User:Zhenlige/EDO impressions: Difference between revisions

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== 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.

*[[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]]: [[2/1|Octaves]]. Equivalent to [[2-limit]] JI, unless you want to temper some other JI intervals into octaves. Not much to talk about.

*[[2edo|2]]: [[sqrt(2)|Half octaves]] aka symmetric [[tritone]]s. 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]] and [[99/70]]~~, the final one~~ giving [[kalismic temperaments|kalismic]]. 2n-edos within 1000 that do not support kalismic are probably bad in [[11-limit]].

*[[2edo|2]]: [[sqrt(2)|Half octaves]] aka symmetric [[tritone]]s. 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 [[17/1|17]] for even EDOs with a good [[3/1|3]]) and [[99/70]] (giving [[kalismic temperaments|kalismic]]). 2n-edos within 1000 that do not support kalismic are probably bad in [[11-limit]].

*[[3edo|3]]: [[12edo]] major thirds. 2.5 subgroup [[augmented (temperament)|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]].

*[[4edo|4]]: [[12edo]] minor thirds.

*[[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 that roughly represents [[3-limit]] or 2.3.7 subgroup.

*[[5edo|5]]: Equalized [[2L 3s|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.

*[[6edo|6]]: [[12edo]] whole tones. Incomplete 12edo. ~~Also a~~ heavily stretched [[didacus]] chain.

*[[6edo|6]]: [[12edo]] whole tones. Incomplete 12edo. A heavily stretched [[didacus]] chain.

*[[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]].

*[[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]].

*[[8edo|8]]: Incomplete [[24edo]].

*[[9edo|9]]: A stack of [[7/6]]. A subset of [[ennealimmal]].

*[[10edo|10]]: A stack of [[13/8]]. A subset of [[130edo]] and [[270edo]].

*[[11edo|11]]: Incomplete [[22edo]].

*[[12edo|12]]: Equalized [[5L 7s|chromatic]] scale. Both [[augmented (temperament)|augmented]] and [[diminished (temperament)|diminished]]. The smallest ~~edo~~ with decent [[7-limit]]. The only reasonable tuning for the [[dominant (temperament)|dominant]] temperament ignoring overall streching. A sharper fifth makes [[garibaldi]] better and a flatter fifth makes [[septimal meantone]] better. The boundary between [[meantone]] and [[schismatic]]. The smallest [[5L 2s|diatonic]] EDO. Efficient at its size. Very excellent [[3/1|3]] as well as [[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 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.

*[[12edo|12]]: Equalized [[5L 7s|chromatic]] scale. Both [[augmented (temperament)|augmented]] and [[diminished (temperament)|diminished]]. The smallest EDO with decent [[3-limit|3-]], [[5-limit|5-]] and [[7-limit]]. The only reasonable tuning for the [[dominant (temperament)|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 [[5L 2s|diatonic]] EDO. Efficient at its size. Very excellent [[3/1|3]] as well as good [[17/1|17]] and [[19/1|19]] for its size, but inaccurate [[5/1|5]] and worse [[7/1|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. Suitable for symmetric scales. Easy to make [[../12neji|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.

*...

*[[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.

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*[[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 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. The upper bound of a good fifth. The best you can get with [[50/49]] tempered out.

*[[22edo|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 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.

*[[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.

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*[[38edo|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.

*...

*[[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 [[miracle]]. The largest problem is its relatively inaccurate [[5/1|5]]. From here on, most EDOs with good [[13-limit]] support [[akea]].

*[[41edo|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]]. Optimal [[magic]] EDO. The [[Kite guitar]] shows its elegance, with many simple intervals evenly spaced. Also [[garibaldi]] and [[miracle]]. The largest problem is its relatively inaccurate [[5/1|5]]. From here on, most EDOs with good [[13-limit]] support [[akea]].

*[[42edo|42]]: Incomplete [[84edo]].

*[[43edo|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]].

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*[[50edo|50]]: [[Meantone]] with a flatter fifth 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]], accurate [[5-limit]], and decent [[7-limit]]. ~~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]]. The smallest EDO that shows the accuracy of [[schismatic]] like [[12edo]] for [[meantone]]. Good for music that emphasizes [[5-limit]].

*...

*[[65edo|65]]: A circle of fifths in [[130edo]].

*...

*[[72edo|72]]: The ultimate extension of [[12edo]] and [[24edo]]. Its [[11-limit]] is very accurate with a slightly ~~sharp~~ tendency that works well with 12edo's flat [[3/1|3]], and some higher limit intervals are also usable. [[Compton]] which is useful in 12edo-based programs. The only notable [[13-limit]] non-[[akea]] EDO around this size. A powerful tool for modulating quartertones, which can be difficult in 24edo itself. Good for [[miracle]]. Suitable for octave stretching if only [[17-limit]] or below is used. Playable by using three 24edo instruments or six 12edo instruments.

*[[72edo|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 [[3/1|3]], and some higher limit intervals are also usable. [[Compton]] which is useful in 12edo-based programs. The only notable [[13-limit]] non-[[akea]] EDO around this size. A powerful tool for modulating quartertones, which can be difficult in 24edo itself. Good for [[miracle]]. Suitable for octave stretching if only [[17-limit]] or below 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.

