401edo: Difference between revisions

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== 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
| 2.3
| {{monzo| 636 -401 }}
| {{monzo| 636 -401 }}
| {{mapping| 401 636 }}
| {{mapping| 401 636 }}
| -0.4060
| &minus;0.4060
| 0.4058
| 0.4058
| 13.56
| 13.56
Line 34: Line 26:
| 15625/15552, {{monzo| 107 -66 -1 }}
| 15625/15552, {{monzo| 107 -66 -1 }}
| {{mapping| 401 636 931 }}
| {{mapping| 401 636 931 }}
| -0.2307
| &minus;0.2307
| 0.4139
| 0.4139
| 13.83
| 13.83
Line 41: Line 33:
| 10976/10935, 15625/15552, 67108864/66706983
| 10976/10935, 15625/15552, 67108864/66706983
| {{mapping| 401 636 931 1126 }}
| {{mapping| 401 636 931 1126 }}
| -0.2400
| &minus;0.2400
| 0.3588
| 0.3588
| 11.99
| 11.99
Line 48: Line 40:
| 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
| &minus;0.1518
| 0.3661
| 0.3661
| 12.23
| 12.23
Line 55: Line 47:
| 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
| &minus;0.1431
| 0.3348
| 0.3348
| 11.19
| 11.19
|}
{{comma basis end}}


=== 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 74: Line 60:
| 75/56
| 75/56
| [[Diatessic]]
| [[Diatessic]]
|}
{{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 ==
Line 83: Line 69:
== Music ==
== Music ==
; [[Francium]]
; [[Francium]]
* [https://www.youtube.com/watch?v=hh-cgGe_zas ''Diana Tessa''] (2023) – diatessic in 403edo tuning
* [https://www.youtube.com/watch?v=hh-cgGe_zas ''Diana Tessa''] (2023) &ndash; diatessic in 403edo tuning


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

Revision as of 03:19, 16 November 2024

← 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

Template:EDO intro

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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 169\401 | 505.74 | 75/56 | Diatessic Template:Rank-2 end Template:Orf

Scales

Music

Francium
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