neutral scale]] music. The smallest EDO that roughly represents [[5-limit]]. Around 3/4-comma [[meantone]].		*[[7edo\|7]]: Equalized [[5L 2s\|diatonic]] scale. [[3-limit]] [[whitewood]]. Similar to [[5edo]], it sounds like out-of-tone [[3L 4s\|neutral scale]] music. The smallest EDO that roughly represents [[5-limit]]. Around [[3/4-comma meantone\|3/4-comma]] [[meantone]].
	*[[8edo\|8]]: Incomplete [[24edo]].		*[[8edo\|8]]: Incomplete [[24edo]].
	*[[9edo\|9]]: A stack of [[7/6]]. A subset of [[ennealimmal]].		*[[9edo\|9]]: A stack of [[7/6]]. A subset of [[ennealimmal]].
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	*[[15edo\|15]]: Better than it seems to be, tho still rough. [[Blackwood]] [[augmented (temperament)\|augmented]]. A heavily stretched [[Carlos Alpha]] scale. The best you can get with blackwood. I don't know why there are “people fond of” such inaccurate temperaments.		*[[15edo\|15]]: Better than it seems to be, tho still rough. [[Blackwood]] [[augmented (temperament)\|augmented]]. A heavily stretched [[Carlos Alpha]] scale. The best you can get with blackwood. I don't know why there are “people fond of” such inaccurate temperaments.
	*...		*...
	*[[17edo\|17]]: A circle of fifths in [[34edo]]. Interesting sharp fifths. The smallest [[5L 2s\|diatonic]] EDO with neutral intervals. The boundary between neogothic and superpyth. Like [[12edo]], its diatonic thirds do not approximate any simple ratios well, and a [[well temperament]] may help. Its [[13/1\|13]] is good, and [[11/1\|11]] and [[7/1\|7]] have a similar precision to 12edo's [[5/1\|5]]. Even more suitable for making [[well temperament]]s than [[12edo]] since there are ~~mor~~ target intervals. It benefits from compression.		*[[17edo\|17]]: A circle of fifths in [[34edo]]. Interesting sharp fifths. The smallest [[5L 2s\|diatonic]] EDO with neutral intervals. The boundary between neogothic and superpyth. Like [[12edo]], its diatonic thirds do not approximate any simple ratios well, and a [[well temperament]] may help. Its [[13/1\|13]] is good, and [[11/1\|11]] and [[7/1\|7]] have a similar precision to 12edo's [[5/1\|5]]. Even more suitable for making [[well temperament]]s than [[12edo]] since there are more target intervals. It benefits from compression.
	*[[18edo\|18]]: Incomplete [[36edo]].		*[[18edo\|18]]: Incomplete [[36edo]].
	*[[19edo\|19]]: A stack of [[5/3]]. Very different tradeoffs from [[12edo]]. Usable but imperfect for many temperaments. [[Meantone]] or [[magic]], but with a too flat fifth. Strangely large minor 2nds off from [[16/15]] even more than 12edo's, kinda between a typical semitone and a neutral second, shaping its melody, making some pieces sound bad. A compressed [[Carlos Beta]] scale. For meantone [[31edo]] is more preferable, and for magic [[41edo]]. The lower bound of a good fifth. It benefits from stretching. Good as a subset of [[enneadecal]].		*[[19edo\|19]]: A stack of [[5/3]]. Very different tradeoffs from [[12edo]]. Usable but imperfect for many temperaments. [[Meantone]] or [[magic]], but with a too flat fifth. Strangely large minor 2nds off from [[16/15]] even more than 12edo's, kinda between a typical semitone and a neutral second, shaping its melody, making some pieces sound bad. A compressed [[Carlos Beta]] scale. For meantone [[31edo]] is more preferable, and for magic [[41edo]]. The lower bound of a good fifth. It benefits from stretching. Good as a subset of [[enneadecal]].

← Older revision		Revision as of 05:34, 12 March 2026
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	*[[22edo\|22]]: The smallest EDO with decent [[11-limit]] and the smallest non-meantone EDO with decent [[5-limit]]. [[Superpyth]] and [[porcupine]]. Close to optimal 2.3.7 [[archy]] with compression. With pure octaves it is almost the best archy can give, since archy highly relies on octave compression. The upper bound of a good fifth. The best you can get with [[50/49]] tempered out.		*[[22edo\|22]]: The smallest EDO with decent [[11-limit]] and the smallest non-meantone EDO with decent [[5-limit]]. [[Superpyth]] and [[porcupine]]. Close to optimal 2.3.7 [[archy]] with compression. With pure octaves it is almost the best archy can give, since archy highly relies on octave compression. The upper bound of a good fifth. The best you can get with [[50/49]] tempered out.
	*[[23edo\|23]]: Incomplete [[46edo]]. The largest EDO without a [[5L 2s\|diatonic]], [[5edo\|blackwood]] or [[7edo\|whitewood]] fifth.		*[[23edo\|23]]: Incomplete [[46edo]]. The largest EDO without a [[5L 2s\|diatonic]], [[5edo\|blackwood]] or [[7edo\|whitewood]] fifth.
	*[[24edo\|24]]: What some non-microtonalists think microtonality is. 12edo with neutrals. Good for prime [[11/1\|11]]. Accurate in subgroup 2.3.11.17.19.		*[[24edo\|24]]: What some non-microtonalists think microtonality is. [[12edo]] with neutrals. Good for prime [[11/1\|11]]. Accurate in subgroup 2.3.11.17.19. Nearly optimal for 2.3.11.19 tempering out [[243/242]] and [[513/512]], where the next EDO that significanly improves is [[89edo]].
	*...		*...
