212edo: Difference between revisions

Line 3:

== Theory ==

212edo is ~~distinctly~~ [[consistent]] in the [[15-odd-limit]] with ~~harmonics~~ of 3 through 13 all tuned flat. 212 = 4 × 53, and it shares the 3rd, 5th, and 13th [[harmonic]]s with [[53edo]], but the mapping differs for 7 and 11.

212edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]] with [[harmonic]]s of 3 through 13 all tuned flat. 212 = 4 × 53, and it shares the [[3/1|3rd]], [[5/1|5th]], and [[13/1|13th]] [[harmonic]]s with [[53edo]], but the mapping differs for [[7/1|7]] and [[11/1|11]].

The equal temperament tempers out the same commas ([[15625/15552]], [[32805/32768]], [[amity comma|1600000/1594323]], etc.) as 53edo in the [[5-limit]]. In the [[7-limit]], it tempers out 2401/2400 ([[breedsma]]), 390625/388962 ([[dimcomp comma]]), and 4802000/4782969 ([[canousma]]). In the [[11-limit]], [[385/384]], [[1375/1372]], [[6250/6237]], [[9801/9800]] and [[14641/14580]]; in the [[13-limit]], [[325/324]], [[625/624]], [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]] and [[10648/10647]].

The equal temperament [[tempering out|tempers out]] the same commas ([[15625/15552]], [[32805/32768]], [[amity comma|1600000/1594323]], etc.) as 53edo in the [[5-limit]]. In the [[7-limit]], it tempers out 2401/2400 ([[breedsma]]), 390625/388962 ([[dimcomp comma]]), and 4802000/4782969 ([[canousma]]). In the [[11-limit]], [[385/384]], [[1375/1372]], [[6250/6237]], [[9801/9800]] and [[14641/14580]]; in the [[13-limit]], [[325/324]], [[625/624]], [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]] and [[10648/10647]].

It is the [[optimal patent val]] for 7- and 13-limit [[quadritikleismic]] temperament, the 7-limit [[Kleismic rank three family #Rank-3 kleismic|rank-3 kleismic]] temperament, and the 13-limit rank-3 [[agni]] temperament. It enables [[marveltwin chords]], [[keenanismic chords]], [[sinbadmic chords]], and [[lambeth chords]] in the 13-odd-limit in addition to [[island chords]] in the 15-odd-limit.

Line 32:

| 2.3.5.7

| 2401/2400, 15625/15552, 32805/32768

| [{{~~val~~| 212 336 492 595 }}]

| {{mapping| 212 336 492 595 }}

| +0.243

| 0.244

Line 39:

| 2.3.5.7.11

| 385/384, 1375/1372, 6250/6237, 14641/14580

| [{{~~val~~| 212 336 492 595 733 }}]

| {{mapping| 212 336 492 595 733 }}

| +0.325

| 0.273

Line 46:

| 2.3.5.7.11.13

| 325/324, 385/384, 625/624, 1375/1372, 10648/10647

| [{{~~val~~| 212 336 492 595 733 784 }}]

| {{mapping| 212 336 492 595 733 784 }}

| +0.396

| 0.296

Line 53:

| 2.3.5.7.11.13.17

| 289/288, 325/324, 385/384, 442/441, 625/624, 10648/10647

| [{{~~val~~| 212 336 492 595 733 784 866 }}] (212g)

| {{mapping| 212 336 492 595 733 784 866 }} (212g)

| +0.447

| 0.301

Line 60:

| 2.3.5.7.11.13.17.19

| 289/288, 325/324, 361/360, 385/384, 442/441, 513/512, 625/624

| [{{~~val~~| 212 336 492 595 733 784 866 900 }}] (212gh)

| {{mapping| 212 336 492 595 733 784 866 900 }} (212gh)

| +0.485

| 0.299

| 5.27

|}

*212et (212gh val) has a lower absolute error in the 19-limit than any previous equal temperaments, past [[193edo|193]] and followed by [[217edo|217]].

* 212et (212gh val) has a lower absolute error in the 19-limit than any previous equal temperaments, past [[193edo|193]] and followed by [[217edo|217]].

