362edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|362}}
{{ED intro}}


== Theory ==
== Theory ==
362et is only consistent to the [[5-odd-limit]], with three mappings possible for the [[7-limit]]:
362edo is [[enfactoring|enfactored]] in the [[3-limit]] and is only [[consistent]] to the [[5-odd-limit]], with two mappings possible for the [[7-limit]]:
* {{val|362 574 841 1016}} (patent val),
* {{val| 362 574 841 1016 }} (patent val),
* {{val|362 574 '''840''' 1016}} (362c),
* {{val| 362 574 841 '''1017''' }} (362d).
* {{val|362 574 841 '''1017'''}} (362d).
 
Using the patent val, it tempers out [[393216/390625]] and {{monzo|25 -48 22}} in the 5-limit; [[4375/4374]], [[458752/455625]] and 11529602/11390625 in the 7-limit, [[support]]ing [[barbados]].
Using the patent val, it tempers out [[393216/390625]] and {{monzo| 25 -48 22 }} in the 5-limit; [[4375/4374]], [[458752/455625]] and 11529602/11390625 in the 7-limit, [[support]]ing [[barbados]].
Using the 362c val, it tempers out [[2109375/2097152]] and {{monzo|14 -22 9}} in the 5-limit; [[2401/2400]], [[10976/10935]] and 390625/387072 in the 7-limit.
 
Using the 362d val, it tempers out 393216/390625 and {{monzo|25 -48 22}} in the 5-limit; [[5120/5103]], 118098/117649 and 1959552/1953125 in the 7-limit.
Using the 362d val, it tempers out 393216/390625 and {{monzo| 25 -48 22 }} in the 5-limit; [[5120/5103]], 118098/117649 and 1959552/1953125 in the 7-limit.


=== Odd harmonics ===
=== Odd harmonics ===
Line 15: Line 15:


=== Subsets and supersets ===
=== Subsets and supersets ===
362 factors into 2 × 181, with [[2edo]] and [[181edo]] as its subset edos. [[1448edo]], which quadruples it, gives a good correction to the harmonic 7.
Since 362 factors into 2 × 181, 372edo has [[2edo]] and [[181edo]] as its subsets. [[1448edo]], which quadruples it, is a strong full 13-limit system.  


== 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|287 -181}}
! rowspan="2" | [[Comma list]]
|{{mapping|362 574}}
! rowspan="2" | [[Mapping]]
| -0.2547
! rowspan="2" | Optimal<br />8ve stretch (¢)
| 0.2547
! colspan="2" | Tuning error
| 7.68
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3.5
| 2.3.5
|393216/390625, {{monzo|25 -48 22}}
| 393216/390625, {{monzo| 25 -48 22 }}
|{{mapping|362 574 841}}
| {{mapping| 362 574 841 }}
| -0.3896
| −0.3896
| 0.2822
| 0.2822
| 8.51
| 8.51
Line 45: Line 39:
=== 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
|-
|-
|1
| 1
|117\362
| 117\362
|387.85
| 387.85
|5/4
| 5/4
|[[Würschmidt}}
| [[Würschmidt]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

Latest revision as of 12:18, 21 February 2025

← 361edo 362edo 363edo →
Prime factorization 2 × 181
Step size 3.31492 ¢ 
Fifth 212\362 (702.762 ¢) (→ 106\181)
Semitones (A1:m2) 36:26 (119.3 ¢ : 86.19 ¢)
Consistency limit 5
Distinct consistency limit 5

362 equal divisions of the octave (abbreviated 362edo or 362ed2), also called 362-tone equal temperament (362tet) or 362 equal temperament (362et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 362 equal parts of about 3.31 ¢ each. Each step represents a frequency ratio of 21/362, or the 362nd root of 2.

Theory

362edo is enfactored in the 3-limit and is only consistent to the 5-odd-limit, with two mappings possible for the 7-limit:

  • ⟨362 574 841 1016] (patent val),
  • ⟨362 574 841 1017] (362d).

Using the patent val, it tempers out 393216/390625 and [25 -48 22⟩ in the 5-limit; 4375/4374, 458752/455625 and 11529602/11390625 in the 7-limit, supporting barbados.

Using the 362d val, it tempers out 393216/390625 and [25 -48 22⟩ in the 5-limit; 5120/5103, 118098/117649 and 1959552/1953125 in the 7-limit.

Odd harmonics

Approximation of odd harmonics in 362edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.81 +1.53 -0.87 +1.61 -1.04 +1.46 -0.98 +1.12 +0.83 -0.06 +1.56
Relative (%) +24.4 +46.2 -26.2 +48.7 -31.4 +44.1 -29.4 +33.8 +25.0 -1.9 +47.1
Steps
(reduced) 		574
(212) 		841
(117) 		1016
(292) 		1148
(62) 		1252
(166) 		1340
(254) 		1414
(328) 		1480
(32) 		1538
(90) 		1590
(142) 		1638
(190)

Subsets and supersets

Since 362 factors into 2 × 181, 372edo has 2edo and 181edo as its subsets. 1448edo, which quadruples it, is a strong full 13-limit system.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 393216/390625, [25 -48 22⟩ [⟨362 574 841]] −0.3896 0.2822 8.51

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 117\362 387.85 5/4 Würschmidt

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

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