57ed12: Difference between revisions

(2 intermediate revisions by 2 users not shown)

Line 1:

57ed12 is similar to [[16edo]], but has the [[12/1]] ~~tuned just instead of the octave~~, which stretches the octave by 7.6{{c}}. It can be used as a tuning for [[~~Mavila~~]] and ~~also~~ approximates [[Pelog]] tunings in Indonesian gamelan music.

== Theory ==

57ed12 is similar to [[16edo]], but has the 12th harmonic tuned just instead of the [[2/1|octave]], which stretches the octave by 7.6{{c}}. It can be used as a tuning for [[mavila]] and has an antidiatonic ([[2L 5s]]) scale which approximates [[Pelog]] tunings in Indonesian gamelan music.

16edo's [[harmonic]]s 3, 11 and 17 are all more than 25 cents away from just, making them unusable for most purposes. 57ed12 improves upon all of their tunings, bringing all those harmonics within 16 cents of just, and bringing 11 and 17 within an impressive 1 cent of just. This dramatically increases the number of [[consonant]] intervals and chords available in the tuning.

The trade-off is that 57ed12's harmonic 7 is significantly worse than 16edo. It has almost 28 cents of error, compared to 6, making it go from very consonant to completely unusable. So, if one intends to use harmonies involving 7's, 57ed12 is not a good choice. But if one is interested in the [[5-limit]], the 2.3.5.11 [[subgroup]], or the no-7's [[13-limit]] or [[17-limit]], then 57ed12 is by far the superior option, thanks to its dramatically improved 3/1 and 11/1.

~~== Theory ==~~

=== Commas ===

As an equal temperament, 57ed12 [[tempering out|tempers out]] 36/35 and 50/49 in the [[7-limit]]; 33/32 and 45/44 in the 11-limit; 65/64, 66/65, and 78/77 in the 13-limit; 51/50 and 85/84 in the 17-limit; 39/38, 57/56, 77/76, and 96/95 in the 19-limit; 69/68 and 92/91 in the 23-limit; 58/57 and 87/85 in the 29-limit; 63/62, 93/92, and 93/91 in the 31-limit; and 75/74 in the 37-limit.

=== Harmonics ===

~~16edo’s [[harmonic]]s 3/~~1, 11/1 ~~and 17/1 are all more than 25 cents away from [[just]], making them unusable for most purposes. 57ed12 improves upon all~~ of ~~their tunings, bringing all those~~ harmonics ~~within 16 cents of just, and bringing 11 and 17 within an impressive 1 cent of just. This dramatically increases the number of [[consonant]] intervals and chords available~~ in ~~the tuning.~~

{{Harmonics in equal|57|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed12 (continued)}}

The trade-off is that 57ed12’s 7/1 is significantly worse than 16edo. It has almost 28 cents of error, compared to 6, making it go from very consonant to completely unusable. So, if one intends to use harmonies involving 7s, 57ed12 ~~is not a good choice. But if one is interested in the~~ [[~~5-limit~~]]~~, the 2.3.5.11 [[subgroup]], or the no-7s~~ [[~~13-limit~~]] ~~or [[17-limit]]s, then 57ed12 is by far the superior option, thanks to its dramatically improved 3/1 and 11/1~~.

=== Subsets and supersets ===

~~{{Harmonics in equal|57|12|1|intervals=prime}}~~

Since 57 factors into primes as {{nowrap| 3 × 19 }}, 57ed12 contains subset ed12's [[3ed12]] and [[19ed12]].