	*[[26edo\|26]]: The fact that [[53edo]] is good indicates that 26- and 27edo are probably bad. A stack of [[7/4]]. Good for 2.7.11 subgroup. Other intervals suck. Since it is relatively small, consistency does not implies high accuracy. [[Meantone]] in [[5-limit]] but tuned terribly. Incomplete [[130edo]].		*[[26edo\|26]]: The fact that [[53edo]] is good indicates that 26- and 27edo are probably bad. A stack of [[7/4]]. Good for 2.7.11 subgroup. Other intervals suck. Since it is relatively small, consistency does not implies high accuracy. [[Meantone]] in [[5-limit]] but tuned terribly. Incomplete [[130edo]].
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	*[[72edo\|72]]: The ultimate extension of [[12edo]] and [[24edo]]. Its [[11-limit]] is very accurate with a slightly flat tendency that works well with 12edo's flat [[3/1\|3]]. A real [[miracle]] (pun intended). The only reasonable way of extending [[compton]] to 11-limit. Some higher limit intervals are also usable. The only notable [[13-limit]] non-[[akea]] EDO around this size. A powerful tool for modulating quartertones, which can be difficult in 24edo itself. Suitable for octave stretching if only [[17-limit]] or below is used. Playable by using three 24edo instruments or six 12edo instruments.		*[[72edo\|72]]: The ultimate extension of [[12edo]] and [[24edo]]. Its [[11-limit]] is very accurate with a slightly flat tendency that works well with 12edo's flat [[3/1\|3]]. A real [[miracle]] (pun intended). The only reasonable way of extending [[compton]] to 11-limit. Some higher limit intervals are also usable. The only notable [[13-limit]] non-[[akea]] EDO around this size. A powerful tool for modulating quartertones, which can be difficult in 24edo itself. Suitable for octave stretching if only [[17-limit]] or below is used. Playable by using three 24edo instruments or six 12edo instruments.
	*...		*...
	*[[74edo\|74]]: [[37edo]] with meantone fifths. Close to [[tungsten meantone]]. The intrinsic error of meantone becomes a problem at this size, making [[9/1\|9]] inconsistent.		*[[74edo\|74]]: [[37edo]] with [[meantone]] fifths. Close to [[tungsten meantone]]. The intrinsic error of meantone becomes a problem at this size, making [[9/1\|9]] inconsistent.
	*...		*...
	*[[77edo\|77]]: Good for [[valentine]]. Its slightly flat [[3/2]] gives a good [[19/1\|19]] via [[boethius~~]]. Containing [[Carlos Alpha~~]]. [[40/27]] as 4\7 aka [[absurdity]]. Usable for high limit JI. At this size even some inconsistent intervals are usable via val mapping since its step size is only ~16 cents.		*[[77edo\|77]]: Good for [[valentine]] therefore containing [[Carlos Alpha]]. Its slightly flat [[3/2]] gives a good [[19/1\|19]] via [[boethius]]. [[40/27]] as 4\[[7edo\|7]] aka [[absurdity]]. Usable for high limit JI. At this size even some inconsistent intervals are usable via val mapping since its step size is only ~16 cents.
	*...		*...
	*[[81edo\|81]]: The [[optimal patent val]] for [[meantone]] and some of its higher-limit extentions, but I won't use such a large EDO for a temperament with relatively low accuracy, and rather use [[golden meantone]] instead, which is simpler and more elegant mathematically.		*[[81edo\|81]]: The [[optimal patent val]] for [[meantone]] and some of its higher-limit extentions, but I won't use such a large EDO for a temperament with relatively low accuracy, and rather use [[golden meantone]] instead, which is simpler and more elegant mathematically.

← Older revision		Revision as of 07:01, 14 March 2026
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	*[[15edo\|15]]: Better than it seems to be, tho still rough. [[Blackwood]] [[augmented (temperament)\|augmented]]. A heavily stretched [[Carlos Alpha]] scale. The best you can get with blackwood. I don't know why there are “people fond of” such inaccurate temperaments.		*[[15edo\|15]]: Better than it seems to be, tho still rough. [[Blackwood]] [[augmented (temperament)\|augmented]]. A heavily stretched [[Carlos Alpha]] scale. The best you can get with blackwood. I don't know why there are “people fond of” such inaccurate temperaments.
	*...		*...
	*[[17edo\|17]]: A circle of fifths in [[34edo]]. Interesting sharp fifths. The smallest [[5L 2s\|diatonic]] EDO with neutral intervals. The boundary between neogothic and superpyth. Like [[12edo]], its diatonic thirds do not approximate any simple ratios well~~, and a [[well temperament]] may help~~. Its [[13/1\|13]] is good, and [[11/1\|11]] and [[7/1\|7]] have a similar precision to 12edo's [[5/1\|5]]. Even more suitable for making [[well temperament]]s than [[12edo]] since there are more target intervals. It benefits from compression.		*[[17edo\|17]]: A circle of fifths in [[34edo]]. Interesting sharp fifths. The smallest [[5L 2s\|diatonic]] EDO with neutral intervals. The boundary between neogothic and superpyth. Like [[12edo]], its diatonic thirds do not approximate any simple ratios well. Its [[13/1\|13]] is good, and [[11/1\|11]] and [[7/1\|7]] have a similar precision to 12edo's [[5/1\|5]], giving decent no-5 [[13-limit]]. Even more suitable for making [[well temperament]]s than [[12edo]] since there are more target intervals. It benefits from compression.
	*[[18edo\|18]]: Incomplete [[36edo]].		*[[18edo\|18]]: Incomplete [[36edo]].
