416edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|416}}
{{ED intro}}


== Theory ==
== Theory ==
416et is consistent to the [[7-odd-limit]] and the [[harmonic]] [[3/1|3]] is about halfway its steps. It is suitable for the 2.9.5.7.11.19.23.29.31.37 [[subgroup]], tempering out [[1540/1539]], [[5632/5625]], [[9801/9800]], 10241/10240, 1045/1044, 26125/26082, 46000/45927, 17600/17577 and 1036/1035. It [[support]]s [[tridecatonic]], [[fermionic]], [[embankment]] and [[polder]].
416et is [[consistent]] to the [[7-odd-limit]], but the error of [[harmonic]] [[3/1|3]] is quite large. Nonetheless, it gives reasonable approximations to harmonics [[5/1|5]], [[7/1|7]], [[9/1|9]], [[11/1|11]], [[19/1|19]], and [[23/1|23]], making it suitable for a 2.9.5.7.11.19.23 [[subgroup]] interpretation, where it notably [[tempering out|tempers out]] [[1331/1330]], [[1540/1539]], [[5632/5625]], [[9801/9800]], and 10241/10240.
 
Using the patent val regardless, it tempers out 321489/320000, 589824/588245, and 703125/702464, [[support]]ing [[tridecatonic]] and [[fermionic]].


=== Odd harmonics ===
=== Odd harmonics ===
Line 9: Line 11:


=== Subsets and supersets ===
=== Subsets and supersets ===
416 factors into 2<sup>5</sup> × 13, with subset edos {{EDOs|2, 4, 8, 13, 16, 26, 32, 52, 104, and 208}}. [[832edo]], which doubles it, gives a good correction to the harmonic 3.
Since 416 factors into 2<sup>5</sup> × 13, 416edo subset edos {{EDOs| 2, 4, 8, 13, 16, 26, 32, 52, 104, and 208 }}. [[832edo]], which doubles it, gives a good correction to the harmonic 3.


== 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" | [[Subgroup]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | [[Mapping]]
! colspan="2" |Tuning Error
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.9
| 2.9
|{{monzo|1319 -416}}
| {{monzo| 1319 -416 }}
|{{mapping|416 1319}}
| {{mapping| 416 1319 }}
| -0.1416
| −0.1416
| 0.1416
| 0.1416
| 4.91
| 4.91
|-
|-
|2.9.5
| 2.9.5
|{{monzo|56 -14 -5}}, {{monzo|-5 -16 24}}
| {{monzo| 56 -14 -5 }}, {{monzo| -5 -16 24 }}
|{{mapping|416 1319 966}}
| {{mapping| 416 1319 966 }}
| -0.1267
| −0.1267
| 0.1175
| 0.1175
| 4.07
| 4.07
|-
|-
|2.9.5.7
| 2.9.5.7
|420175/419904, 102760448/102515625, 1280000000/1275989841
| 420175/419904, 102760448/102515625, {{monzo| 14 -6 7 -4 }}
|{{mapping|416 1319 966 1168}}
| {{mapping| 416 1319 966 1168 }}
| -0.1310
| −0.1310
| 0.1021
| 0.1021
| 3.54
| 3.54
|-
|-
|2.9.5.7.11
| 2.9.5.7.11
|5632/5625, 9801/9800, 41503/41472, 774400000/771895089
| 5632/5625, 9801/9800, 41503/41472, 774400000/771895089
|{{mapping|416 1319 966 1168 1439}}
| {{mapping| 416 1319 966 1168 1439 }}
| -0.0842
| −0.0842
| 0.1308
| 0.1308
| 4.53
| 4.53
|}
|}
== Music ==
; [[birdshite stalactite]]
* [https://www.youtube.com/watch?v=nmUCjbfGpKU ''palate cleanser''] (2024) – tetracot[13] in 416edo tuning

Latest revision as of 12:16, 21 February 2025

← 415edo 416edo 417edo →
Prime factorization 25 × 13
Step size 2.88462 ¢ 
Fifth 243\416 (700.962 ¢)
Semitones (A1:m2) 37:33 (106.7 ¢ : 95.19 ¢)
Dual sharp fifth 244\416 (703.846 ¢) (→ 61\104)
Dual flat fifth 243\416 (700.962 ¢)
Dual major 2nd 71\416 (204.808 ¢)
Consistency limit 7
Distinct consistency limit 7

416 equal divisions of the octave (abbreviated 416edo or 416ed2), also called 416-tone equal temperament (416tet) or 416 equal temperament (416et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 416 equal parts of about 2.88 ¢ each. Each step represents a frequency ratio of 21/416, or the 416th root of 2.

Theory

416et is consistent to the 7-odd-limit, but the error of harmonic 3 is quite large. Nonetheless, it gives reasonable approximations to harmonics 5, 7, 9, 11, 19, and 23, making it suitable for a 2.9.5.7.11.19.23 subgroup interpretation, where it notably tempers out 1331/1330, 1540/1539, 5632/5625, 9801/9800, and 10241/10240.

Using the patent val regardless, it tempers out 321489/320000, 589824/588245, and 703125/702464, supporting tridecatonic and fermionic.

Odd harmonics

Approximation of odd harmonics in 416edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.99 +0.22 +0.40 +0.90 -0.36 -1.10 -0.77 -1.11 -0.40 -0.59 +0.57
Relative (%) -34.4 +7.8 +14.0 +31.1 -12.4 -38.3 -26.6 -38.5 -13.8 -20.4 +19.8
Steps
(reduced) 		659
(243) 		966
(134) 		1168
(336) 		1319
(71) 		1439
(191) 		1539
(291) 		1625
(377) 		1700
(36) 		1767
(103) 		1827
(163) 		1882
(218)

Subsets and supersets

Since 416 factors into 25 × 13, 416edo subset edos 2, 4, 8, 13, 16, 26, 32, 52, 104, and 208. 832edo, which doubles it, gives a good correction to the harmonic 3.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.9 [1319 -416⟩ [⟨416 1319]] −0.1416 0.1416 4.91
2.9.5 [56 -14 -5⟩, [-5 -16 24⟩ [⟨416 1319 966]] −0.1267 0.1175 4.07
2.9.5.7 420175/419904, 102760448/102515625, [14 -6 7 -4⟩ [⟨416 1319 966 1168]] −0.1310 0.1021 3.54
2.9.5.7.11 5632/5625, 9801/9800, 41503/41472, 774400000/771895089 [⟨416 1319 966 1168 1439]] −0.0842 0.1308 4.53

Music

birdshite stalactite
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