400edo: Difference between revisions

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m →Theory: such a giant EDO, if anyone is considering it seriously for general use, deserves analysis in higher limits, so that it should not be collapsed; however, for those interested ive collapsed prime tables > 101
ArrowHead294 (talk | contribs)
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Line 22: Line 22:
=== Selected intervals ===
=== Selected intervals ===
{| class="wikitable center-1"
{| class="wikitable center-1"
|+
|-
! Step
! Step
! Eliora's Naming System
! Eliora's Naming System
Line 81: Line 81:


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| -7 22 -12 }}, {{monzo| 47 -15 -10 }}
| {{monzo| -7 22 -12 }}, {{monzo| 47 -15 -10 }}
| {{mapping| 400 634 929 }}
| {{mapping| 400 634 929 }}
| -0.1080
| &minus;0.1080
| 0.1331
| 0.1331
| 4.44
| 4.44
Line 101: Line 93:
| 2401/2400, 1959552/1953125, 14348907/14336000
| 2401/2400, 1959552/1953125, 14348907/14336000
| {{mapping| 400 634 929 1123 }}
| {{mapping| 400 634 929 1123 }}
| -0.0965
| &minus;0.0965
| 0.1170
| 0.1170
| 3.90
| 3.90
Line 108: Line 100:
| 2401/2400, 5632/5625, 9801/9800, 46656/46585
| 2401/2400, 5632/5625, 9801/9800, 46656/46585
| {{mapping| 400 634 929 1123 1384 }}
| {{mapping| 400 634 929 1123 1384 }}
| -0.1166
| &minus;0.1166
| 0.1121
| 0.1121
| 3.74
| 3.74
Line 115: Line 107:
| 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325
| 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325
| {{mapping| 400 634 929 1123 1384 1480 }}
| {{mapping| 400 634 929 1123 1384 1480 }}
| -0.0734
| &minus;0.0734
| 0.1407
| 0.1407
| 4.69
| 4.69
Line 122: Line 114:
| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 4096/4095
| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 4096/4095
| {{mapping| 400 634 929 1123 1384 1480 1635 }}
| {{mapping| 400 634 929 1123 1384 1480 1635 }}
| -0.0645
| &minus;0.0645
| 0.1321
| 0.1321
| 4.40
| 4.40
Line 129: Line 121:
| 676/675, 936/935, 969/968, 1001/1000, 1156/1155, 1216/1215, 1716/1715
| 676/675, 936/935, 969/968, 1001/1000, 1156/1155, 1216/1215, 1716/1715
| {{mapping| 400 634 929 1123 1384 1480 1635 1699 }}
| {{mapping| 400 634 929 1123 1384 1480 1635 1699 }}
| -0.0413
| &minus;0.0413
| 0.1380
| 0.1380
| 4.60
| 4.60
|}
{{comma basis end}}
* 400et has lower absolute errors than any previous equal temperaments in the 17- and 19-limit. It is the first to beat [[354edo|354]] in the 17-limit, and [[311edo|311]] in the 19-limit; it is bettered by [[422edo|422]] in either subgroup.  
* 400et has lower absolute errors than any previous equal temperaments in the 17- and 19-limit. It is the first to beat [[354edo|354]] in the 17-limit, and [[311edo|311]] in the 19-limit; it is bettered by [[422edo|422]] in either subgroup.  