	*[[19edo\|19]]: A stack of [[5/3]]. Very different tradeoffs from [[12edo]]. Usable but imperfect for many temperaments. [[Meantone]] or [[magic]], but with a too flat fifth. Strangely large minor 2nds off from [[16/15]] even more than 12edo's, kinda between a typical semitone and a neutral second, shaping its melody, making some pieces sound bad. A compressed [[Carlos Beta]] scale. For meantone [[31edo]] is more preferable, and for magic [[41edo]]. The lower bound of a good fifth. It benefits from stretching. Good as a subset of [[enneadecal]].		*[[19edo\|19]]: A stack of [[5/3]]. Close to [[1/3-comma meantone\|1/3-comma]] [[meantone]]. Very different tradeoffs from [[12edo]]. Usable but imperfect for many temperaments. Meantone or [[magic]], but with a too flat fifth. Strangely large minor 2nds off from [[16/15]] even more than 12edo's, kinda between a typical semitone and a neutral second, shaping its melody, making some pieces sound bad. A compressed [[Carlos Beta]] scale. For meantone [[31edo]] is more preferable, and for magic [[41edo]]. The lower bound of a good fifth. It benefits from stretching. Good as a subset of [[enneadecal]].
	*[[20edo\|20]]: The fact that [[41edo]] is good indicates that 20- and 21edo are probably bad. Does anyone really think it is OK for a chord to contain a lot of inconsistent mappings involving 3?		*[[20edo\|20]]: The fact that [[41edo]] is good indicates that 20- and 21edo are probably bad. Does anyone really think it is OK for a chord to contain a lot of inconsistent mappings involving 3?
	*...		*...
	*[[22edo\|22]]: The smallest EDO with decent [[11-limit]] and the smallest non-meantone EDO with decent [[5-limit]]. [[Superpyth]] and [[porcupine]]. Close to optimal 2.3.7 [[archy]] with compression. With pure octaves it is almost the best archy can give, since archy highly relies on octave compression. The upper bound of a good fifth. The best you can get with [[50/49]] tempered out.		*[[22edo\|22]]: The smallest EDO with decent [[11-limit]] and the smallest non-meantone EDO with decent [[5-limit]]. [[Superpyth]] and [[porcupine]]. Close to optimal 2.3.7 [[archy]] with compression. With pure octaves it is almost the best archy and its extensions can give, since archy highly relies on octave compression. The upper bound of a good fifth. The best you can get with [[50/49]] tempered out.
	*[[23edo\|23]]: Incomplete [[46edo]]. The largest EDO without a [[5L 2s\|diatonic]], [[5edo\|blackwood]] or [[7edo\|whitewood]] fifth.		*[[23edo\|23]]: Incomplete [[46edo]]. The largest EDO without a [[5L 2s\|diatonic]], [[5edo\|blackwood]] or [[7edo\|whitewood]] fifth.
	*[[24edo\|24]]: What some non-microtonalists think microtonality is. [[12edo]] with neutrals~~. Good for prime~~ [[11/1\|11]]. Accurate in subgroup 2.3.11.17.19. Nearly optimal for 2.3.11.19 tempering out [[243/242]] and [[513/512]], where the next EDO that ~~significanly~~ improves is [[89edo]].		*[[24edo\|24]]: What some non-microtonalists think microtonality is. [[12edo]] with neutrals, giving good [[11/1\|11]]. Accurate in subgroup 2.3.11.17.19. Nearly optimal for 2.3.11.19 tempering out [[243/242]] and [[513/512]], where the next EDO that significantly improves is [[89edo]].
	*...		*...
	*[[26edo\|26]]: The fact that [[53edo]] is good indicates that 26- and 27edo are probably bad. A stack of [[7/4]]. Good for 2.7.11 subgroup. Other intervals suck. Since it is relatively small, consistency does not implies high accuracy. [[Meantone]] in [[5-limit]] but tuned terribly. Incomplete [[130edo]].		*[[26edo\|26]]: The fact that [[53edo]] is good indicates that 26- and 27edo are probably bad. A stack of [[7/4]]. Good for 2.7.11 subgroup. Other intervals suck. Since it is relatively small, consistency does not implies high accuracy. [[Meantone]] in [[5-limit]] but tuned terribly. Incomplete [[130edo]].
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	*[[29edo\|29]]: A circle of fifths in [[mystery]], which supports [[pele]] (and is close to its optimal tuning), a convenient temperament with [[5/1\|5]], [[7/1\|7]], [[11/1\|11]] and [[13/1\|13]] on the same chain of fifths. Near pure [[13/11]]. The smallest EDO with a better [[3/2]] than [[12edo]].		*[[29edo\|29]]: A circle of fifths in [[mystery]], which supports [[pele]] (and is close to its optimal tuning), a convenient temperament with [[5/1\|5]], [[7/1\|7]], [[11/1\|11]] and [[13/1\|13]] on the same chain of fifths. Near pure [[13/11]]. The smallest EDO with a better [[3/2]] than [[12edo]].
	*...		*...
	*[[31edo\|31]]: Ideal for pure-octave [[meantone]], combining lots of [[11-limit]] extensions in a single tuning. The meantone flat fifth makes its neutral thirds close to [[11/9]] (tho there is the JI subgroup problem since [[11/1\|11]] and [[9/1\|9]] themselves are not so good). The intersection of [[valentine]] and [[miracle]]. IMO the best meantone EDO. Nearly the best meantone can give for high limit. For other temperaments its flat fifth may be a drawback, making 9 bad. A possibly problematic interval is its [[14/11]]~[[9/7]] which is off from both JI intervals.		*[[31edo\|31]]: Ideal for pure-octave [[meantone]], combining lots of [[11-limit]] extensions in a single tuning. Near optimal 2.5.7 [[didacus]]. The meantone flat fifth makes its neutral thirds close to [[11/9]] (tho there is the JI subgroup problem since [[11/1\|11]] and [[9/1\|9]] themselves are not so good). The intersection of [[valentine]] and [[miracle]]. IMO the best meantone EDO. Nearly the best meantone can give for high limit. For other temperaments its flat fifth may be a drawback, making 9 bad. A possibly problematic interval is its [[14/11]]~[[9/7]] which is off from both JI intervals.
