212edo: Difference between revisions

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 harmonics with 53edo, but the mapping differs for 7 and 11.

The equal temperament tempers out the same commas (15625/15552, 32805/32768, 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 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.

The 212gh val shows some potential beyond 15-odd-limit. Also, using 212bb val (where fifth is flattened by single step, approximately 1/4 comma) gives a tuning almost identical to the POTE tuning for 5-limit meantone. This is related to the fact that 212edo splits steps of 53edo, which are mapped to a syntonic comma, in four.

Prime harmonics

Approximation of prime harmonics in 212edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-0.07	-1.41	-0.90	-2.26	-2.79	+2.59	+2.49	+0.03	+0.61	-1.64
Error	Relative (%)	+0.0	-1.2	-24.9	-15.9	-40.0	-49.3	+45.8	+43.9	+0.5	+10.8	-29.0
Steps (reduced)		212 (0)	336 (124)	492 (68)	595 (171)	733 (97)	784 (148)	867 (19)	901 (53)	959 (111)	1030 (182)	1050 (202)

Subsets and supersets

Since 212 factors into 2² × 53, 212edo has subset edos 2, 4, 53, and 106. As such, it can be used to tune the period-53 cartography temperament and the period-106 boiler temperment.

A step of 212edo is exactly 50 türk sents.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Absolute (¢)	Relative (%)
2.3.5.7	2401/2400, 15625/15552, 32805/32768	[⟨212 336 492 595]]	+0.243	0.244	4.30
2.3.5.7.11	385/384, 1375/1372, 6250/6237, 14641/14580	[⟨212 336 492 595 733]]	+0.325	0.273	4.82
2.3.5.7.11.13	325/324, 385/384, 625/624, 1375/1372, 10648/10647	[⟨212 336 492 595 733 784]]	+0.396	0.296	5.23
2.3.5.7.11.13.17	289/288, 325/324, 385/384, 442/441, 625/624, 10648/10647	[⟨212 336 492 595 733 784 866]] (212g)	+0.447	0.301	5.32
2.3.5.7.11.13.17.19	289/288, 325/324, 361/360, 385/384, 442/441, 513/512, 625/624	[⟨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 193 and followed by 217.

Rank-2 temperaments

Note: temperaments supported by 53et are not included.

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	15\212	84.91	21/20	Amicable / amorous / pseudoamical
1	31\212	175.47	448/405	Sesquiquartififths
1	41\212	232.08	8/7	Quadrawell
1	67\212	379.25	56/45	Marthirds
2	11\212	62.26	28/27	Eagle
2	15\212	84.91	21/20	Floral
2	31\212	175.47	448/405	Bisesqui
2	97\212 (9\212)	549.06 (50.94)	11/8 (36/35)	Kleischismic
4	56\212 (3\212)	316.98 (16.98)	6/5 (126/125)	Quadritikleismic
4	88\212 (18\212)	498.11 (101.89)	4/3 (35/33)	Quadrant
53	41\212 (1\212)	232.08 (5.66)	8/7 (225/224)	Schismerc / cartography

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Music

Eliora

Etude in Amicable, Bisesqui, and 53edo

@@ 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]]

212edo: Difference between revisions

Revision as of 09:04, 3 April 2024

Contents

Theory

Prime harmonics

Subsets and supersets

Regular temperament properties

Rank-2 temperaments

Music

Navigation menu

212edo: Difference between revisions

Revision as of 09:04, 3 April 2024

Theory

Prime harmonics

Subsets and supersets

Regular temperament properties

Rank-2 temperaments

Music

Navigation menu

Search