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 [[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.

*[[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. AFAIK 12edo and [[17edo]] are the most suitable EDOs for making well 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.

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*[[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 [[11/1|11]] and [[9/1|9]] themselves are not so good). ~~Also~~ [[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.

*[[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 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.

*...

*[[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.

@@ Line 25: / Line 25: @@
 *[[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 [[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.
+*[[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. AFAIK 12edo and [[17edo]] are the most suitable EDOs for making well 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.
@@ Line 43: / Line 43: @@
 *[[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 [[11/1|11]] and [[9/1|9]] themselves are not so good). Also [[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.
+*[[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 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.
 *...
 *[[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.