	*...		*...
	*[[34edo\|34]]: [[17edo]] with [[5/1\|5]] and [[17/1\|17]] added, making a good 2.3.5.13.17 system. A slightly stretched [[Carlos Gamma]] scale.		*[[34edo\|34]]: [[17edo]] with [[5/1\|5]] and [[17/1\|17]] added, making a good 2.3.5.13.17 system. A slightly stretched [[Carlos Gamma]] scale.
	*[[35edo\|35]]: The largest non-[[5L 2s\|diatonic]] EDO.		*[[35edo\|35]]: The largest non-[[5L 2s\|diatonic]] EDO.
	*[[36edo\|36]]: Good for 2.3.7.13.17.19.23.29 subroup. Avoid [[5/1\|5]] here because it is almost completely missed. Otherwise incomplete [[72edo]].		*[[36edo\|36]]: [[12edo]] with [[7/1\|7]] added. Good for 2.3.7.13.17.19.23.29 subroup. Avoid [[5/1\|5]] here because it is almost completely missed. Otherwise incomplete [[72edo]].
	*[[37edo\|37]]: A strong no-[[3/1\|3]] system, which is kinda hard to use since the only isoharmonic chords are subsets of the 3n+1 and 3n+2 series, and there are not many useful scales.		*[[37edo\|37]]: A strong no-[[3/1\|3]] system, which is kinda hard to use since the only isoharmonic chords are subsets of the 3n+1 and 3n+2 series, and there are not many useful scales.
	*[[38edo\|38]]: [[19edo]] with neutrals. Near pure [[11/9]]. Doubling such a coarse EDO won't give anything very notable, and the acceptable error of 19edo really becomes a problem at this size.		*[[38edo\|38]]: [[19edo]] with neutrals. Near pure [[11/9]]. Doubling such a coarse EDO won't give anything very notable, and the acceptable error of 19edo really becomes a problem at this size.
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	*[[42edo\|42]]: Incomplete [[84edo]].		*[[42edo\|42]]: Incomplete [[84edo]].
	*[[43edo\|43]]: Close to [[1/5-comma meantone\|1/5-comma]] [[meantone]] which gives pure [[15/8]]. Not very notable besides that. Its fifth is too sharp for [[septimal meantone]].		*[[43edo\|43]]: Close to [[1/5-comma meantone\|1/5-comma]] [[meantone]] which gives pure [[15/8]]. Not very notable besides that. Its fifth is too sharp for [[septimal meantone]].
	*[[44edo\|44]]: Like [[38edo]], doubling a coarse EDO won't give anything very notable.		*[[44edo\|44]]: [[22edo]] with neutrals. Like [[38edo]], doubling a coarse EDO won't give anything very notable.
	*...		*...
	*[[46edo\|46]]: [[13-limit]] [[diaschismic]] and [[valentine]]. Near pure [[11/7]]. It has quartertones similar to [[22edo]] but approximates JI intervals more accurately. Its 30::36 are all 2 steps apart.		*[[46edo\|46]]: [[13-limit]] [[diaschismic]] and [[valentine]]. Near pure [[11/7]]. It has quartertones with mappings similar to [[22edo]] but approximates JI intervals more accurately. Its 30::36 are all 2 steps apart.
	*[[47edo\|47]]: Incomplete [[94edo]].		*[[47edo\|47]]: Incomplete [[94edo]].
	*[[48edo\|48]]: A not-so-good multiple of [[12edo]].		*[[48edo\|48]]: A not-so-good multiple of [[12edo]].
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	*[[50edo\|50]]: [[Meantone]] with a flatter fifth than [[31edo]], but I usually use [[golden meantone]] (with slight octave stretching) for this range, instead of using a large EDO for an inaccurate temperament.		*[[50edo\|50]]: [[Meantone]] with a flatter fifth than [[31edo]], but I usually use [[golden meantone]] (with slight octave stretching) for this range, instead of using a large EDO for an inaccurate temperament.
	*...		*...
	*[[53edo\|53]]: A stack of [[3/2]]. Almost just [[3/1\|3]], accurate [[5-limit]], and decent [[7-limit]]. The smallest EDO that shows the accuracy of [[schismatic]] like [[12edo]] for [[meantone]]. Good for music that emphasizes [[5-limit]].		*[[53edo\|53]]: A stack of [[3/2]]. Almost just [[3/1\|3]], accurate [[5-limit]], and decent [[7-limit]]. The smallest EDO that shows the accuracy of [[schismatic]] like [[12edo]] for [[meantone]]. Good for music that emphasizes 5-limit.
	*...		*...
	*[[60edo\|60]]: [[12edo]] with [[13/1\|13]] and [[magic]]-tempered [[5/1\|5]] added. 5, [[7/1\|7]] and [[11/1\|11]] are quite off. Streching can help but it damages the good 13.		*[[60edo\|60]]: [[12edo]] with [[13/1\|13]] and [[magic]]-tempered [[5/1\|5]] added. 5, [[7/1\|7]] and [[11/1\|11]] are quite off. Streching can help but it damages the good 13.
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	*[[65edo\|65]]: A circle of fifths in [[130edo]].		*[[65edo\|65]]: A circle of fifths in [[130edo]].
	*...		*...
	*[[68edo\|68]]: Good for no-11 [[23-limit]]. I may explore it some day.		*[[68edo\|68]]: Good for no-[[11/1\|11]] [[23-limit]]. It is amazing that [[17edo]]'s [[3-limit]] still works at this size. I may explore it some day.
	*...		*...
