358edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|358}}
{{ED intro}}


== Theory ==
== Theory ==
358et is consistent to the [[7-odd-limit]] and the [[harmonic]] 3 is about halfway its steps. It is suitable for use with the 2.9.5.7.11.17.23.29.37.41 [[subgroup]]. Using the patent val, it tempers out 283115520/282475249, 26873856/26796875, 48828125/48771072, [[2100875/2097152]] and 200120949/200000000 in the 7-limit; 20155392/20131375, 234375/234256, 29296875/29218112, 46656/46585, 1953125/1951488, 226492416/226474325, [[540/539]], 107495424/107421875, 5788125/5767168, 5767168/5764801, 1375/1372, [[3025/3024]], 766656/765625, 4108797/4096000 and 644204/643125 in the 11-limit. It supports [[hypnos]] and [[lee]].
358edo is [[consistent]] to the [[7-odd-limit]], but the [[harmonic]] [[3/1|3]] is about halfway its steps. It is suitable for use with the 2.9.5.7.13 [[subgroup]] or even better the 2.9.15.7.13 subgroup.  
 
In the 2.9.5.7.13 subgroup, the equal temperament [[tempering out|tempers out]] [[4096/4095]], [[13720/13689]], 59150/59049, 60025/59904, 142884/142805, and [[390625/388962]].  
 
The [[patent val]] [[support]]s [[hypnos]] in the 11-limit and [[lee]] in the 2.3.7 subgroup.


=== Odd harmonics ===
=== Odd harmonics ===
Line 13: Line 17:
== 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]] (%)
|-
|-
|2.9
! rowspan="2" | [[Subgroup]]
|{{monzo|1135 -358}}
! rowspan="2" | [[Comma list]]
|{{mapping|358 1135}}
! rowspan="2" | [[Mapping]]
| -0.0882
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.9
| {{monzo| 1135 -358 }}
| {{mapping| 358 1135 }}
| −0.0882
| 0.0882
| 0.0882
| 2.63
| 2.63
|-
|-
|2.9.5
| 2.9.5
|{{monzo|3 -9 11}}, {{monzo|-98 17 19}}
| {{monzo| 3 -9 11 }}, {{monzo| -98 17 19 }}
|{{mapping|358 1135 831}}
| {{mapping| 358 1135 831 }}
| +0.0616
| +0.0616
| 0.2238
| 0.2238
| 6.68
| 6.68
|-
|-
|2.9.5.7
| 2.9.5.7
|390625/388962, 4802000/4782969, 2100875/2097152
| 390625/388962, 2100875/2097152, 4802000/4782969
|{{mapping|358 1135 831 1005}}
| {{mapping| 358 1135 831 1005 }}
| +0.0561
| +0.0561
| 0.1941
| 0.1941
| 5.79
| 5.79
|}
|}

Latest revision as of 20:17, 6 May 2025

← 357edo 358edo 359edo →
Prime factorization 2 × 179
Step size 3.35196 ¢ 
Fifth 209\358 (700.559 ¢)
Semitones (A1:m2) 31:29 (103.9 ¢ : 97.21 ¢)
Dual sharp fifth 210\358 (703.911 ¢) (→ 105\179)
Dual flat fifth 209\358 (700.559 ¢)
Dual major 2nd 61\358 (204.469 ¢)
Consistency limit 7
Distinct consistency limit 7

358 equal divisions of the octave (abbreviated 358edo or 358ed2), also called 358-tone equal temperament (358tet) or 358 equal temperament (358et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 358 equal parts of about 3.35 ¢ each. Each step represents a frequency ratio of 21/358, or the 358th root of 2.

Theory

358edo is consistent to the 7-odd-limit, but the harmonic 3 is about halfway its steps. It is suitable for use with the 2.9.5.7.13 subgroup or even better the 2.9.15.7.13 subgroup.

In the 2.9.5.7.13 subgroup, the equal temperament tempers out 4096/4095, 13720/13689, 59150/59049, 60025/59904, 142884/142805, and 390625/388962.

The patent val supports hypnos in the 11-limit and lee in the 2.3.7 subgroup.

Odd harmonics

Approximation of odd harmonics in 358edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.40 -0.84 -0.11 +0.56 -1.60 +0.81 +1.12 -1.04 +0.81 -1.51 -1.46
Relative (%) -41.7 -25.0 -3.3 +16.7 -47.7 +24.3 +33.3 -31.2 +24.2 -45.0 -43.5
Steps
(reduced) 		567
(209) 		831
(115) 		1005
(289) 		1135
(61) 		1238
(164) 		1325
(251) 		1399
(325) 		1463
(31) 		1521
(89) 		1572
(140) 		1619
(187)

Subset and supersets

358 factors into 2 × 179, with 2edo and 179edo as its subset edos. 716edo, 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 [1135 -358⟩ [⟨358 1135]] −0.0882 0.0882 2.63
2.9.5 [3 -9 11⟩, [-98 17 19⟩ [⟨358 1135 831]] +0.0616 0.2238 6.68
2.9.5.7 390625/388962, 2100875/2097152, 4802000/4782969 [⟨358 1135 831 1005]] +0.0561 0.1941 5.79
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