~~16edo for comparison:~~

~~{{Harmonics in equal|16|2|1|intervals=prime}}~~

== Intervals ==

~~{{todo|expand}}~~

== See also ==

* [[16edo]] – relative edo

* [[25edt]] – relative edt

* [[41ed6]] – relative ed6

~~[[Category:16edo]]~~

~~[[Category:Edonoi]]~~

[[Category:Mavila]]

[[Category:Pelog]]

@@ Line 1: / Line 1: @@
-{{Infobox ET}}{{todo|expand}}
+{{Infobox ET}}
 {{ED intro}}
-ed12 is similar to [[16edo]], but has the [[12/1]] tuned just instead of the octave, which stretches the octave by 7.6{{c}}. It can be used as a tuning for [[Mavila]] and also approximates [[Pelog]] tunings in Indonesian gamelan music.
+== Theory ==
+ed12 is similar to [[16edo]], but has the 12th harmonic tuned just instead of the [[2/1|octave]], which stretches the octave by 7.6{{c}}. It can be used as a tuning for [[mavila]] and has an antidiatonic ([[2L&nbsp;5s]]) scale which approximates [[Pelog]] tunings in Indonesian gamelan music.
+edo's [[harmonic]]s 3, 11 and 17 are all more than 25 cents away from just, making them unusable for most purposes. 57ed12 improves upon all of their tunings, bringing all those harmonics within 16 cents of just, and bringing 11 and 17 within an impressive 1 cent of just. This dramatically increases the number of [[consonant]] intervals and chords available in the tuning.
+The trade-off is that 57ed12's harmonic 7 is significantly worse than 16edo. It has almost 28 cents of error, compared to 6, making it go from very consonant to completely unusable. So, if one intends to use harmonies involving 7's, 57ed12 is not a good choice. But if one is interested in the [[5-limit]], the 2.3.5.11 [[subgroup]], or the no-7's [[13-limit]] or [[17-limit]], then 57ed12 is by far the superior option, thanks to its dramatically improved 3/1 and 11/1.
-== Theory ==
 === Commas ===
 As an equal temperament, 57ed12 [[tempering out|tempers out]] 36/35 and 50/49 in the [[7-limit]]; 33/32 and 45/44 in the 11-limit; 65/64, 66/65, and 78/77 in the 13-limit; 51/50 and 85/84 in the 17-limit; 39/38, 57/56, 77/76, and 96/95 in the 19-limit; 69/68 and 92/91 in the 23-limit; 58/57 and 87/85 in the 29-limit; 63/62, 93/92, and 93/91 in the 31-limit; and 75/74 in the 37-limit.
 === Harmonics ===
-edo’s [[harmonic]]s 3/1, 11/1 and 17/1 are all more than 25 cents away from [[just]], making them unusable for most purposes. 57ed12 improves upon all of their tunings, bringing all those harmonics within 16 cents of just, and bringing 11 and 17 within an impressive 1 cent of just. This dramatically increases the number of [[consonant]] intervals and chords available in the tuning.
+{{Harmonics in equal|57|12|1|intervals=integer|columns=11}}
+{{Harmonics in equal|57|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed12 (continued)}}
-The trade-off is that 57ed12’s 7/1 is significantly worse than 16edo. It has almost 28 cents of error, compared to 6, making it go from very consonant to completely unusable. So, if one intends to use harmonies involving 7s, 57ed12 is not a good choice. But if one is interested in the [[5-limit]], the 2.3.5.11 [[subgroup]], or the no-7s [[13-limit]] or [[17-limit]]s, then 57ed12 is by far the superior option, thanks to its dramatically improved 3/1 and 11/1.
+=== Subsets and supersets ===
-{{Harmonics in equal|57|12|1|intervals=prime}}
+Since 57 factors into primes as {{nowrap| 3 × 19 }}, 57ed12 contains subset ed12's [[3ed12]] and [[19ed12]].
-edo for comparison:
-{{Harmonics in equal|16|2|1|intervals=prime}}
 == Intervals ==
 {{Interval table}}
-{{todo|expand}}
+== See also ==
+* [[16edo]] – relative edo
+* [[25edt]] – relative edt
+* [[41ed6]] – relative ed6
-[[Category:16edo]]
-[[Category:Edonoi]]
 [[Category:Mavila]]
 [[Category:Pelog]]