	*[[72edo\|72]]: The ultimate extension of [[12edo]] and [[24edo]]. Its [[11-limit]] is very accurate with a slightly flat tendency that works well with 12edo's flat [[3/1\|3]]. A real [[miracle]] (pun intended). The only reasonable way of extending [[compton]] to 11-limit. Some higher limit intervals are also usable. The only notable [[13-limit]] non-[[akea]] EDO around this size. ~~A powerful tool for modulating quartertones, which can be difficult~~ in 24edo ~~itself~~. Suitable for octave stretching if only [[17-limit]] or below is used. Playable by using three 24edo instruments or six 12edo instruments.		*[[72edo\|72]]: The ultimate extension of [[12edo]] and [[24edo]]. Its [[11-limit]] is very accurate with a slightly flat tendency that works well with 12edo's flat [[3/1\|3]]. A real [[miracle]] (pun intended). The only reasonable way of extending [[compton]] to 11-limit. Some higher limit intervals are also usable. The only notable [[13-limit]] non-[[akea]] EDO around this size. It gives many useful interpretations of the quartertone in 24edo, now besides ~[[33/32]] it is also ~[[36/35]] and ~[[128/125]]. Suitable for octave stretching if only [[17-limit]] or below is used. Playable by using three 24edo instruments or six 12edo instruments.
	*...		*...
	*[[74edo\|74]]: [[37edo]] with [[meantone]] fifths. Close to [[tungsten meantone]]. The intrinsic error of meantone becomes a problem at this size, making [[9/1\|9]] inconsistent.		*[[74edo\|74]]: [[37edo]] with [[meantone]] fifths. Close to [[tungsten meantone]]. The intrinsic error of meantone becomes a problem at this size, making [[9/1\|9]] inconsistent.

← Older revision		Revision as of 12:45, 13 March 2026
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	*[[41edo\|41]]: Prime steps in an octave and highly composite steps in a fifth, opposite from [[12edo]], thus good for fifth-dividing temperaments. Containing [[Bohlen-Pierce scale]]. Good for [[magic]]. The [[Kite guitar]] shows its elegance, with many simple intervals evenly spaced. Also [[garibaldi]] and [[miracle]]. The largest problem is its relatively inaccurate [[5/1\|5]]. From here on, most EDOs with good [[13-limit]] support [[akea]].		*[[41edo\|41]]: Prime steps in an octave and highly composite steps in a fifth, opposite from [[12edo]], thus good for fifth-dividing temperaments. Containing [[Bohlen-Pierce scale]]. Good for [[magic]]. The [[Kite guitar]] shows its elegance, with many simple intervals evenly spaced. Also [[garibaldi]] and [[miracle]]. The largest problem is its relatively inaccurate [[5/1\|5]]. From here on, most EDOs with good [[13-limit]] support [[akea]].
	*[[42edo\|42]]: Incomplete [[84edo]].		*[[42edo\|42]]: Incomplete [[84edo]].
	*[[43edo\|43]]: Close to 1/5-comma [[meantone]] which gives pure [[15/8]]. Not very notable besides that. Its fifth is too sharp for [[septimal meantone]].		*[[43edo\|43]]: Close to [[1/5-comma meantone\|1/5-comma]] [[meantone]] which gives pure [[15/8]]. Not very notable besides that. Its fifth is too sharp for [[septimal meantone]].
	*[[44edo\|44]]: Like [[38edo]], doubling a coarse EDO won't give anything very notable.		*[[44edo\|44]]: Like [[38edo]], doubling a coarse EDO won't give anything very notable.
	*...		*...
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	*[[53edo\|53]]: A stack of [[3/2]]. Almost just [[3/1\|3]], accurate [[5-limit]], and decent [[7-limit]]. The smallest EDO that shows the accuracy of [[schismatic]] like [[12edo]] for [[meantone]]. Good for music that emphasizes [[5-limit]].		*[[53edo\|53]]: A stack of [[3/2]]. Almost just [[3/1\|3]], accurate [[5-limit]], and decent [[7-limit]]. The smallest EDO that shows the accuracy of [[schismatic]] like [[12edo]] for [[meantone]]. Good for music that emphasizes [[5-limit]].
	*...		*...
	*[[60edo\|60]]: [[12edo]] with [[13/1\|13]] and [[magic]]-tempered [[5/1\|5]] added.		*[[60edo\|60]]: [[12edo]] with [[13/1\|13]] and [[magic]]-tempered [[5/1\|5]] added. 5, [[7/1\|7]] and [[11/1\|11]] are quite off. Streching can help but it damages the good 13.
	*...		*...
	*[[65edo\|65]]: A circle of fifths in [[130edo]].		*[[65edo\|65]]: A circle of fifths in [[130edo]].
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	*[[77edo\|77]]: Good for [[valentine]] therefore containing [[Carlos Alpha]]. Its slightly flat [[3/2]] gives a good [[19/1\|19]] via [[boethius]]. [[40/27]] as 4\[[7edo\|7]] aka [[absurdity]]. Usable for high limit JI. At this size even some inconsistent intervals are usable via val mapping since its step size is only ~16 cents.		*[[77edo\|77]]: Good for [[valentine]] therefore containing [[Carlos Alpha]]. Its slightly flat [[3/2]] gives a good [[19/1\|19]] via [[boethius]]. [[40/27]] as 4\[[7edo\|7]] aka [[absurdity]]. Usable for high limit JI. At this size even some inconsistent intervals are usable via val mapping since its step size is only ~16 cents.
	*...		*...
	*[[81edo\|81]]: The [[optimal patent val]] for [[meantone]] and some of its higher-limit extentions, but I ~~won't use such a large EDO for a temperament with relatively low accuracy, and~~ rather use [[golden meantone]] instead~~, which is simpler and more elegant mathematically~~.		*[[81edo\|81]]: The [[optimal patent val]] for [[meantone]] and some of its higher-limit extentions, but like [[50edo]], I will rather use [[golden meantone]] instead.
	*...		*...