=== Rank-2 temperaments ===

Line 73:

Line 74:

|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

Line 144:

Line 145:

| [[Schismerc]] / [[cartography]]

|}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Music ==

; [[Eliora]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is distinctly [[consistent]] in the [[15-odd-limit]] with harmonics of 3 through 13 all tuned flat. 212 = 4 × 53, and it shares the 3rd, 5th, and 13th [[harmonic]]s with [[53edo]], but the mapping differs for 7 and 11.
+edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]] with [[harmonic]]s of 3 through 13 all tuned flat. 212 = 4 × 53, and it shares the [[3/1|3rd]], [[5/1|5th]], and [[13/1|13th]] [[harmonic]]s with [[53edo]], but the mapping differs for [[7/1|7]] and [[11/1|11]].
-The equal temperament tempers out the same commas ([[15625/15552]], [[32805/32768]], [[amity comma|1600000/1594323]], etc.) as 53edo in the [[5-limit]]. In the [[7-limit]], it tempers out 2401/2400 ([[breedsma]]), 390625/388962 ([[dimcomp comma]]), and 4802000/4782969 ([[canousma]]). In the [[11-limit]], [[385/384]], [[1375/1372]], [[6250/6237]], [[9801/9800]] and [[14641/14580]]; in the [[13-limit]], [[325/324]], [[625/624]], [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]] and [[10648/10647]].
+The equal temperament [[tempering out|tempers out]] the same commas ([[15625/15552]], [[32805/32768]], [[amity comma|1600000/1594323]], etc.) as 53edo in the [[5-limit]]. In the [[7-limit]], it tempers out 2401/2400 ([[breedsma]]), 390625/388962 ([[dimcomp comma]]), and 4802000/4782969 ([[canousma]]). In the [[11-limit]], [[385/384]], [[1375/1372]], [[6250/6237]], [[9801/9800]] and [[14641/14580]]; in the [[13-limit]], [[325/324]], [[625/624]], [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]] and [[10648/10647]].
 It is the [[optimal patent val]] for 7- and 13-limit [[quadritikleismic]] temperament, the 7-limit [[Kleismic rank three family #Rank-3 kleismic|rank-3 kleismic]] temperament, and the 13-limit rank-3 [[agni]] temperament. It enables [[marveltwin chords]], [[keenanismic chords]], [[sinbadmic chords]], and [[lambeth chords]] in the 13-odd-limit in addition to [[island chords]] in the 15-odd-limit.
@@ Line 32: / Line 32: @@
 | 2.3.5.7
 | 2401/2400, 15625/15552, 32805/32768
-| [{{val| 212 336 492 595 }}]
+| {{mapping| 212 336 492 595 }}
 | +0.243
 | 0.244
@@ Line 39: / Line 39: @@
 | 2.3.5.7.11
 | 385/384, 1375/1372, 6250/6237, 14641/14580
-| [{{val| 212 336 492 595 733 }}]
+| {{mapping| 212 336 492 595 733 }}
 | +0.325
 | 0.273
@@ Line 46: / Line 46: @@
 | 2.3.5.7.11.13
 | 325/324, 385/384, 625/624, 1375/1372, 10648/10647
-| [{{val| 212 336 492 595 733 784 }}]
+| {{mapping| 212 336 492 595 733 784 }}
 | +0.396
 | 0.296
@@ Line 53: / Line 53: @@
 | 2.3.5.7.11.13.17
 | 289/288, 325/324, 385/384, 442/441, 625/624, 10648/10647
-| [{{val| 212 336 492 595 733 784 866 }}] (212g)
+| {{mapping| 212 336 492 595 733 784 866 }} (212g)
 | +0.447
 | 0.301
@@ Line 60: / Line 60: @@
 | 2.3.5.7.11.13.17.19
 | 289/288, 325/324, 361/360, 385/384, 442/441, 513/512, 625/624
-| [{{val| 212 336 492 595 733 784 866 900 }}] (212gh)
+| {{mapping| 212 336 492 595 733 784 866 900 }} (212gh)
 | +0.485
 | 0.299
 | 5.27
 |}
-*212et (212gh val) has a lower absolute error in the 19-limit than any previous equal temperaments, past [[193edo|193]] and followed by [[217edo|217]].
+* 212et (212gh val) has a lower absolute error in the 19-limit than any previous equal temperaments, past [[193edo|193]] and followed by [[217edo|217]].
 === Rank-2 temperaments ===
@@ Line 73: / Line 74: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 144: / Line 145: @@
 | [[Schismerc]] / [[cartography]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 == Music ==
 ; [[Eliora]]