User:Zhenlige/EDO impressions: Difference between revisions

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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 ~~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 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.

*...

*[[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.

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*...

*[[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]].

*[[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]].

*[[44edo|44]]: Like [[38edo]], doubling a rough EDO won't give something very notable.

*...

*[[46edo|46]]: [[13-limit]] [[diaschismic]] and [[valentine]]. Near pure [[11/7]]. It has quartertones similar to [[22edo]] but approximates JI intervals more accurately. Its 30::36 are all 2 steps apart.

*[[47edo|47]]: Incomplete [[94edo]].

*[[48edo|48]]: A not-so-good multiple of [[12edo]].

*...

*[[50edo|50]]: ~~Flatter meantone~~ than [[31edo]], but I usually use [[golden meantone]] (with slight octave stretching) for this range.

*[[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]].

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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 [[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.

*[[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.

*...

*[[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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*...

*[[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.

*...

*[[84edo|84]]: An alternate [[compton]] EDO besides [[72edo]], with better [[5/1|5]] and [[13/1|13]] with the expense of [[11/1|11]]. It has a sharp tendency instead of 72edo's flat.

*...

*[[87edo|87]]: Good [[mystery]] EDO. Useful for high limit JI. Playable by using three [[29edo]] instruments.

@@ Line 41: / Line 41: @@
 *[[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 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 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.
 *...
 *[[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 50: / Line 50: @@
 *...
 *[[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]].
+*[[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]].
+*[[44edo|44]]: Like [[38edo]], doubling a rough EDO won't give something very notable.
 *...
 *[[46edo|46]]: [[13-limit]] [[diaschismic]] and [[valentine]]. Near pure [[11/7]]. It has quartertones similar to [[22edo]] but approximates JI intervals more accurately. Its 30::36 are all 2 steps apart.
+*[[47edo|47]]: Incomplete [[94edo]].
+*[[48edo|48]]: A not-so-good multiple of [[12edo]].
 *...
-*[[50edo|50]]: Flatter meantone than [[31edo]], but I usually use [[golden meantone]] (with slight octave stretching) for this range.
+*[[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]].
@@ Line 59: / Line 64: @@
 *[[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 [[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.
+*[[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.
 *...
 *[[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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 *...
 *[[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.
+*...
+*[[84edo|84]]: An alternate [[compton]] EDO besides [[72edo]], with better [[5/1|5]] and [[13/1|13]] with the expense of [[11/1|11]]. It has a sharp tendency instead of 72edo's flat.
 *...
 *[[87edo|87]]: Good [[mystery]] EDO. Useful for high limit JI. Playable by using three [[29edo]] instruments.