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 181: Line 167:
|-
|-
| 5
| 5
| 123\400<br>(37\400)
| 123\400<br />(37\400)
| 369.00<br>(111.00)
| 369.00<br />(111.00)
| 1024/891<br>(16/15)
| 1024/891<br />(16/15)
| [[Quintosec]]
| [[Quintosec]]
|-
|-
| 10
| 10
| 83\400<br>(3\400)
| 83\400<br />(3\400)
| 249.00<br>(9.00)
| 249.00<br />(9.00)
| 15/13<br>(176/175)
| 15/13<br />(176/175)
| [[Decoid]]
| [[Decoid]]
|-
|-
| 80
| 80
| 166\400<br>(1\400)
| 166\400<br />(1\400)
| 498.00<br>(3.00)
| 498.00<br />(3.00)
| 4/3<br>(245/243)
| 4/3<br />(245/243)
| [[Octogintic]]
| [[Octogintic]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


== Scales ==
== Scales ==

Revision as of 05:50, 16 November 2024

← 399edo 400edo 401edo →
Prime factorization 24 × 52
Step size 3 ¢ 
Fifth 234\400 (702 ¢) (→ 117\200)
Semitones (A1:m2) 38:30 (114 ¢ : 90 ¢)
Consistency limit 21
Distinct consistency limit 21

Template:EDO intro

Theory

400edo is a strong 17- and 19-limit system, distinctly consistent to the 21-odd-limit. It shares its excellent harmonic 3 with 200edo, which is a semiconvergent, while correcting the higher harmonics to near-just qualities.

The equal temperament tempers out the unidecma, [-7 22 -12⟩, and the quintosec comma, [47 -15 -10⟩, in the 5-limit; 2401/2400, 1959552/1953125, and 14348907/14336000 in the 7-limit; 5632/5625, 9801/9800, 117649/117612, and 131072/130977 in the 11-limit; 676/675, 1001/1000, 1716/1715, 2080/2079, 4096/4095, 4225/4224 and 39366/39325 in the 13-limit, supporting the decoid temperament and the quinmite temperament. It tempers out 936/935, 1156/1155, 2058/2057, 2601/2600, 4914/4913 and 24576/24565 in the 17-limit, and 969/968, 1216/1215, 1521/1520, and 1729/1728 in the 19-limit.

Prime harmonics

Approximation of prime harmonics in 400edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41
Error Absolute (¢) +0.00 +0.04 +0.69 +0.17 +0.68 -0.53 +0.04 -0.51 -1.27 -0.58 +0.96 +0.66 -0.06
Relative (%) +0.0 +1.5 +22.9 +5.8 +22.7 -17.6 +1.5 -17.1 -42.5 -19.2 +32.1 +21.9 -2.1
Steps
(reduced) 		400
(0) 		634
(234) 		929
(129) 		1123
(323) 		1384
(184) 		1480
(280) 		1635
(35) 		1699
(99) 		1809
(209) 		1943
(343) 		1982
(382) 		2084
(84) 		2143
(143)
Approximation of prime harmonics in 400edo (continued)
Harmonic 43 47 53 59 61 67 71 73 79 83 89 97 101
Error Absolute (¢) +1.48 +0.49 -0.50 -0.17 -0.88 -1.31 +0.30 +0.21 +1.46 -0.05 -0.88 +0.10 -0.85
Relative (%) +49.4 +16.4 -16.8 -5.7 -29.5 -43.6 +10.1 +7.0 +48.8 -1.6 -29.3 +3.5 -28.5
Steps
(reduced) 		2171
(171) 		2222
(222) 		2291
(291) 		2353
(353) 		2372
(372) 		2426
(26) 		2460
(60) 		2476
(76) 		2522
(122) 		2550
(150) 		2590
(190) 		2640
(240) 		2663
(263)
Approximation of prime harmonics in 400edo (103 to 157)
Harmonic 103 107 109 113 127 131 137 139 149 151 157
Error Absolute (¢) +1.20 +1.24 -0.82 -0.21 -1.42 -1.11 -0.64 +1.27 +1.00 -1.09 +0.46
Relative (%) +40.0 +41.3 -27.4 -7.2 -47.4 -36.9 -21.3 +42.4 +33.3 -36.2 +15.2
Steps
(reduced) 		2675
(275) 		2697
(297) 		2707
(307) 		2728
(328) 		2795
(395) 		2813
(13) 		2839
(39) 		2848
(48) 		2888
(88) 		2895
(95) 		2918
(118)
Approximation of prime harmonics in 400edo (163 to 229)
Harmonic 163 167 173 179 181 191 193 197 199 211 223 227 229
Error Absolute (¢) -1.47 -1.45 +0.45 +1.42 +0.18 +0.09 +0.05 +0.54 +1.05 -1.32 -1.08 +1.14 +0.96
Relative (%) -49.1 -48.2 +14.9 +47.4 +6.2 +2.8 +1.7 +17.9 +35.0 -44.0 -36.0 +38.1 +31.8
Steps
(reduced) 		2939
(139) 		2953
(153) 		2974
(174) 		2994
(194) 		3000
(200) 		3031
(231) 		3037
(237) 		3049
(249) 		3055
(255) 		3088
(288) 		3120
(320) 		3131
(331) 		3136
(336)

