212edo: Difference between revisions

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{{Infobox ET

~~| Prime factorization = 22 × 53~~

~~| Step size = 5.66038¢~~

~~| Fifth = 124\212 (701.89¢) (→ [[53edo|31\53]])~~

~~| Semitones = 20:16 (113.21¢ : 90.57¢)~~

~~| Consistency = 15~~

}}

The '''212 equal divisions of the octave''' ('''212edo'''), or the '''212(-tone) equal temperament''' ('''212tet''', '''212et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 212 [[equal]] parts of about 5.66 [[cent]]s each.

== 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. 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.

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 primes as {{nowrap| 22 × 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.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | Subgroup

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal ~~8ve~~ stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

Line 33:

Line 36:

| 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 40:

Line 43:

| 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 47:

Line 50:

| 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 54:

Line 57:

| 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 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

| [{{~~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

|-

| 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]].

=== Rank-2 temperaments ===

Note: temperaments supported by 53et are not included.

{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods per ~~octave~~

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! Associated ratio*

! Temperaments

|-

Line 120:

Line 133:

|-

| 2

| 97\212 (9\212)

| 97\212 (9\212)

| 549.06 (50.94)

| 549.06 (50.94)

| 11/8 (36/35)

| 11/8 (36/35)

| [[Kleischismic]]

|-

| 4

| 56\212 (3\212)

| 56\212 (3\212)

| 316.98 (16.98)

| 316.98 (16.98)

| 6/5 (126/125)

| 6/5 (126/125)

| [[Quadritikleismic]]

|-

| 4

| 88\212 (18\212)

| 88\212 (18\212)

| 498.11 (101.89)

| 498.11 (101.89)

| 4/3 (35/33)

| 4/3 (35/33)

| [[Quadrant]]

|-

| 53

| 41\212 (1\~~198~~)

| 41\212 (1\212)

| 232.08 (5.66)

| 232.08 (5.66)

| 8/7 (225/224)

| 8/7 (225/224)

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

|}

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

== Music ==

; [[Eliora]]

* [https://www.youtube.com/watch?v=bvCeNcxUDnA ''Etude in Amicable, Bisesqui, and 53edo'']

~~[[Category:Equal divisions of the octave|###]] ~~

[[Category:53edo]]

[[Category:Agni]]

[[Category:Kleismic]]

[[Category:~~Agni~~]]

[[Category:Listen]]

[[Category:Quadritikleismic]]

@@ Line 1: / Line 1: @@
-{{Infobox ET
+{{Infobox ET}}
-| Prime factorization = 2<sup>2</sup> × 53
+{{ED intro}}
-| Step size = 5.66038¢
-| Fifth = 124\212 (701.89¢) (→ [[53edo|31\53]])
-| Semitones = 20:16 (113.21¢ : 90.57¢)
-| Consistency = 15
-}}
-The '''212 equal divisions of the octave''' ('''212edo'''), or the '''212(-tone) equal temperament''' ('''212tet''', '''212et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 212 [[equal]] parts of about 5.66 [[cent]]s each.
 == 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. 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.
+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 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.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+|-
+! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal 8ve <br>stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 33: / Line 36: @@
 | 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 40: / Line 43: @@
 | 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 47: / Line 50: @@
 | 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 54: / Line 57: @@
 | 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 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
-| [{{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
+|-
+| 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]].
+=== Rank-2 temperaments ===
 Note: temperaments supported by 53et are not included.
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per octave
+|-
-! Generator<br>(reduced)
+! Periods<br />per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 120: / Line 133: @@
 |-
 | 2
-| 97\212<br>(9\212)
+| 97\212<br />(9\212)
-| 549.06<br>(50.94)
+| 549.06<br />(50.94)
-| 11/8<br>(36/35)
+| 11/8<br />(36/35)
 | [[Kleischismic]]
 |-
 | 4
-| 56\212<br>(3\212)
+| 56\212<br />(3\212)
-| 316.98<br>(16.98)
+| 316.98<br />(16.98)
-| 6/5<br>(126/125)
+| 6/5<br />(126/125)
 | [[Quadritikleismic]]
 |-
 | 4
-| 88\212<br>(18\212)
+| 88\212<br />(18\212)
-| 498.11<br>(101.89)
+| 498.11<br />(101.89)
-| 4/3<br>(35/33)
+| 4/3<br />(35/33)
 | [[Quadrant]]
 |-
 | 53
-| 41\212<br>(1\198)
+| 41\212<br />(1\212)
-| 232.08<br>(5.66)
+| 232.08<br />(5.66)
-| 8/7<br>(225/224)
+| 8/7<br />(225/224)
 | [[Schismerc]] / [[cartography]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+== Music ==
+; [[Eliora]]
+* [https://www.youtube.com/watch?v=bvCeNcxUDnA ''Etude in Amicable, Bisesqui, and 53edo'']
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:53edo]]
+[[Category:Agni]]
 [[Category:Kleismic]]
-[[Category:Agni]]
+[[Category:Listen]]
 [[Category:Quadritikleismic]]