422edo: Difference between revisions

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== Theory ==
== Theory ==
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. The equal temperament [[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 provides the [[optimal patent val]]s for [[gamera]] in the 7-limit and [[hemigamera]] in the 13-limit. Other notable temperaments it supports are [[vishnu]], [[semisupermajor]], and [[countritonic]].  
422edo is a [[zeta peak edo]], though not zeta integral nor zeta gap. It is [[distinctly consistent]] through the [[27-odd-limit]], with [[harmonic]]s of 3 through 23 all tuned sharp. It [[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 provides the [[optimal patent val]]s for [[gamera]] in the 7-limit and [[hemigamera]] in the 13-limit. Other notable temperaments it supports are [[vishnu]], [[semisupermajor]], and [[countritonic]].  


=== Prime harmonics ===
=== Prime harmonics ===
Line 26: Line 26:
| {{monzo| 669 -422 }}
| {{monzo| 669 -422 }}
| {{mapping| 422 669 }}
| {{mapping| 422 669 }}
| −0.1308
| −0.1308
| 0.1308
| 0.1308
| 4.60
| 4.60
Line 33: Line 33:
| {{monzo| 23 6 -14 }}, {{monzo| 33 -34 9 }}
| {{monzo| 23 6 -14 }}, {{monzo| 33 -34 9 }}
| {{mapping| 422 669 980 }}
| {{mapping| 422 669 980 }}
| −0.1469
| −0.1469
| 0.1092
| 0.1092
| 3.84
| 3.84
Line 40: Line 40:
| 4375/4374, 589824/588245, 29360128/29296875
| 4375/4374, 589824/588245, 29360128/29296875
| {{mapping| 422 669 980 1185 }}
| {{mapping| 422 669 980 1185 }}
| −0.1852
| −0.1852
| 0.1155
| 0.1155
| 4.06
| 4.06
Line 47: Line 47:
| 3025/3024, 4375/4374, 5632/5625, 589824/588245
| 3025/3024, 4375/4374, 5632/5625, 589824/588245
| {{mapping| 422 669 980 1185 1460 }}
| {{mapping| 422 669 980 1185 1460 }}
| −0.1679
| −0.1679
| 0.1090
| 0.1090
| 3.83
| 3.83
Line 54: Line 54:
| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625
| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625
| {{mapping| 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 61: Line 61:
| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430
| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430
| {{mapping| 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 68: Line 68:
| 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
| {{mapping| 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
Line 75: Line 75:
| 736/735, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1496/1495, 1716/1715
| 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 }}
| {{mapping| 422 669 980 1185 1460 1562 1725 1793 1909 }}
| −0.1675
| −0.1675
| 0.1142
| 0.1142
| 4.02
| 4.02
Line 133: Line 133:
| [[Semisupermajor]]
| [[Semisupermajor]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


[[Category:Countritonic]]
[[Category:Countritonic]]
[[Category:Gamera]]
[[Category:Gamera]]
[[Category:Vishnu]]
[[Category:Vishnu]]

Revision as of 14:02, 16 January 2025

← 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

Template:EDO intro

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. 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 provides the optimal patent vals for gamera in the 7-limit and hemigamera in the 13-limit. Other notable temperaments it supports are vishnu, semisupermajor, and countritonic.

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)

Subsets and supersets

Since 422 factors into 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 Countritonic
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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