422edo: Difference between revisions

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{{novelty}}{{stub}}{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|422}}
{{ED intro}}


== Theory ==
== Theory ==
422edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is distinctly [[consistent]] through the [[27-odd-limit]], with harmonics of 3 through 23 all tuned sharp. In the 5-limit it tempers out the [[vishnuzma]], {{monzo| 23 6 -14 }}; and in the 7-limit [[4375/4374]] and 589824/588245 so that it [[support]]s the [[gamera]] temperament, and provides its [[optimal patent val]], and also supports the [[vishnu]] temperament.  
422edo is a [[zeta peak edo]], though not zeta integral nor zeta gap. It is [[consistency|distinctly consistent]] through the [[27-odd-limit]], with [[harmonic]]s of 3 through 23 all tuned sharp. As an equal temperament, it [[tempering out|tempers out]] the [[vishnuzma]], {{monzo| 23 6 -14 }} and the countritonic comma, {{monzo| 33 -34 9 }}, in the 5-limit; [[4375/4374]] and [[589824/588245]] in the 7-limit; [[3025/3024]], [[5632/5625]], and [[9801/9800]] in the 11-limit; [[1716/1715]], [[2080/2079]], and [[2200/2197]] in the 13-limit; [[1156/1155]], [[1275/1274]], and [[2431/2430]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]], and [[2432/2431]] in the 19-limit; and [[736/735]], [[1496/1495]], and [[1863/1862]] in the 23-limit. It [[support]]s and provides the [[optimal patent val]]s for [[gamera]] in the 7-limit, [[hemigamera]] in the 13-limit, and [[ragitritonic]] in the 11- and 13-limit. Other notable temperaments it supports are [[vishnu]] and [[semisupermajor]].  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|422|columns=11}}
{{Harmonics in equal|422|columns=11}}
{{Harmonics in equal|422|columns=11|start=12|collapsed=1|title=Approximation of prime harmonics in 422edo (continued)}}
{{Harmonics in equal|422|columns=11|start=23|collapsed=1|title=Approximation of prime harmonics in 422edo (continued)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
422edo has subset edos [[2edo]] and [[211edo]].  
Since 422 factors into primes as {{nowrap| 2 × 211 }}, 422edo has subset edos [[2edo]] and [[211edo]].


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 23: Line 26:
|-
|-
| 2.3
| 2.3
| {{monzo| 669 -422 }}
| {{Monzo| 669 -422 }}
| [{{val| 422 669 }}]
| {{Mapping| 422 669 }}
| -0.1308
| −0.1308
| 0.1308
| 0.1308
| 4.60
| 4.60
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 23 6 -14 }}, {{monzo| 33 -34 9 }}
| {{Monzo| 23 6 -14 }}, {{monzo| 33 -34 9 }}
| [{{val| 422 669 980 }}]
| {{Mapping| 422 669 980 }}
| -0.1469
| −0.1469
| 0.1092
| 0.1092
| 3.84
| 3.84
Line 38: Line 41:
| 2.3.5.7
| 2.3.5.7
| 4375/4374, 589824/588245, 29360128/29296875
| 4375/4374, 589824/588245, 29360128/29296875
| [{{val| 422 669 980 1185 }}]
| {{Mapping| 422 669 980 1185 }}
| -0.1852
| −0.1852
| 0.1155
| 0.1155
| 4.06
| 4.06
Line 45: Line 48:
| 2.3.5.7.11
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5632/5625, 589824/588245
| 3025/3024, 4375/4374, 5632/5625, 589824/588245
| [{{val| 422 669 980 1185 1460 }}]
| {{Mapping| 422 669 980 1185 1460 }}
| -0.1679
| −0.1679
| 0.1090
| 0.1090
| 3.83
| 3.83
Line 52: Line 55:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625
| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625
| [{{val| 422 669 980 1185 1460 1562 }}]
| {{Mapping| 422 669 980 1185 1460 1562 }}
| -0.1930
| −0.1930
| 0.1142
| 0.1142
| 4.02
| 4.02
Line 59: Line 62:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430
| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430
| [{{val| 422 669 980 1185 1460 1562 1725 }}]
| {{Mapping| 422 669 980 1185 1460 1562 1725 }}
| -0.1744
| −0.1744
| 0.1151
| 0.1151
| 4.05
| 4.05
Line 66: Line 69:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197
| 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197
| [{{val| 422 669 980 1185 1460 1562 1725 1793 }}]
| {{Mapping| 422 669 980 1185 1460 1562 1725 1793 }}
| -0.1839
| −0.1839
| 0.1106
| 0.1106
| 3.89
| 3.89
|-
| 2.3.5.7.11.13.17.19.23
| 736/735, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1496/1495, 1716/1715
| {{Mapping| 422 669 980 1185 1460 1562 1725 1793 1909 }}
| −0.1675
| 0.1142
| 4.02
|}
|}
* 422et has lower absolute errors than any previous equal temperaments in the 17-, 19- and 23-limit. In the 17- and 19-limit it beats [[400edo|400]] and is bettered by [[460edo|460]]. In the 23-limit it beats [[373edo|373g]] and is bettered by [[525edo|525]].  
* 422et has lower absolute errors than any previous equal temperaments in the 17-, 19-, and 23-limit. In the 17- and 19-limit it beats [[400edo|400]] and is bettered by [[460edo|460]]. In the 23-limit it beats [[373edo|373g]] and is bettered by [[525edo|525]].  