@@ Line 10: / Line 10: @@
 == 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.
 *[[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]]: [[2/1|Octaves]]. Equivalent to [[2-limit]] JI, unless you want to temper some other JI intervals into octaves. Not much to talk about.
-*[[2edo|2]]: [[sqrt(2)|Half octaves]] aka symmetric [[tritone]]s. 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]] and [[99/70]], the final one giving [[kalismic temperaments|kalismic]]. 2n-edos within 1000 that do not support kalismic are probably bad in [[11-limit]].
+*[[2edo|2]]: [[sqrt(2)|Half octaves]] aka symmetric [[tritone]]s. 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 [[17/1|17]] for even EDOs with a good [[3/1|3]]) and [[99/70]] (giving [[kalismic temperaments|kalismic]]). 2n-edos within 1000 that do not support kalismic are probably bad in [[11-limit]].
 *[[3edo|3]]: [[12edo]] major thirds. 2.5 subgroup [[augmented (temperament)|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]].
 *[[4edo|4]]: [[12edo]] minor thirds.
-*[[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 that roughly represents [[3-limit]] or 2.3.7 subgroup.
+*[[5edo|5]]: Equalized [[2L 3s|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.
-*[[6edo|6]]: [[12edo]] whole tones. Incomplete 12edo. Also a heavily stretched [[didacus]] chain.
+*[[6edo|6]]: [[12edo]] whole tones. Incomplete 12edo. A heavily stretched [[didacus]] chain.
-*[[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]].
+*[[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]].
 *[[8edo|8]]: Incomplete [[24edo]].
 *[[9edo|9]]: A stack of [[7/6]]. A subset of [[ennealimmal]].
 *[[10edo|10]]: A stack of [[13/8]]. A subset of [[130edo]] and [[270edo]].
 *[[11edo|11]]: Incomplete [[22edo]].
-*[[12edo|12]]: Equalized [[5L 7s|chromatic]] scale. Both [[augmented (temperament)|augmented]] and [[diminished (temperament)|diminished]]. The smallest edo with decent [[7-limit]]. The only reasonable tuning for the [[dominant (temperament)|dominant]] temperament ignoring overall streching. A sharper fifth makes [[garibaldi]] better and a flatter fifth makes [[septimal meantone]] better. The boundary between [[meantone]] and [[schismatic]]. The smallest [[5L 2s|diatonic]] EDO. Efficient at its size. Very excellent [[3/1|3]] as well as [[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 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.
+*[[12edo|12]]: Equalized [[5L 7s|chromatic]] scale. Both [[augmented (temperament)|augmented]] and [[diminished (temperament)|diminished]]. The smallest EDO with decent [[3-limit|3-]], [[5-limit|5-]] and [[7-limit]]. The only reasonable tuning for the [[dominant (temperament)|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 [[5L 2s|diatonic]] EDO. Efficient at its size. Very excellent [[3/1|3]] as well as good [[17/1|17]] and [[19/1|19]] for its size, but inaccurate [[5/1|5]] and worse [[7/1|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. Suitable for symmetric scales. Easy to make [[../12neji|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.
 *...
 *[[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.
@@ Line 32: / Line 34: @@
 *[[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 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. The upper bound of a good fifth. The best you can get with [[50/49]] tempered out.
+*[[22edo|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 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.
 *[[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.
@@ Line 49: / Line 51: @@
 *[[38edo|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.
 *...
-*[[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 [[miracle]]. The largest problem is its relatively inaccurate [[5/1|5]]. From here on, most EDOs with good [[13-limit]] support [[akea]].
+*[[41edo|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]]. Optimal [[magic]] EDO. The [[Kite guitar]] shows its elegance, with many simple intervals evenly spaced. Also [[garibaldi]] and [[miracle]]. The largest problem is its relatively inaccurate [[5/1|5]]. From here on, most EDOs with good [[13-limit]] support [[akea]].
 *[[42edo|42]]: Incomplete [[84edo]].
 *[[43edo|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]].
@@ Line 60: / Line 62: @@
 *[[50edo|50]]: [[Meantone]] with a flatter fifth 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]], accurate [[5-limit]], and decent [[7-limit]]. 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]]. The smallest EDO that shows the accuracy of [[schismatic]] like [[12edo]] for [[meantone]]. Good for music that emphasizes [[5-limit]].
 *...
 *[[65edo|65]]: A circle of fifths in [[130edo]].
 *...
-*[[72edo|72]]: The ultimate extension of [[12edo]] and [[24edo]]. Its [[11-limit]] is very accurate with a slightly sharp tendency that works well with 12edo's flat [[3/1|3]], and some higher limit intervals are also usable. [[Compton]] which is useful in 12edo-based programs. The only notable [[13-limit]] non-[[akea]] EDO around this size. A powerful tool for modulating quartertones, which can be difficult in 24edo itself. Good for [[miracle]]. Suitable for octave stretching if only [[17-limit]] or below is used. Playable by using three 24edo instruments or six 12edo instruments.
+*[[72edo|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 [[3/1|3]], and some higher limit intervals are also usable. [[Compton]] which is useful in 12edo-based programs. The only notable [[13-limit]] non-[[akea]] EDO around this size. A powerful tool for modulating quartertones, which can be difficult in 24edo itself. Good for [[miracle]]. Suitable for octave stretching if only [[17-limit]] or below 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.