	*[[84edo\|84]]: An alternate [[compton]] EDO besides [[72edo]], with better [[5/1\|5]] and [[13/1\|13]] with the expense of [[11/1\|11]]. It has a sharp tendency instead of 72edo's flat.		*[[84edo\|84]]: An alternate [[compton]] EDO besides [[72edo]], with better [[5/1\|5]] and [[13/1\|13]] with the expense of [[11/1\|11]]. It has a sharp tendency instead of 72edo's flat.
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	*[[118edo\|118]]: The relationship of [[53edo\|53]]-118-[[171edo]] for [[schismatic]] is similar to [[12edo\|12]]-[[19edo\|19]]-[[31edo]] for meantone. 53 and 12 are the simplest reasonable EDO with very mildly tempered fifths, 118 and 19 are better over all but slightly overtempered (outside 5-odd-limit [[diamond tradeoff]]), and 171 and 31 are ideal. So like 19, I won't appreciate it much.		*[[118edo\|118]]: The relationship of [[53edo\|53]]-118-[[171edo]] for [[schismatic]] is similar to [[12edo\|12]]-[[19edo\|19]]-[[31edo]] for meantone. 53 and 12 are the simplest reasonable EDO with very mildly tempered fifths, 118 and 19 are better over all but slightly overtempered (outside 5-odd-limit [[diamond tradeoff]]), and 171 and 31 are ideal. So like 19, I won't appreciate it much.
	*...		*...
	*[[130edo\|130]]: Good [[13-limit]] EDO. It doesn't support [[akea]] and has a flat [[3/1\|3]] instead of sharp, unlike smaller 13-limit EDOs.		*[[130edo\|130]]: Good [[13-limit]] EDO. It doesn't support [[akea]] and has a flat [[3/1\|3]] instead of sharp, unlike smaller 13-limit EDOs. It may need more exploration.
	*...		*...
	*[[171edo\|171]]: The ultimate EDO for approximating [[7-limit]] JI. It suggests very slight stretching. If you don't need some ridiculous high precision or specific microtemperaments, there is no need to go any further. [[Schismatic]], [[gammic]], [[ennealimmal]] and [[enneadecal]]. Containing a better [[Carlos Gamma]] scale than [[34edo]]. At this size level, EDOs are more like free pitch, rather than either JI or a stable temperament. Ideal for free-pitch-like music that emphasizes 7-limit.		*[[171edo\|171]]: The ultimate EDO for approximating [[7-limit]] JI. It suggests very slight stretching. If you don't need some ridiculous high precision or specific microtemperaments, there is no need to go any further. [[Schismatic]], [[gammic]], [[ennealimmal]] and [[enneadecal]]. Containing a better [[Carlos Gamma]] scale than [[34edo]]. At this size level, EDOs are more like free pitch, rather than either JI or a stable temperament. Ideal for free-pitch-like music that emphasizes 7-limit.

← Older revision		Revision as of 02:07, 13 March 2026
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	*[[5edo\|5]]: Equalized [[2L 3s\|pentatonic]] scale. [[3-limit]] [[blackwood]]. Kinda familiar but everything is distorted. To me as a Chinese, it sounds like out-of-tone traditional Chinese music. A heavily streched [[slendric]] chain. The smallest EDO that roughly represents [[3-limit]] or 2.3.7 subgroup.		*[[5edo\|5]]: Equalized [[2L 3s\|pentatonic]] scale. [[3-limit]] [[blackwood]]. Kinda familiar but everything is distorted. To me as a Chinese, it sounds like out-of-tone traditional Chinese music. A heavily streched [[slendric]] chain. The smallest EDO that roughly represents [[3-limit]] or 2.3.7 subgroup.
	*[[6edo\|6]]: [[12edo]] whole tones. Incomplete 12edo. A heavily stretched [[didacus]] chain.		*[[6edo\|6]]: [[12edo]] whole tones. Incomplete 12edo. A heavily stretched [[didacus]] chain.
	*[[7edo\|7]]: Equalized [[5L 2s\|diatonic]] scale. [[3-limit]] [[whitewood]]. Similar to [[5edo]], it sounds like out-of-tone [[3L 4s\|neutral scale]] music. The smallest EDO that roughly represents [[5-limit]].		*[[7edo\|7]]: Equalized [[5L 2s\|diatonic]] scale. [[3-limit]] [[whitewood]]. Similar to [[5edo]], it sounds like out-of-tone [[3L 4s\|neutral scale]] music. The smallest EDO that roughly represents [[5-limit]]. Around 3/4-comma [[meantone]].
	*[[8edo\|8]]: Incomplete [[24edo]].		*[[8edo\|8]]: Incomplete [[24edo]].
	*[[9edo\|9]]: A stack of [[7/6]]. A subset of [[ennealimmal]].		*[[9edo\|9]]: A stack of [[7/6]]. A subset of [[ennealimmal]].
	*[[10edo\|10]]: A stack of [[13/8]]. A subset of [[130edo]] and [[270edo]].		*[[10edo\|10]]: A stack of [[13/8]]. A subset of [[130edo]] and [[270edo]]. It is a zeta peak but most approximations it gives are pretty rough.
	*[[11edo\|11]]: Incomplete [[22edo]].		*[[11edo\|11]]: Incomplete [[22edo]].
