User:Zhenlige/EDO impressions: Difference between revisions

*If (2n+1)-edo is good, then n-edo and (n+1)-edo are probably bad, or at least heavily inconsistent, because many intervals near the half octave (especially [[3/2]] and possibly [[5/4]]) will be off.

'''Note: the version here may be edited frequently. A stabler version is on [[Collection of EDO impressions]].'''

 

Most descriptions assume pure octaves because I haven't got enough experience with equal-step tunings with tempered octaves.

 

*[[0edo|0]]: A fancy way to say “no melody”. The only tuning of the [[Single-pitch tuning|Om]] temperament. Important in theory, useless in practice.

*[[1edo|1]]: [[2/1|Octaves]]. Equivalent to [[2-limit]] JI, unless you want to temper some other JI intervals into octaves. Not much to talk about.

*[[2edo|2]]: [[sqrt(2)|Half octaves]] aka symmetric [[tritone]]s. The key to tritone substitution, although sometimes asymmetric tritones or even non-tritone intervals can also be used. Some notable approximations are [[7/5]], [[17/12]] (giving [[17/1|17]] for even EDOs with a good [[3/1|3]]) and [[99/70]] (giving [[kalismic temperaments|kalismic]]). 2n-edos within 1000 that do not support kalismic are probably bad in [[11-limit]].

*[[3edo|3]]: [[12edo]] major thirds. 2.5 subgroup [[augmented (temperament)|augmented]]. The smallest EDO with decent 2.5 subgroup. Treating its steps as [[63/50]] gives [[landscape]]. 3n-edos within 1000 that do not support landscape are probably bad in [[7-limit]].

*[[4edo|4]]: [[12edo]] minor thirds.

*[[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|3/4-comma]] [[meantone]].

*[[9edo|9]]: A stack of [[7/6]]. A subset of [[ennealimmal]].

*[[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.

*[[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.

*[[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.

*[[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 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, 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]].

*[[27edo|27]]: Worse than both [[12edo]] and [[22edo]] for [[5-limit]]. It generally sounds worse than 22edo. Its fifth sucks and its diatonic scale makes little sense. Its [[7/3]] and [[7/5]] are good, but 3, 5 and 7 are off. When I hear its ~[[10:12:15]] and ~[[6:7:9]] chord I feel the fifth is obviously off. Use [[108edo]] (not [[landscape]]) to make it a true [[7-limit]] EDO, and [[270edo]] is excellent.

*[[28edo|28]]: [[Whitewood]] [[diminished (temperament)|diminished]]. Kinda opposite from [[15edo]]. The best you can get with whitewood.

*[[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. 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.

*[[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]].

*[[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]]. 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|1/5-comma]] [[meantone]] which gives pure [[15/8]]. Not very notable besides that. Its fifth is too sharp for [[septimal meantone]].

*[[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 with mappings similar to [[22edo]] but approximates JI intervals more accurately. Its 30::36 are all 2 steps apart.

*[[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.

*[[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.

*[[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]].

*[[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.

*[[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.

*[[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]].

*[[99edo|99]]: Efficient near-[[Logarithmic approximants#Argent tuning|argent]] EDO. It suggests slight compression. Good for [[hemififths]]. It completely misses [[11/1|11]] and [[13/1|13]].

*[[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.

*[[224edo|224]]: Like [[171edo]] but with a slightly sharper (and closer to just) fifth, worse [[7-limit]] but better [[13-limit]].

*[[270edo|270]]: Better than [[224edo]] if [[schismatic]] is not required. Ideal for free-pitch-like music that emphasizes [[13-limit]].

*[[311edo|311]]: Good for very high limit JI.