User:Zhenlige/EDO impressions: Difference between revisions

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

 

*[[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]]: Equivalent to [[2-limit]] JI, unless you want to temper some other JI intervals into octaves. Not much about harmony. Possibly useful for a transition between different tunings.

*[[2edo|2]]: [[sqrt(2)|Half octaves]] aka symmetric [[tritones]]. Some notable approximations are [[7/5]], [[17/12]] and [[99/70]]. The key to tritone substitution, although sometimes asymmetric tritones or even non-tritone intervals can also be used.

*[[3edo|3]]: [[12edo]] augmented chords. 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]] diminished seventh chords.

*[[5edo|5]]: Equalized [[2L 3s|pentatonic]] scale. [[3-limit]] [[blackwood]]. Kinda familiar but everything is warped. To me as a Chinese, it sounds like out-of-tone traditional Chinese music. The smallest EDO containing an interval that roughly represents [[3-limit]] or 2.3.7 subgroup.

*[[7edo|7]]: Equalized [[5L 2s|diatonic]] scale. 3-limit [[whitewood]]. It sounds like out-of-tone [[3L 4s|neutral scale]] music. The smallest EDO that roughly represents [[5-limit]].

*[[12edo|12]]: Equalized [[5L 7s|chromatic]] scale. Both [[augmented (temperament)|augmented]] and [[diminished (temperament)|diminished]]. The only reasonable tuning for the [[dominant (temperament)|dominant]] temperament. The boundary between [[meantone]] and [[schismatic]]. The smallest [[5L 2s|diatonic]] EDO. Efficient at its size. Very excellent [[3/1|3]] as well as prime [[17/1|17]] and [[19/1|19]] for its size, but inaccurate [[5/1|5]] and worse [[7/1|7]]. Suitable for symmetric scales. Easy to make [[../12neji|accurate NEJIs]]. Its 2.3.17.19 subgroup really 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.

*[[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.

*[[19edo|19]]: 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. A stack of [[5/3]]. Good as a subset of [[enneadecal]].

*[[22edo|22]]: The smallest non-meantone EDO with reasonable [[5-limit]]. [[Superpyth]] and [[porcupine]]. The only reasonable superpyth EDO. 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.

*[[26edo|26]]: 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]]: A stack of [[7/6]]. Worse than both [[12edo]] and [[22edo]] for [[5-limit]]. 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]] to make it a true [[7-limit]] EDO, and [[270edo]] is excellent. The fact that [[53edo]] is good indicates that 26- and 27edo are probably bad.

*[[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. 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 close). Also [[valentine]] and [[miracle]]. IMO the best meantone EDO. 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.

*[[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]].

*[[37edo|37]]: Good for no-[[3/1|3]] [[13-limit]].

*[[38edo|38]]: [[19edo]] with neutrals. Near pure [[11/9]]. The acceptable error of 19edo really becomes a problem at this size.

*[[41edo|41]]: Prime octave and highly composite 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 [[neutral]]. The largest problem is its inaccurate [[5/1|5]].

*[[46edo|46]]: [[13-limit]] [[diaschismic]] and [[valentine]]. It has quartertones similar to [[22edo]] but approximates JI intervals more accurately.

*[[53edo|53]]: A stack of [[3/2]]. Almost just [[3/1|3]], accurate [[5-limit]], and decent [[7-limit]]. Good for 5-limit [[schismatic]] with occasional [[garibaldi]] [[7/1|7]].

*[[72edo|72]]: The ultimate extension of [[12edo]] and [[24edo]] with accurate [[11-limit]] and usable higher limit. A powerful tool for modulating quartertones, which can be difficult in [[24edo]] itself. Good for [[miracle]]. Suitable for octave stretching if only [[17-limit]] is used. Playable by using three [[24edo]] instruments or six [[12edo]] instruments.

*[[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.

*[[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.

*[[87edo|87]]: Good [[mystery]] EDO. Useful for high limit JI. Playable by using three [[29edo]] instruments.

*[[99edo|99]]: Efficient near-[[Logarithmic approximants#Argent tuning|argent]] EDO. It suggests slight compression. Good for [[hemififths]].

*[[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.

*[[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.

*[[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.