	*[[12edo\|12]]: Equalized [[5L 7s\|chromatic]] scale. Both [[augmented (temperament)\|augmented]] and [[diminished (temperament)\|diminished]]. The smallest EDO with decent [[3-limit\|3-]], [[5-limit\|5-]] and [[7-limit]]. The only reasonable tuning for the [[dominant (temperament)\|dominant]] temperament, ignoring the difference of overall streching. A sharper fifth makes [[garibaldi]] better and a flatter fifth makes [[septimal meantone]] better. The boundary between [[meantone]] and [[schismatic]]. The smallest [[5L 2s\|diatonic]] EDO. Efficient at its size. Very excellent [[3/1\|3]] as well as good [[17/1\|17]] and [[19/1\|19]] for its size, but inaccurate [[5/1\|5]] and worse [[7/1\|7]]. Since its 7 is around twice as off as 5, adding a comma-sized generator gives [[7-limit]] [[compton]], a simple and accurate rank-2 temperament which is friendly to 12edo-based softwares. Suitable for symmetric scales. Easy to make [[../12neji\|accurate NEJIs]]. Its 2.3.17.19 subgroup deserves more exploration (something “xenharmonic” but not “microtonal”). Its thirds do not accurately approximate common JI intervals. A [[well temperament]] can make some of them do better. AFAIK 12edo and [[17edo]] are the most suitable EDOs for making well temperaments.		*[[12edo\|12]]: Equalized [[5L 7s\|chromatic]] scale. Both [[augmented (temperament)\|augmented]] and [[diminished (temperament)\|diminished]]. The smallest EDO with decent [[3-limit\|3-]], [[5-limit\|5-]] and [[7-limit]]. The only reasonable tuning for the [[dominant (temperament)\|dominant]] temperament, ignoring the difference of overall streching. A sharper fifth makes [[garibaldi]] better and a flatter fifth makes [[septimal meantone]] better. The boundary between [[meantone]] and [[schismatic]]. The smallest [[5L 2s\|diatonic]] EDO. Efficient at its size. Very excellent [[3/1\|3]] as well as good [[17/1\|17]] and [[19/1\|19]] for its size, but inaccurate [[5/1\|5]] and worse [[7/1\|7]]. Since its 7 is around twice as off as 5, adding a comma-sized generator gives [[7-limit]] [[compton]], a simple and accurate rank-2 temperament which is friendly to 12edo-based softwares. Suitable for symmetric scales. Easy to make [[../12neji\|accurate NEJIs]]. Its 2.3.17.19 subgroup deserves more exploration (something “xenharmonic” but not “microtonal”). Its thirds do not accurately approximate common JI intervals. A [[well temperament]] can make some of them do better. AFAIK 12edo and [[17edo]] are the most suitable EDOs for making well temperaments.
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	*[[15edo\|15]]: Better than it seems to be, tho still rough. [[Blackwood]] [[augmented (temperament)\|augmented]]. A heavily stretched [[Carlos Alpha]] scale. The best you can get with blackwood. I don't know why there are “people fond of” such inaccurate temperaments.		*[[15edo\|15]]: Better than it seems to be, tho still rough. [[Blackwood]] [[augmented (temperament)\|augmented]]. A heavily stretched [[Carlos Alpha]] scale. The best you can get with blackwood. I don't know why there are “people fond of” such inaccurate temperaments.
	*...		*...
	*[[17edo\|17]]: A circle of fifths in [[34edo]]. Interesting sharp fifths. The smallest [[5L 2s\|diatonic]] EDO with neutral intervals. The boundary between neogothic and superpyth. Like [[12edo]], its diatonic thirds do not approximate any simple ratios well, and a [[well temperament]] may help. Its [[13/1\|13]] is good, and [[11/1\|11]] and [[7/1\|7]] have a similar precision to 12edo's [[5/1\|5]]. It benefits from compression.		*[[17edo\|17]]: A circle of fifths in [[34edo]]. Interesting sharp fifths. The smallest [[5L 2s\|diatonic]] EDO with neutral intervals. The boundary between neogothic and superpyth. Like [[12edo]], its diatonic thirds do not approximate any simple ratios well, and a [[well temperament]] may help. Its [[13/1\|13]] is good, and [[11/1\|11]] and [[7/1\|7]] have a similar precision to 12edo's [[5/1\|5]]. Even more suitable for making [[well temperament]]s than [[12edo]] since there are mor target intervals. It benefits from compression.
	*[[18edo\|18]]: Incomplete [[36edo]].		*[[18edo\|18]]: Incomplete [[36edo]].
	*[[19edo\|19]]: A stack of [[5/3]]. Very different tradeoffs from [[12edo]]. Usable but imperfect for many temperaments. [[Meantone]] or [[magic]], but with a too flat fifth. Strangely large minor 2nds off from [[16/15]] even more than 12edo's. A compressed [[Carlos Beta]] scale. For meantone [[31edo]] is more preferable, and for magic [[41edo]]. The lower bound of a good fifth. It benefits from stretching. Good as a subset of [[enneadecal]].		*[[19edo\|19]]: A stack of [[5/3]]. Very different tradeoffs from [[12edo]]. Usable but imperfect for many temperaments. [[Meantone]] or [[magic]], but with a too flat fifth. Strangely large minor 2nds off from [[16/15]] even more than 12edo's, kinda between a typical semitone and a neutral second, shaping its melody, making some pieces sound bad. A compressed [[Carlos Beta]] scale. For meantone [[31edo]] is more preferable, and for magic [[41edo]]. The lower bound of a good fifth. It benefits from stretching. Good as a subset of [[enneadecal]].
	*[[20edo\|20]]: The fact that [[41edo]] is good indicates that 20- and 21edo are probably bad. Does anyone really think it is OK for a chord to contain a lot of inconsistent mappings involving 3?		*[[20edo\|20]]: The fact that [[41edo]] is good indicates that 20- and 21edo are probably bad. Does anyone really think it is OK for a chord to contain a lot of inconsistent mappings involving 3?
	*...		*...
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	*[[29edo\|29]]: A circle of fifths in [[mystery]], which supports [[pele]] (and is close to its optimal tuning), a convenient temperament with [[5/1\|5]], [[7/1\|7]], [[11/1\|11]] and [[13/1\|13]] on the same chain of fifths. Near pure [[13/11]]. The smallest EDO with a better [[3/2]] than [[12edo]].		*[[29edo\|29]]: A circle of fifths in [[mystery]], which supports [[pele]] (and is close to its optimal tuning), a convenient temperament with [[5/1\|5]], [[7/1\|7]], [[11/1\|11]] and [[13/1\|13]] on the same chain of fifths. Near pure [[13/11]]. The smallest EDO with a better [[3/2]] than [[12edo]].