Subsets and supersets

Since 400 factors into 24 × 52, 400edo has subset edos 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200.

Interval table

All intervals

see Table of 400edo intervals

Selected intervals

Step Eliora's Naming System Associated Ratio
0 unison 1/1
28 5/12-meantone semitone 6561/6250
33 small septendecimal semitone 18/17, 55/52
35 septendecimal semitone 17/16
37 diatonic semitone 16/15
99 undevicesimal minor third 19/16
100 symmetric minor third
200 symmetric tritone 99/70, 140/99
231 Gregorian leap week fifth 525/352, 3/2 / (81/80)^(5/12)
234 perfect fifth 3/2
323 harmonic seventh 7/4
372 5/12-meantone seventh 12500/6561
400 octave 2/1

Regular temperament properties

Template:Comma basis begin |- | 2.3.5 | [-7 22 -12⟩, [47 -15 -10⟩ | [⟨400 634 929]] | −0.1080 | 0.1331 | 4.44 |- | 2.3.5.7 | 2401/2400, 1959552/1953125, 14348907/14336000 | [⟨400 634 929 1123]] | −0.0965 | 0.1170 | 3.90 |- | 2.3.5.7.11 | 2401/2400, 5632/5625, 9801/9800, 46656/46585 | [⟨400 634 929 1123 1384]] | −0.1166 | 0.1121 | 3.74 |- | 2.3.5.7.11.13 | 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325 | [⟨400 634 929 1123 1384 1480]] | −0.0734 | 0.1407 | 4.69 |- | 2.3.5.7.11.13.17 | 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 4096/4095 | [⟨400 634 929 1123 1384 1480 1635]] | −0.0645 | 0.1321 | 4.40 |- | 2.3.5.7.11.13.17.19 | 676/675, 936/935, 969/968, 1001/1000, 1156/1155, 1216/1215, 1716/1715 | [⟨400 634 929 1123 1384 1480 1635 1699]] | −0.0413 | 0.1380 | 4.60 Template:Comma basis end

  • 400et has lower absolute errors than any previous equal temperaments in the 17- and 19-limit. It is the first to beat 354 in the 17-limit, and 311 in the 19-limit; it is bettered by 422 in either subgroup.

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 83\400 | 249.00 | [-26 18 -1⟩ | Monzismic |- | 1 | 33\400 | 99.00 | 18/17 | Gregorian leap day |- | 1 | 101\400 | 303.00 | 25/21 | Quinmite |- | 1 | 153\400 | 459.00 | 125/96 | Majvamic |- | 1 | 169\400 | 507.00 | 525/352 | Gregorian leap week |- | 2 | 61\400 | 183.00 | 10/9 | Unidecmic |- | 5 | 123\400
(37\400) | 369.00
(111.00) | 1024/891
(16/15) | Quintosec |- | 10 | 83\400
(3\400) | 249.00
(9.00) | 15/13
(176/175) | Decoid |- | 80 | 166\400
(1\400) | 498.00
(3.00) | 4/3
(245/243) | Octogintic Template:Rank-2 end Template:Orf

Scales

Music

Eliora
Francium
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