401edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Francium (talk | contribs)
4,827 edits
m +categories
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
(5 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|401}}
{{ED intro}}


== Theory ==
== Theory ==
401edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps. Otherwise, it has a reasonable approximation to harmonics [[5/1|5]], [[7/1|7]], [[9/1|9]], [[11/1|11]], and [[13/1|13]], making it suitable for a 2.9.5.7.11.13 [[subgroup]] interpretation.  
401edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps. Otherwise, it has a reasonable approximation to harmonics [[5/1|5]], [[7/1|7]], [[9/1|9]], [[11/1|11]], and [[13/1|13]], making it suitable for a 2.9.5.7.11.13 [[subgroup]] interpretation.  


Using the [[patent val]] nonetheless, the equal temperament [[tempering out|tempers out]] 283115520/282475249 and [[703125/702464]] in the 7-limit; 35156250/35153041, 2097152/2096325, 117649/117612, 226492416/226474325, 9765625/9732096, 42875/42768, 1375/1372, [[5632/5625]], 15488/15435, 202397184/201768035, 102487/102400 and 805255/802816 in the 11-limit. It provides the [[optimal patent val]] for [[diatessic]], the 140 & 261 temperament.
Using the [[patent val]] nonetheless, the equal temperament [[tempering out|tempers out]] 283115520/282475249 and [[703125/702464]] in the 7-limit; 35156250/35153041, 2097152/2096325, 117649/117612, 226492416/226474325, 9765625/9732096, 42875/42768, 1375/1372, [[5632/5625]], 15488/15435, 202397184/201768035, 102487/102400 and 805255/802816 in the 11-limit. It provides the [[optimal patent val]] for [[diatessic]], the {{nowrap|140 & 261}} temperament.


=== Odd harmonics ===
=== Odd harmonics ===
Line 15: Line 15:
== 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 27: Line 28:
| {{monzo| 636 -401 }}
| {{monzo| 636 -401 }}
| {{mapping| 401 636 }}
| {{mapping| 401 636 }}
| -0.4060
| −0.4060
| 0.4058
| 0.4058
| 13.56
| 13.56
Line 34: Line 35:
| 15625/15552, {{monzo| 107 -66 -1 }}
| 15625/15552, {{monzo| 107 -66 -1 }}
| {{mapping| 401 636 931 }}
| {{mapping| 401 636 931 }}
| -0.2307
| −0.2307
| 0.4139
| 0.4139
| 13.83
| 13.83
Line 41: Line 42:
| 10976/10935, 15625/15552, 67108864/66706983
| 10976/10935, 15625/15552, 67108864/66706983
| {{mapping| 401 636 931 1126 }}
| {{mapping| 401 636 931 1126 }}
| -0.2400
| −0.2400
| 0.3588
| 0.3588
| 11.99
| 11.99
Line 48: Line 49:
| 2200/2187, 1375/1372, 5632/5625, 102487/102400
| 2200/2187, 1375/1372, 5632/5625, 102487/102400
| {{mapping| 401​ 636 ​931 ​1126 ​1387 ​}}
| {{mapping| 401​ 636 ​931 ​1126 ​1387 ​}}
| -0.1518
| −0.1518
| 0.3661
| 0.3661
| 12.23
| 12.23
Line 55: Line 56:
| 325/324, 352/351, 625/624, 1375/1372, 3276800/3270267
| 325/324, 352/351, 625/624, 1375/1372, 3276800/3270267
| {{mapping| 401 ​636 ​931 ​1126 ​1387 ​1484 }}
| {{mapping| 401 ​636 ​931 ​1126 ​1387 ​1484 }}
| -0.1431
| −0.1431
| 0.3348
| 0.3348
| 11.19
| 11.19
Line 62: Line 63:
=== 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*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
Line 75: Line 77:
| [[Diatessic]]
| [[Diatessic]]
|}
|}
<nowiki>*</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 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


== Scales ==
== Scales ==
Line 86: Line 88:


[[Category:Listen]]
[[Category:Listen]]
[[Category:Diatessic]]

Latest revision as of 13:31, 13 March 2026

← 400edo 401edo 402edo →
Prime factorization 401 (prime)
Step size 2.99252 ¢ 
Fifth 235\401 (703.242 ¢)
Semitones (A1:m2) 41:28 (122.7 ¢ : 83.79 ¢)
Dual sharp fifth 235\401 (703.242 ¢)
Dual flat fifth 234\401 (700.249 ¢)
Dual major 2nd 68\401 (203.491 ¢)
Consistency limit 3
Distinct consistency limit 3

401 equal divisions of the octave (abbreviated 401edo or 401ed2), also called 401-tone equal temperament (401tet) or 401 equal temperament (401et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 401 equal parts of about 2.99 ¢ each. Each step represents a frequency ratio of 21/401, or the 401st root of 2.

Theory

401edo is inconsistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. Otherwise, it has a reasonable approximation to harmonics 5, 7, 9, 11, and 13, making it suitable for a 2.9.5.7.11.13 subgroup interpretation.

Using the patent val nonetheless, the equal temperament tempers out 283115520/282475249 and 703125/702464 in the 7-limit; 35156250/35153041, 2097152/2096325, 117649/117612, 226492416/226474325, 9765625/9732096, 42875/42768, 1375/1372, 5632/5625, 15488/15435, 202397184/201768035, 102487/102400 and 805255/802816 in the 11-limit. It provides the optimal patent val for diatessic, the 140 & 261 temperament.

Odd harmonics

Approximation of odd harmonics in 401edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.29 -0.28 +0.75 -0.42 -0.69 +0.37 +1.01 -0.22 -1.25 -0.96 +0.15
Relative (%) +43.0 -9.3 +25.1 -14.0 -23.2 +12.4 +33.7 -7.3 -41.9 -31.9 +5.2
Steps
(reduced) 		636
(235) 		931
(129) 		1126
(324) 		1271
(68) 		1387
(184) 		1484
(281) 		1567
(364) 		1639
(35) 		1703
(99) 		1761
(157) 		1814
(210)

Subsets and supersets

401edo is the 79th prime edo

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [636 -401⟩ [⟨401 636]] −0.4060 0.4058 13.56
2.3.5 15625/15552, [107 -66 -1⟩ [⟨401 636 931]] −0.2307 0.4139 13.83
2.3.5.7 10976/10935, 15625/15552, 67108864/66706983 [⟨401 636 931 1126]] −0.2400 0.3588 11.99
2.3.5.7.11 2200/2187, 1375/1372, 5632/5625, 102487/102400 [⟨401​ 636 ​931 ​1126 ​1387 ​]] −0.1518 0.3661 12.23
2.3.5.7.11.13 325/324, 352/351, 625/624, 1375/1372, 3276800/3270267 [⟨401 ​636 ​931 ​1126 ​1387 ​1484]] −0.1431 0.3348 11.19

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 169\401 505.74 75/56 Diatessic

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

Scales

Music

Francium
Retrieved from "https://en.xen.wiki/index.php?title=401edo&oldid=225780"