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| 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 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
Line 98: Line 109:
| 435.07
| 435.07
| 9/7
| 9/7
| [[Supermajor]]
| [[Supermajor (temperament)|Supermajor]]
|-
| 1
| 207\422
| 588.63
| 128/91
| [[Ragitritonic]]
|-
|-
| 2
| 2
Line 118: Line 135:
| [[Semisupermajor]]
| [[Semisupermajor]]
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


[[Category:Ragitritonic]]
[[Category:Gamera]]
[[Category:Gamera]]
[[Category:Vishnu]]
[[Category:Vishnu]]

Latest revision as of 11:11, 20 May 2026

← 421edo 422edo 423edo →
Prime factorization 2 × 211
Step size 2.8436 ¢ 
Fifth 247\422 (702.37 ¢)
Semitones (A1:m2) 41:31 (116.6 ¢ : 88.15 ¢)
Consistency limit 27
Distinct consistency limit 27

422 equal divisions of the octave (abbreviated 422edo or 422ed2), also called 422-tone equal temperament (422tet) or 422 equal temperament (422et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 422 equal parts of about 2.84 ¢ each. Each step represents a frequency ratio of 21/422, or the 422nd root of 2.

Theory

422edo is a zeta peak edo, though not zeta integral nor zeta gap. It is distinctly consistent through the 27-odd-limit, with harmonics of 3 through 23 all tuned sharp. As an equal temperament, it tempers out the vishnuzma, [23 6 -14⟩ and the countritonic comma, [33 -34 9⟩, in the 5-limit; 4375/4374 and 589824/588245 in the 7-limit; 3025/3024, 5632/5625, and 9801/9800 in the 11-limit; 1716/1715, 2080/2079, and 2200/2197 in the 13-limit; 1156/1155, 1275/1274, and 2431/2430 in the 17-limit; 1216/1215, 1331/1330, 1445/1444, and 2432/2431 in the 19-limit; and 736/735, 1496/1495, and 1863/1862 in the 23-limit. It supports and provides the optimal patent vals for gamera in the 7-limit, hemigamera in the 13-limit, and ragitritonic in the 11- and 13-limit. Other notable temperaments it supports are vishnu and semisupermajor.

Prime harmonics

Approximation of prime harmonics in 422edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.41 +0.42 +0.84 +0.34 +1.18 +0.26 +1.07 +0.16 -0.19 +0.94
Relative (%) +0.0 +14.6 +14.6 +29.6 +12.0 +41.4 +9.1 +37.5 +5.7 -6.8 +32.9
Steps
(reduced) 		422
(0) 		669
(247) 		980
(136) 		1185
(341) 		1460
(194) 		1562
(296) 		1725
(37) 		1793
(105) 		1909
(221) 		2050
(362) 		2091
(403)
Approximation of prime harmonics in 422edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) -1.11 +0.32 +0.33 -0.10 -0.52 -1.35 +0.65 +0.31 -0.55 -0.30 -0.56
Relative (%) -38.9 +11.3 +11.6 -3.6 -18.2 -47.5 +22.9 +11.0 -19.3 -10.6 -19.5
Steps
(reduced) 		2198
(88) 		2261
(151) 		2290
(180) 		2344
(234) 		2417
(307) 		2482
(372) 		2503
(393) 		2560
(28) 		2595
(63) 		2612
(80) 		2660
(128)
Approximation of prime harmonics in 422edo (continued)
Harmonic 83 89 97 101 103 107 109 113 127 131 137
Error Absolute (¢) -0.76 +0.68 -0.46 +0.67 +0.84 +0.29 -0.49 -0.33 -0.64 -0.30 -1.05
Relative (%) -26.7 +24.0 -16.3 +23.5 +29.7 +10.1 -17.4 -11.6 -22.5 -10.5 -37.0
Steps
(reduced) 		2690
(158) 		2733
(201) 		2785
(253) 		2810
(278) 		2822
(290) 		2845
(313) 		2856
(324) 		2878
(346) 		2949
(417) 		2968
(14) 		2995
(41)

Subsets and supersets

Since 422 factors into primes as 2 × 211, 422edo has subset edos 2edo and 211edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [669 -422⟩ [⟨422 669]] −0.1308 0.1308 4.60
2.3.5 [23 6 -14⟩, [33 -34 9⟩ [⟨422 669 980]] −0.1469 0.1092 3.84
2.3.5.7 4375/4374, 589824/588245, 29360128/29296875 [⟨422 669 980 1185]] −0.1852 0.1155 4.06
2.3.5.7.11 3025/3024, 4375/4374, 5632/5625, 589824/588245 [⟨422 669 980 1185 1460]] −0.1679 0.1090 3.83
2.3.5.7.11.13 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625 [⟨422 669 980 1185 1460 1562]] −0.1930 0.1142 4.02
2.3.5.7.11.13.17 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430 [⟨422 669 980 1185 1460 1562 1725]] −0.1744 0.1151 4.05
2.3.5.7.11.13.17.19 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197 [⟨422 669 980 1185 1460 1562 1725 1793]] −0.1839 0.1106 3.89
2.3.5.7.11.13.17.19.23 736/735, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1496/1495, 1716/1715 [⟨422 669 980 1185 1460 1562 1725 1793 1909]] −0.1675 0.1142 4.02
  • 422et has lower absolute errors than any previous equal temperaments in the 17-, 19-, and 23-limit. In the 17- and 19-limit it beats 400 and is bettered by 460. In the 23-limit it beats 373g and is bettered by 525.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 81\422 230.33 8/7 Gamera
1 111\422 315.64 6/5 Egads
1 153\422 435.07 9/7 Supermajor
1 207\422 588.63 128/91 Ragitritonic
2 25\422 71.09 25/24 Vishnu / acyuta
2 81\422 230.33 8/7 Hemigamera
2 153\422
(58\422) 		435.07
(164.93) 		9/7
(11/10) 		Semisupermajor

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

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