212edo: Difference between revisions

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== Theory ==

212edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]] with [[harmonic]]s of 3 through 13 all tuned flat. 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]].

It [[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 [[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.

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.

To the 13-limit we may add the [[prime harmonic|prime]] [[23/1|23]] without introducing too much extra error, tempering out [[484/483]] and [[507/506]]. The 212gh val shows some potential if the full [[23-limit]] is desired, where it notably tempers out [[289/288]] and [[361/360]]. 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 ===

=== Octave stretch ===

212edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[124edf]] or [[336edt]]. This improves the approximated harmonics 5, 7, 11, 13 and brings the flat mappings of 17 and 19 closer; only the 23 becomes less accurate as it is tuned sharp already.

=== Subsets and supersets ===

Since 212 factors into {{~~factorisation~~|~~212~~}}, 212edo has subset edos {{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.

Since 212 factors into primes as {{nowrap| 2<sup>2</sup> × 53 }}, 212edo has subset edos {{EDOs| 2, 4, 53, and 106 }}. As such, it can be used to tune the 53rd-octave [[cartography]] temperament and the 106th-octave [[boiler]] temperment.

A step of 212edo is exactly 50 [[türk sent]]s.

Line 33:

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| 2.3.5.7

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

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

| {{Mapping| 212 336 492 595 }}

| +0.243

| 0.244

Line 40:

Line 43:

| 2.3.5.7.11

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

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

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

| +0.325

| 0.273

Line 47:

Line 50:

| 2.3.5.7.11.13

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

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

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

| +0.396

| 0.296

Line 54:

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| 2.3.5.7.11.13.17

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

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

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

| +0.447

| 0.301

Line 61:

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| 2.3.5.7.11.13.17.19

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

| {{~~mapping~~| 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

|-

| 2.3.5.7.11.13.17.19.23

| 289/288, 323/322, 325/324, 361/360, 385/384, 442/441, 484/483, 507/506

| {{Mapping| 212 336 492 595 733 784 866 900 959 }} (212gh)

| +0.430

| 0.321

| 5.67

|}

* 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]].

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|212}}
+{{ED intro}}
 == Theory ==
 edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]] with [[harmonic]]s of 3 through 13 all tuned flat. 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]].
-It [[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 [[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.
-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.
+To the 13-limit we may add the [[prime harmonic|prime]] [[23/1|23]] without introducing too much extra error, tempering out [[484/483]] and [[507/506]]. The 212gh val shows some potential if the full [[23-limit]] is desired, where it notably tempers out [[289/288]] and [[361/360]]. 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 ===
-{{Harmonics in equal|212|columns=11}}
+{{Harmonics in equal|212}}
+=== Octave stretch ===
+edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[124edf]] or [[336edt]]. This improves the approximated harmonics 5, 7, 11, 13 and brings the flat mappings of 17 and 19 closer; only the 23 becomes less accurate as it is tuned sharp already.
 === Subsets and supersets ===
-Since 212 factors into {{factorisation|212}}, 212edo has subset edos {{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.
+Since 212 factors into primes as {{nowrap| 2<sup>2</sup> × 53 }}, 212edo has subset edos {{EDOs| 2, 4, 53, and 106 }}. As such, it can be used to tune the 53rd-octave [[cartography]] temperament and the 106th-octave [[boiler]] temperment.
 A step of 212edo is exactly 50 [[türk sent]]s.
@@ Line 33: / Line 36: @@
 | 2.3.5.7
 | 2401/2400, 15625/15552, 32805/32768
-| {{mapping| 212 336 492 595 }}
+| {{Mapping| 212 336 492 595 }}
 | +0.243
 | 0.244
@@ Line 40: / Line 43: @@
 | 2.3.5.7.11
 | 385/384, 1375/1372, 6250/6237, 14641/14580
-| {{mapping| 212 336 492 595 733 }}
+| {{Mapping| 212 336 492 595 733 }}
 | +0.325
 | 0.273
@@ Line 47: / Line 50: @@
 | 2.3.5.7.11.13
 | 325/324, 385/384, 625/624, 1375/1372, 10648/10647
-| {{mapping| 212 336 492 595 733 784 }}
+| {{Mapping| 212 336 492 595 733 784 }}
 | +0.396
 | 0.296
@@ Line 54: / Line 57: @@
 | 2.3.5.7.11.13.17
 | 289/288, 325/324, 385/384, 442/441, 625/624, 10648/10647
-| {{mapping| 212 336 492 595 733 784 866 }} (212g)
+| {{Mapping| 212 336 492 595 733 784 866 }} (212g)
 | +0.447
 | 0.301
@@ Line 61: / Line 64: @@
 | 2.3.5.7.11.13.17.19
 | 289/288, 325/324, 361/360, 385/384, 442/441, 513/512, 625/624
-| {{mapping| 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
+|-
+| 2.3.5.7.11.13.17.19.23
+| 289/288, 323/322, 325/324, 361/360, 385/384, 442/441, 484/483, 507/506
+| {{Mapping| 212 336 492 595 733 784 866 900 959 }} (212gh)
+| +0.430
+| 0.321
+| 5.67
 |}
 * 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]].