User:Zhenlige/EDO impressions: Difference between revisions

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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.

*[[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]].

*[[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.

*...

*[[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. 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]].

*[[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?

*...

*[[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.

*[[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]].

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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]].

*...

*[[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.

*[[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.

*[[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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*[[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]].

*[[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]]: [[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]].

*[[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.

*...

*[[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]].

*...

*[[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.

*[[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. 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.

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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.

*...

*[[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.

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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.

*...

*[[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.

@@ Line 20: / Line 20: @@
 *[[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.
-*[[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]].
 *[[9edo|9]]: A stack of [[7/6]]. A subset of [[ennealimmal]].
@@ Line 29: / Line 29: @@
 *[[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. 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]].
-*[[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?
 *...
-*[[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.
-*[[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]].
@@ Line 43: / Line 43: @@
 *[[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.
 *[[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.
 *[[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.
@@ Line 53: / Line 53: @@
 *[[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]].
-*[[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]]: [[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]].
 *[[48edo|48]]: A not-so-good multiple of [[12edo]].
@@ Line 62: / Line 62: @@
 *[[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.
+*[[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]].
 *...
-*[[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.
+*[[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. 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.
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 *[[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.
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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.
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-*[[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.
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 *[[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.
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-*[[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.
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 *[[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.