	*...		*...
	*[[31edo\|31]]: Ideal for pure-octave [[meantone]], combining lots of [[11-limit]] extensions in a single tuning. The meantone flat fifth makes its neutral thirds close to [[11/9]] (tho there is the JI subgroup problem since [[11/1\|11]] and [[9/1\|9]] themselves are not so good). The intersection of [[valentine]] and [[miracle]]. IMO the best meantone EDO. Nearly the best meantone can give for high limit. For other temperaments its flat fifth may be a drawback, making 9 bad.		*[[31edo\|31]]: Ideal for pure-octave [[meantone]], combining lots of [[11-limit]] extensions in a single tuning. The meantone flat fifth makes its neutral thirds close to [[11/9]] (tho there is the JI subgroup problem since [[11/1\|11]] and [[9/1\|9]] themselves are not so good). The intersection of [[valentine]] and [[miracle]]. IMO the best meantone EDO. Nearly the best meantone can give for high limit. For other temperaments its flat fifth may be a drawback, making 9 bad. A possibly problematic interval is its [[14/11]]~[[9/7]] which is off from both JI intervals.
	*...		*...
	*[[34edo\|34]]: [[17edo]] with [[5/1\|5]] and [[17/1\|17]] added, making a good 2.3.5.13.17 system. A slightly stretched [[Carlos Gamma]] scale.		*[[34edo\|34]]: [[17edo]] with [[5/1\|5]] and [[17/1\|17]] added, making a good 2.3.5.13.17 system. A slightly stretched [[Carlos Gamma]] scale.
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	*[[48edo\|48]]: A not-so-good multiple of [[12edo]].		*[[48edo\|48]]: A not-so-good multiple of [[12edo]].
	*[[49edo\|49]]: Close to optimal [[7-limit]] and [[11-limit]] [[superpyth]] with compression (with the octave around 1197 cents). Useless if you don't specifically want superpyth.		*[[49edo\|49]]: Close to optimal [[7-limit]] and [[11-limit]] [[superpyth]] with compression (with the octave around 1197 cents). Useless if you don't specifically want superpyth.
	*[[50edo\|50]]: [[Meantone]] with a flatter fifth than [[31edo]], but I usually use [[golden meantone]] (with slight octave stretching) for this range.		*[[50edo\|50]]: [[Meantone]] with a flatter fifth than [[31edo]], but I usually use [[golden meantone]] (with slight octave stretching) for this range, instead of using a large EDO for an inaccurate temperament.
	*...		*...
	*[[53edo\|53]]: A stack of [[3/2]]. Almost just [[3/1\|3]], accurate [[5-limit]], and decent [[7-limit]]. The smallest EDO that shows the accuracy of [[schismatic]] like [[12edo]] for [[meantone]]. Good for music that emphasizes [[5-limit]].		*[[53edo\|53]]: A stack of [[3/2]]. Almost just [[3/1\|3]], accurate [[5-limit]], and decent [[7-limit]]. The smallest EDO that shows the accuracy of [[schismatic]] like [[12edo]] for [[meantone]]. Good for music that emphasizes [[5-limit]].
			*...
			*[[60edo\|60]]: [[12edo]] with [[13/1\|13]] and [[magic]]-tempered [[5/1\|5]] added.
	*...		*...
	*[[65edo\|65]]: A circle of fifths in [[130edo]].		*[[65edo\|65]]: A circle of fifths in [[130edo]].
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	*...		*...
	*[[118edo\|118]]: The relationship of [[53edo\|53]]-118-[[171edo]] for [[schismatic]] is similar to [[12edo\|12]]-[[19edo\|19]]-[[31edo]] for meantone. 53 and 12 are the simplest reasonable EDO with very mildly tempered fifths, 118 and 19 are better over all but slightly overtempered (outside 5-odd-limit [[diamond tradeoff]]), and 171 and 31 are ideal. So like 19, I won't appreciate it much.		*[[118edo\|118]]: The relationship of [[53edo\|53]]-118-[[171edo]] for [[schismatic]] is similar to [[12edo\|12]]-[[19edo\|19]]-[[31edo]] for meantone. 53 and 12 are the simplest reasonable EDO with very mildly tempered fifths, 118 and 19 are better over all but slightly overtempered (outside 5-odd-limit [[diamond tradeoff]]), and 171 and 31 are ideal. So like 19, I won't appreciate it much.
			*...
			*[[130edo\|130]]: Good [[13-limit]] EDO. It doesn't support [[akea]] and has a flat [[3/1\|3]] instead of sharp, unlike smaller 13-limit EDOs.
	*...		*...
	*[[171edo\|171]]: The ultimate EDO for approximating [[7-limit]] JI. It suggests very slight stretching. If you don't need some ridiculous high precision or specific microtemperaments, there is no need to go any further. [[Schismatic]], [[gammic]], [[ennealimmal]] and [[enneadecal]]. Containing a better [[Carlos Gamma]] scale than [[34edo]]. At this size level, EDOs are more like free pitch, rather than either JI or a stable temperament. Ideal for free-pitch-like music that emphasizes 7-limit.		*[[171edo\|171]]: The ultimate EDO for approximating [[7-limit]] JI. It suggests very slight stretching. If you don't need some ridiculous high precision or specific microtemperaments, there is no need to go any further. [[Schismatic]], [[gammic]], [[ennealimmal]] and [[enneadecal]]. Containing a better [[Carlos Gamma]] scale than [[34edo]]. At this size level, EDOs are more like free pitch, rather than either JI or a stable temperament. Ideal for free-pitch-like music that emphasizes 7-limit.