User:Zhenlige/EDO impressions

Warning: This page contains a lot of personal opinions and emotions, and may make some readers feel offensive.

General patterns

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.
If m-edo and n-edo are good, and most intervals are either accurate enough or off in the opposite directions, then (m+n)-edo is probably good.
The JI subgroup an EDO supports matters. An EDO whose subgroup is not prime (especially when it is fractional) is better viewed as a subset of a larger temperament (either another EDO or a higher-rank temperament), than as a temperament itself.

Details

In terms of accuracy, assuming harmonic overtone.

0: A fancy way to say “pitchless”. The only tuning of the Om temperament. Important in theory, useless in practice.
1: Equivalent to 2-limit JI (unless you want to temper some higher-rank JI subgroups into octaves, which I think nobody will do). Not much about harmony. Possibly useful for a transition between different tunings.
2: Half octaves aka symmetric tritones. The key to tritone substitution, although sometimes asymmetric tritones or even non-tritone intervals can also be used.
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.
4: 12edo diminished seventh chords.
5: Equalized 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 resembles 3/2. Not very noticeable harmonically.
6: Incomplete 12edo. Also a heavily stretched didacus chain.
7: Equalized diatonic scale. 3-limit whitewood. Similar to 5edo. It sounds like out-of-tone neutral scale music. The smallest EDO that roughly represents 5-limit.
8: Incomplete 24edo.
9: A subset of ennealimmal.
10: A stack of 13/8. A subset of 130edo and 270edo.
11: Incomplete 22edo.
12: Equalized chromatic scale. Both augmented and diminished. The only reasonable tuning for dominant. The boundary between meantone and schismatic. The smallest diatonic EDO. It deserves its position. A good tuning for almost all types of music, tho sometimes not perfect. Very excellent 3 as well as prime 17 and 19 for its size, but inaccurate 5 and worse 7. Suitable for symmetric scales. Easy to make 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, but you can make a well temperament to make some of them do.
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15: Better than it seems to be, tho still rough. Blackwood 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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17: A circle of fifths in 34edo. Interesting sharp fifths. The smallest diatonic EDO with neutral intervals. The boundary between neogothic and superpyth. Like 12edo, its thirds do not approximate any simple ratios well. I don't know how to write harmonies here. Only 3-limit seems to be good but still worse than 12.
18: Incomplete 36edo.
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. 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.
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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22: The smallest non-meantone EDO with reasonable 5-limit. Good superpyth and porcupine tuning. The only reasonable superpyth EDO. The upper bound of a good fifth.
23: Incomplete 46edo. The largest EDO without a diatonic, blackwood or whitewood fifth.
24: What some non-microtonalists think microtonality is. 12edo with neutrals. Good for prime 11. Accurate in subgroup 2.3.11.17.19.
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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 but tuned terribly. Incomplete 130edo.
27: A stack of 7/6. Worse than both 12- and 22edo for 5-limit. 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. Use 108edo to make it a true 7-limit EDO, and 270edo is excellent. When I hear its ~10:12:15 and ~6:7:9 chord I feel the fifth is obviously off. I have listened to both 22edo and 27edo and I feel the former is better. Compared to 22edo, it is like fixing 7 by ruining 5 and partly 3. There are beatings here and there. It really needs compression. The fact that 53edo is good indicates that 26- and 27edo are probably bad.
28: Whitewood diminished. Kinda opposite from 15edo. The best you can get with whitewood.
29: A circle of fifths in mystery, which supports pele (and is close to its optimal tuning), a convenient temperament with 5, 7, 11 and 13 on the same chain of fifths. The smallest EDO with a better fifth than 12edo. Not so useful on its own.
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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 and 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.
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34: 17edo with prime 5, but no 7. A slightly stretched Carlos Gamma scale.
35: The largest non-diatonic EDO.
36: Good for 2.3.7.13.17.19.23.29 subroup. Avoid 5 here because it is almost completely missed. Otherwise incomplete 72edo.
37: Everything but prime 3.
38: 19edo with neutrals. Near pure 11/9. The acceptable error of 19edo really becomes a problem at this size.
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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 equidistantly spaced. I have no idea how other similar-sized EDOs (namely 53) can be applied on a fretted string instrument. It's a pity that its 5 is not very good. Also garibaldi and neutral.
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46: Efficient parapyth EDO.
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50: Flatter meantone than 31edo, but I usually use golden meantone (with slight octave stretching) for this range.
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53: A stack of 3/2. Almost just 3, and accurate 7-limit. Its structure is not very elegant tho. Good for 5-limit schismatic with occasional garibaldi 7.
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65: A circle of fifths in 130edo.
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72: The ultimate extension of 12- and 24edo. A powerful tool for modulating quartertones, whis is difficult in 24edo itself. Good for miracle. The relative error of primes is within 1/3 steps up to large primes except a few including 13, 53 and 59. Suitable for octave stretching in 17-limit. Playable by using three 24edo instruments or six 12edo instruments.
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74: 37edo with meantone fifths. Close to tungsten meantone. The intrinsic error of meantone becomes a problem at this size, making 9 inconsistent.
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77: Good for valentine and accurate boethius. Containing Carlos Alpha. 40/27 as 4\7 aka absurdity. Usable for high-limit JI. At this size even the inconsistent intervals are usable via val mapping since its step size is only ~16 cents, so 50% is not a very large error.
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81: The optimal patent val for meantone and some of its higher-limit extentions, but does anyone really want to use such a large EDO for a temperament with relatively low accuracy? Since it severely loses the convenience of EDOs, I will use golden meantone instead, which is simpler and more elegant mathematically.
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87: Good mystery EDO. Useful for high-limit JI. Playable by using three 29edo instruments.
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94: Good for high-limit JI with the garibaldi structure similar to 41edo and 53edo. Containing Carlos Beta.
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99: Efficient near-argent EDO. Suggests slight compression. Good for hemififths.
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111: 37edo with 3 added.
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118: The relationship of 53-118-171edo for schismatic is similar to 12-19-31edo for meantone. 53 and 12 are the simplest reasonable EDO with very mildly tempered fifths, 118 and 19 are better over all but a bit 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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171: The ultimate EDO for approximating 7-limit JI. If you don't need some ridiculous high precision or specific microtemperaments, there is no need to go any further. Good as 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.
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224: Like 171edo but with a slightly sharper (and closer to just) fifth, worse 7-limit but better 13-limit.
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270: Better than 224edo if schismatic is not required. Ideal for free-pitch-like music that emphasizes 13-limit.
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311: Good for very high-limit JI.