421edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|421}} == Theory == 421et is only consistent to the 3-odd-limit, with its harmonic 5 being way too sharp. It is suitable for the 2.3.7.11.13.29...."
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|421}}
{{ED intro}}


== Theory ==
== Theory ==
421et is only consistent to the [[3-odd-limit]], with its harmonic 5 being way too sharp. It is suitable for the 2.3.7.11.13.29.37 [[subgroup]], tempering out 638/637, 53361/53248, 88209/87808, [[5292/5291]], 24192/24167 and 85293/85184.
421edo is in[[consistent]] to the [[5-odd-limit]], with its [[harmonic]] [[5/1|5]] being way too sharp. To start with, consider the following [[breed]]s:
* {{val| 421 667 '''977''' 1182 }} (421c)
* {{val| 421 667 '''977''' '''1181''' }} (421cd)
* {{val| 421 667 978 1182 }} ([[patent val]])
 
The 421c val [[tempering out|tempers out]] [[4375/4374]] and [[2100875/2097152]], [[support]]ing [[mitonic]].
 
The 421cd val tempers out [[1029/1024]] and 823543/820125.
 
The 421 val tempers out [[2401/2400]] and [[3136/3125]], supporting [[hemiwürschmidt]].
 
Omitting harmonic 5, it is suitable for the 2.3.7.11.13.29.37 [[subgroup]], where it tempers out 638/637, [[5292/5291]], 24192/24167, 53361/53248, 88209/87808 and 85293/85184.


=== Odd harmonics ===
=== Odd harmonics ===
Line 13: Line 24:
== 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" |[[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
! rowspan="2" | [[Subgroup]]
|{{monzo|-667 421}}
! rowspan="2" | [[Comma list]]
|{{mapping|421 667}}
! rowspan="2" | [[Mapping]]
| 0.2421
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -667 421 }}
| {{mapping| 421 667 }}
| +0.2421
| 0.2421
| 0.2421
| 8.49
| 8.49
|-
|-
|2.3.7
| 2.3.7
|{{monzo|-44 26 1}}, {{monzo|37 5 -16}}
| {{monzo| -44 26 1 }}, {{monzo| 37 5 -16 }}
|{{mapping|421 667 1182}}
| {{mapping| 421 667 1182 }}
| 0.1263
| +0.1263
| 0.2567
| 0.2567
| 9.01
| 9.01
|-
|-
|2.3.7.11
| 2.3.7.11
|88209/87808, 2893401/2883584, 208971104256/208422380089
| 88209/87808, 2893401/2883584, 208971104256/208422380089
|{{mapping|421 667 1182 1456}}
| {{mapping| 421 667 1182 1456 }}
| 0.1814
| +0.1814
| 0.2419
| 0.2419
| 8.49
| 8.49
|-
|-
|2.3.7.11.13
| 2.3.7.11.13
|53361/53248, 88209/87808, 24192/24167, 85293/85184
| 24192/24167, 53361/53248, 85293/85184, 88209/87808
|{{mapping|421 667 1182 1456 1558}}
| {{mapping| 421 667 1182 1456 1558 }}
| 0.1274
| +0.1274
| 0.2418
| 0.2418
|8.48
| 8.48
|}
|}
== Music ==
; [[Francium]]
* "Cuckoo Muskrat" from ''Cursed Cuckoo Creations'' (2024) – [https://open.spotify.com/track/0Gqr9upShTnDW0UXWbHoMK Spotify] | [https://francium223.bandcamp.com/track/cuckoo-muskrat Bandcamp] | [https://www.youtube.com/watch?v=YFjcW-tiC-A YouTube]
* "Smash" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/4BScJ5SmSlJWMohlDBvbkw Spotify] | [https://francium223.bandcamp.com/track/smash Bandcamp] | [https://www.youtube.com/watch?v=t-80EKfzxW8 YouTube]
[[Category:Listen]]

Latest revision as of 12:47, 21 February 2025

← 420edo 421edo 422edo →
Prime factorization 421 (prime)
Step size 2.85036 ¢ 
Fifth 246\421 (701.188 ¢)
Semitones (A1:m2) 38:33 (108.3 ¢ : 94.06 ¢)
Consistency limit 3
Distinct consistency limit 3

421 equal divisions of the octave (abbreviated 421edo or 421ed2), also called 421-tone equal temperament (421tet) or 421 equal temperament (421et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 421 equal parts of about 2.85 ¢ each. Each step represents a frequency ratio of 21/421, or the 421st root of 2.

Theory

421edo is inconsistent to the 5-odd-limit, with its harmonic 5 being way too sharp. To start with, consider the following breeds:

  • ⟨421 667 977 1182] (421c)
  • ⟨421 667 977 1181] (421cd)
  • ⟨421 667 978 1182] (patent val)

The 421c val tempers out 4375/4374 and 2100875/2097152, supporting mitonic.

The 421cd val tempers out 1029/1024 and 823543/820125.

The 421 val tempers out 2401/2400 and 3136/3125, supporting hemiwürschmidt.

Omitting harmonic 5, it is suitable for the 2.3.7.11.13.29.37 subgroup, where it tempers out 638/637, 5292/5291, 24192/24167, 53361/53248, 88209/87808 and 85293/85184.

Odd harmonics

Approximation of odd harmonics in 421edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.77 +1.33 +0.30 +1.32 -1.20 +0.33 +0.57 +0.51 -1.08 -0.47 -1.20
Relative (%) -26.9 +46.8 +10.4 +46.2 -42.1 +11.5 +19.9 +17.8 -37.7 -16.6 -42.0
Steps
(reduced) 		667
(246) 		978
(136) 		1182
(340) 		1335
(72) 		1456
(193) 		1558
(295) 		1645
(382) 		1721
(37) 		1788
(104) 		1849
(165) 		1904
(220)

Subsets and supersets

421edo is the 82nd prime edo. 1263edo, which triples it, gives a good correction to the harmonic 5.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-667 421⟩ [⟨421 667]] +0.2421 0.2421 8.49
2.3.7 [-44 26 1⟩, [37 5 -16⟩ [⟨421 667 1182]] +0.1263 0.2567 9.01
2.3.7.11 88209/87808, 2893401/2883584, 208971104256/208422380089 [⟨421 667 1182 1456]] +0.1814 0.2419 8.49
2.3.7.11.13 24192/24167, 53361/53248, 85293/85184, 88209/87808 [⟨421 667 1182 1456 1558]] +0.1274 0.2418 8.48

Music

Francium
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