User:Zhenlige/EDO impressions

Also available in: Collection of 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

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.

0: A fancy way to say “no melody”. The only tuning of the Om temperament. Important in theory, useless in practice.
1: Octaves. Equivalent to 2-limit JI, unless you want to temper some other JI intervals into octaves. Not much to talk about.
2: Half octaves aka symmetric tritones. 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 for even EDOs with a good 3) and 99/70 (giving kalismic). 2n-edos within 1000 that do not support kalismic are probably bad in 11-limit.
3: 12edo major thirds. 2.5 subgroup 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.
4: 12edo minor thirds.
5: Equalized 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.
6: 12edo whole tones. Incomplete 12edo. 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. Around 3/4-comma meantone.
8: Incomplete 24edo.
9: A stack of 7/6. A subset of ennealimmal.
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.
11: Incomplete 22edo.
12: Equalized chromatic scale. Both augmented and diminished. The smallest EDO with decent 3-, 5- and 7-limit. The only reasonable tuning for the 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 diatonic EDO. Efficient at its size. Very excellent 3 as well as good 17 and 19 for its size, but inaccurate 5 and worse 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 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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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 diatonic thirds do not approximate any simple ratios well. Its 13 is good, and 11 and 7 have a similar precision to 12edo's 5, giving decent no-5 13-limit. Even more suitable for making well temperaments than 12edo since there are more target intervals. It benefits from compression.
18: Incomplete 36edo.
19: A stack of 5/3. Close to 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.
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 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.
23: Incomplete 46edo. The largest EDO without a diatonic, blackwood or whitewood fifth.
24: What some non-microtonalists think microtonality is. 12edo with neutrals, giving good 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.
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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.
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.
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. Near pure 13/11. The smallest EDO with a better 3/2 than 12edo.
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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 and 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.
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34: 17edo with 5 and 17 added, making a good 2.3.5.13.17 system. A slightly stretched Carlos Gamma scale.
35: The largest non-diatonic EDO.
36: 12edo with 7 added. Good for 2.3.7.13.17.19.23.29 subroup. Avoid 5 here because it is almost completely missed. Otherwise incomplete 72edo.
37: A strong no-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.
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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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. From here on, most EDOs with good 13-limit support akea.
42: Incomplete 84edo.
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.
44: 22edo with neutrals. Like 38edo, doubling a coarse EDO won't give anything very notable.
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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.
47: Incomplete 94edo.
48: A not-so-good multiple of 12edo.
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.
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.
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53: A stack of 3/2. Almost just 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.
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60: 12edo with 13 and magic-tempered 5 added. 5, 7 and 11 are quite off. Streching can help but it damages the good 13.
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65: A circle of fifths in 130edo.
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68: Good for no-11 23-limit. It is amazing that 17edo's 3-limit still works at this size. I may explore it some day.
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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. 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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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 therefore containing Carlos Alpha. Its slightly flat 3/2 gives a good 19 via boethius. 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.
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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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84: An alternate compton EDO besides 72edo, with better 5 and 13 with the expense of 11. It has a sharp tendency instead of 72edo's flat.
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87: Good mystery EDO. Useful for high limit JI. Playable by using three 29edo instruments.
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89: Like 77edo, the slightly flat fifth gives good 19. In addition, since 89edo has neutrals, it gives good 11 too, together making a strong 2.3.11.19 system after 24edo. Other intervals are just meh.
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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. It suggests slight compression. Good for hemififths. It completely misses 11 and 13.
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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 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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130: Good 13-limit EDO. It doesn't support akea and has a flat 3 instead of sharp, unlike smaller 13-limit EDOs. It may need more exploration.
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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.
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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.