User:Zhenlige/EDO impressions

In terms of accuracy, assuming harmonic overtone.

• 0: A fancy way to say “pitchless”.
• 1: Equivalent to 2-limit JI. Not much about harmony. Possibly useful for a transition between different tunings.
• 2: Equally-divided tritones.
• 3: 12edo augmented chords.
• 4: 12edo diminished seventh chords.
• 5: Equalized pentatonic scale. Kinda familiar but everything is warped. The first 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. Similar to 5edo.
• 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. It deserves its position. A good tuning for almost all types of music, tho sometimes not perfect. Very excellent 3/2 as well as prime 17 and 19 for its size, but inaccurate 5/4 and worse 7/4. Suitable for symmetric scales. Easy to make accurate NEJIs. Its 2.3.17.19 subgroup really deserves more exploration (something “xenharmonic” but not “microtonal”).
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• 15: Better than it seems to be, tho still rough. A heavily stretched Carlos Alpha scale.
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• 17: A circle of fifths in 34edo. Interesting sharp fifths. The fact that its thirds do not approximate any simple ratios well is a pity. (or benefit? I don't know)
• 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. Also a compressed Carlos Beta. For meantone 31edo is more preferable, and for magic 41edo. The lower bound of a good fifth. 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 an inconsistent 27 can be used with a 3 or 9?
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• 22: The simplest non-meantone EDO with reasonable 5-limit. Good superpyth and porcupine tuning. The upper bound of a good fifth.
• 23: Incomplete 46edo.
• 24: 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. Meantone but tuned terribly. Incomplete 130edo.
• 27: Worse than both 12- and 22edo for 5-limit. It is distinct in 7-limit tho, but that is like thinking 0.1 is better represented as 1 than 0. The fact that 53edo is good indicates that 26- and 27edo are probably bad.
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• 31: Ideal for pure-octave meantone, combining lots of 11-limit extensions in a single tuning. Also valentine and miracle. IMO the best meantone EDO. For other temperaments its flat fifth may be a drawback.
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• 34: 17edo with prime 5, but no 7. Also a stretched Carlos Gamma.
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• 36: Good for 2.3.7.13.17.19.23.29 subroup. 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. Good for magic. The Kite guitar shows its elegance, with many simple intervals equidistantly spaced. Also good garibaldi and neutral.
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• 46: Efficient neogothic 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: Almost just 3/2, 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: An excellent extension of 12- and 24edo. 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.
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• 74: 37edo with meantone fifths. 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.
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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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• 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 use it much.
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• 171: Ideal for approximating 7-limit JI. Good as schismatic, gammic, ennealimmal and enneadecal. 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. For 11-limit maybe doubling is a good choice, and for 13-limit quadrupling.
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• 224: like 171 but with a slightly sharper (and closer to just) fifth, worse 7-limit but better 13-limit. Ideal for free-pitch-like music that emphasizes 13-limit.