User:Zhenlige/EDO impressions: Difference between revisions

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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.

0: A fancy way to say “no melody”. 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 other JI intervals into octaves. Not much about harmony. Possibly useful for a transition between different tunings.
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.
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 represents 3-limit or 2.3.7 subgroup.
6: Incomplete 12edo. Also a heavily stretched didacus chain.
7: Equalized diatonic scale. 3-limit whitewood. 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 the dominant temperament. The boundary between meantone and schismatic. The smallest diatonic EDO. Efficient at its size. 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. A well temperament can make some of them do better.
...
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.
...
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, and a well temperament may help. Its 13 is good, and 11 and 7 have a similar precision to 12edo's 5. It benefits from compression.
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 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.
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?
...
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.
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.
...
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.
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.
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.
...
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.
...
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: Good for 2.3.7.13.17.19.23.29 subroup. Avoid 5 here because it is almost completely missed. Otherwise incomplete 72edo.
37: Good for no-3 13-limit.
38: 19edo with neutrals. Near pure 11/9. The acceptable error of 19edo really becomes a problem at this size.
...
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.
...
46: 13-limit diaschismic and valentine. It has quartertones similar to 22edo but approximates JI intervals more accurately.
...
50: Flatter meantone than 31edo, but I usually use golden meantone (with slight octave stretching) for this range.
...
53: A stack of 3/2. Almost just 3, accurate 5-limit, and decent 7-limit. Good for 5-limit schismatic with occasional garibaldi 7.
...
65: A circle of fifths in 130edo.
...
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.
...
74: 37edo with meantone fifths. Close to tungsten meantone. The intrinsic error of meantone becomes a problem at this size, making 9 inconsistent.
...
77: Good for valentine. Its slightly flat 3/2 gives a good 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.
...
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.
...
87: Good mystery EDO. Useful for high limit JI. Playable by using three 29edo instruments.
...
94: Good for high-limit JI with the garibaldi structure similar to 41edo and 53edo. Containing Carlos Beta.
...
99: Efficient near-argent EDO. It suggests slight compression. Good for hemififths.
...
111: 37edo with 3 added.
...
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.
...
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.
...
224: Like 171edo but with a slightly sharper (and closer to just) fifth, worse 7-limit but better 13-limit.
...
270: Better than 224edo if schismatic is not required. Ideal for free-pitch-like music that emphasizes 13-limit.
...
311: Good for very high limit JI.

@@ Line 9: / Line 9: @@
 == Details ==
-In terms of accuracy, assuming harmonic overtone.
+'''Note: the version here may be edited frequently. A stabler version is on [[Collection of EDO impressions]].'''
-*[[0edo|0]]: A fancy way to say “pitchless”. 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 higher-rank JI subgroups into octaves, which I think nobody will do). Not much about harmony. Possibly useful for a transition between different tunings.
+*[[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.
-*[[2edo|2]]: Half octaves aka symmetric tritones. The key to tritone substitution, although sometimes asymmetric tritones or even non-tritone intervals can also be used.
+*[[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.
-*[[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.
+*[[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.
-*[[4edo|4]]: 12edo diminished seventh chords.
+*[[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]].
-*[[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 resembles [[3/2]]. Not very noticeable harmonically.
+*[[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.
 *[[6edo|6]]: Incomplete [[12edo]]. Also 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]].
+*[[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]].
 *[[8edo|8]]: Incomplete [[24edo]].
 *[[9edo|9]]: A subset of [[ennealimmal]].
 *[[10edo|10]]: A stack of [[13/8]]. A subset of [[130edo]] and [[270edo]].
 *[[11edo|11]]: Incomplete [[22edo]].
-*[[12edo|12]]: Equalized [[chromatic]] scale. Both [[augmented]] and [[diminished]]. The only reasonable tuning for [[dominant (temperament)|dominant]]. The boundary between [[meantone]] and [[schismatic]]. The smallest [[5L 2s|diatonic]] EDO. It deserves its position. A good tuning for almost all types of music, tho sometimes not perfect. 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, but you can make a [[well temperament]] to make some of them do.
+*[[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]]. 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.
+*[[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 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.
+*[[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.
 *[[18edo|18]]: Incomplete [[36edo]].
-*[[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. 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]].
+*[[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]].
 *[[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 non-meantone EDO with reasonable 5-limit. Good [[superpyth]] and [[porcupine]] tuning. The only reasonable superpyth EDO. The upper bound of a good fifth.
+*[[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 diatonic, blackwood or whitewood fifth.
+*[[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. Good for prime [[11/1|11]]. Accurate in subgroup 2.3.11.17.19.
 *...
-*[[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 but tuned terribly. Incomplete [[130edo]].
+*[[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 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.
+*[[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]]. Kinda opposite from [[15edo]]. The best you can get with [[whitewood]].
+*[[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, 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.
+*[[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 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.
+*[[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 prime [[5/1|5]], but no [[7/1|7]]. A slightly stretched [[Carlos Gamma]] scale.
+*[[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-diatonic EDO.
+*[[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.
+*[[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]]: Everything but prime [[3/1|3]].
+*[[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.
+*[[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 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 [[5/1|5]] is not very good. Also [[garibaldi]] and [[neutral]].
+*[[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]]: Efficient [[parapyth]] EDO.
+*[[46edo|46]]: [[13-limit]] [[diaschismic]] and [[valentine]]. It has quartertones similar to [[22edo]] but approximates JI intervals more accurately.
 *...
 *[[50edo|50]]: Flatter meantone than [[31edo]], but I usually use [[golden meantone]] (with slight octave stretching) for this range.
 *...
-*[[53edo|53]]: A stack of [[3/2]]. Almost just [[3/1|3]], and accurate [[7-limit]]. Its structure is not very elegant tho. Good for 5-limit [[schismatic]] with occasional [[garibaldi]] [[7/1|7]].
+*[[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]].
 *...
 *[[65edo|65]]: A circle of fifths in [[130edo]].
 *...
-*[[72edo|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.
+*[[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.
 *...
 *[[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.
 *...
-*[[77edo|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.
+*[[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 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.
+*[[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.
+*[[87edo|87]]: Good [[mystery]] EDO. Useful for high limit JI. Playable by using three [[29edo]] instruments.
 *...
 *[[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 temperament|argent]] EDO. Suggests slight compression. Good for [[hemififths]].
+*[[99edo|99]]: Efficient near-[[Logarithmic approximants#Argent tuning|argent]] EDO. It suggests slight compression. Good for [[hemififths]].
 *...
 *[[111edo|111]]: [[37edo]] with [[3/1|3]] added.
 *...
-*[[118edo|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.
+*[[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. 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.
+*[[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]].
+*[[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.
+*[[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.
+*[[311edo|311]]: Good for very high limit JI.
 [[Category:Impression]]

User:Zhenlige/EDO impressions: Difference between revisions

Revision as of 16:17, 24 February 2026

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