User:Zhenlige/EDO impressions: Difference between revisions

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*[[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 ~~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 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 [[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.

*...

*[[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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*[[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 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]].

*[[19edo|19]]: A stack of [[5/3]]. 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. 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?

*...

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*[[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 evenly spaced. Also [[garibaldi]] and [[~~neutral~~]]. The largest problem is its inaccurate [[5/1|5]].

*[[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]].

*...

*[[46edo|46]]: [[13-limit]] [[diaschismic]] and [[valentine]]. It has quartertones similar to [[22edo]] but approximates JI intervals more accurately.

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

*...

*[[72edo|72]]: The ultimate extension of [[12edo]] and [[24edo]] with accurate [[11-limit]] and usable higher limit. The only notable 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]] is used. 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. 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]] 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.

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*[[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 tuning|argent]] EDO. It suggests slight compression. Good for [[hemififths]].

*[[99edo|99]]: Efficient near-[[Logarithmic approximants#Argent tuning|argent]] EDO. It suggests slight compression. Good for [[hemififths]]. It completely misses [[11/1|11]] and [[13/1|13]].

*...

*[[111edo|111]]: [[37edo]] with [[3/1|3]] added.

@@ Line 23: / Line 23: @@
 *[[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 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 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 [[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.
 *...
 *[[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 29: / Line 29: @@
 *[[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 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]].
+*[[19edo|19]]: A stack of [[5/3]]. 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. 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?
 *...
@@ Line 49: / Line 49: @@
 *[[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 evenly spaced. Also [[garibaldi]] and [[neutral]]. The largest problem is its inaccurate [[5/1|5]].
+*[[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]].
 *...
 *[[46edo|46]]: [[13-limit]] [[diaschismic]] and [[valentine]]. It has quartertones similar to [[22edo]] but approximates JI intervals more accurately.
@@ Line 59: / Line 59: @@
 *[[65edo|65]]: A circle of fifths in [[130edo]].
 *...
-*[[72edo|72]]: The ultimate extension of [[12edo]] and [[24edo]] with accurate [[11-limit]] and usable higher limit. The only notable 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]] is used. 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. 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]] 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 71: / Line 71: @@
 *[[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 tuning|argent]] EDO. It suggests slight compression. Good for [[hemififths]].
+*[[99edo|99]]: Efficient near-[[Logarithmic approximants#Argent tuning|argent]] EDO. It suggests slight compression. Good for [[hemififths]]. It completely misses [[11/1|11]] and [[13/1|13]].
 *...
 *[[111edo|111]]: [[37edo]] with [[3/1|3]] added.