402edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|402}}
{{ED intro}}


== Theory ==
== Theory ==
402et is only consistent to the 5-limit, tempering out the [[semicomma]]. There are three possible mappings in the 7-limit:
402edo is only [[consistent]] to the [[5-odd-limit]]. There are three possible mappings in the 7-limit:
* {{val|402 637 933 1129}} (patent val)
* {{val| 402 637 933 1129 }} ([[patent val]])
* {{val|402 637 '''934''' 1129}} (402c)
* {{val| 402 637 933 '''1128''' }} (402d)
* {{val|402 637 933 '''1128'''}} (402d)
* {{val| 402 637 '''934''' 1129 }} (402c)


Using the patent val, it tempers out 4375/4374, 7381125/7340032 and 3200000/3176523 in the 7-limit. It supports [[ragismic]] and [[abigail]].
Using the patent val, it tempers out the [[semicomma]] in the 5-limit and [[4375/4374]], 7381125/7340032, and 3200000/3176523 in the 7-limit, [[support]]ing [[Abigail]].


Using the 402c val, it tempers out 3136/3125, 321489/320000 and 13060694016/12867859375 in the 7-limit. It supports [[bischismic]] and [[parahemwuer]].
Using the 402d val, it tempers out [[250047/250000]], and 1500625/1492992 and 2460375/2458624 in the 7-limit.  


Using the 402d val, it tempers out 250047/250000, 1500625/1492992 and 2460375/2458624 in the 7-limit. It supports the [[landscape]] temperament.
Using the 402c val, it tempers out the [[schisma]] in the 5-limit; and [[3136/3125]], 321489/320000, and 13060694016/12867859375 in the 7-limit, supporting [[bischismic]].


=== Odd harmonics ===
=== Odd harmonics ===
Line 18: Line 18:


=== Subsets and supersets ===
=== Subsets and supersets ===
402 factors into 2 × 3 × 67, with subset edos {{EDOs|2, 3, 6, 67, 134, and 201}}. [[804edo]], which doubles it, gives a good correction to the harmonic 7.
Since 402 factors into {{nowrap|2 × 3 × 67}}, 402edo has subset edos {{EDOs| 2, 3, 6, 67, 134, and 201 }}. [[804edo]], which doubles it, gives a good correction to the harmonics 5 and 7.


== 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|-637 402}}
! rowspan="2" | [[Comma list]]
|{{mapping|402 637}}
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -637 402 }}
| {{mapping| 402 637 }}
| 0.1459
| 0.1459
| 0.1459
| 0.1459
| 4.89
| 4.89
|-
|-
|2.3.5
| 2.3.5
|2109375/2097152, {{monzo|25 -48 22}}
| 2109375/2097152, {{monzo| 25 -48 22 }}
|{{mapping|402 637 933}}
| {{mapping| 402 637 933 }}
| 0.2752
| 0.2752
| 0.2182
| 0.2182
Line 48: Line 49:
=== 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
|-
! Generator<br>(reduced)*
! Periods<br />per 8ve
! Cents<br>(reduced)*
! Generator*
! Associated<br>Ratio*
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|91\402
| 91\402
|271.64
| 271.64
|75/64
| 75/64
|[[Orson]]
| [[Orson]] (402)
|-
| 1
| 115\402
| 343.28
| 8000/6561
| [[Raider]] (402)
|-
|-
|1
| 2
|115\402
| 70\402
|343.28
| 208.96
|8000/6561
| 44/39
|[[Raider]]
| [[Abigail]] (402)
|}
|-
| 2
| 167\402<br />(34\402)
| 498.51<br />(101.49)
| 4/3<br />(200/189)
| [[Bischismic]] (402c, 7-limit)
|}
|}
<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 06:24, 21 February 2025

← 401edo 402edo 403edo →
Prime factorization 2 × 3 × 67
Step size 2.98507 ¢ 
Fifth 235\402 (701.493 ¢)
Semitones (A1:m2) 37:31 (110.4 ¢ : 92.54 ¢)
Consistency limit 5
Distinct consistency limit 5

402 equal divisions of the octave (abbreviated 402edo or 402ed2), also called 402-tone equal temperament (402tet) or 402 equal temperament (402et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 402 equal parts of about 2.99 ¢ each. Each step represents a frequency ratio of 21/402, or the 402nd root of 2.

Theory

402edo is only consistent to the 5-odd-limit. There are three possible mappings in the 7-limit:

  • ⟨402 637 933 1129] (patent val)
  • ⟨402 637 933 1128] (402d)
  • ⟨402 637 934 1129] (402c)

Using the patent val, it tempers out the semicomma in the 5-limit and 4375/4374, 7381125/7340032, and 3200000/3176523 in the 7-limit, supporting Abigail.

Using the 402d val, it tempers out 250047/250000, and 1500625/1492992 and 2460375/2458624 in the 7-limit.

Using the 402c val, it tempers out the schisma in the 5-limit; and 3136/3125, 321489/320000, and 13060694016/12867859375 in the 7-limit, supporting bischismic.

Odd harmonics

Approximation of odd harmonics in 402edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.46 -1.24 +1.32 -0.92 +0.92 +1.26 +1.28 -0.48 +0.99 +0.86 -1.41
Relative (%) -15.5 -41.5 +44.3 -31.0 +30.8 +42.3 +43.0 -16.0 +33.3 +28.8 -47.2
Steps
(reduced) 		637
(235) 		933
(129) 		1129
(325) 		1274
(68) 		1391
(185) 		1488
(282) 		1571
(365) 		1643
(35) 		1708
(100) 		1766
(158) 		1818
(210)

Subsets and supersets

Since 402 factors into 2 × 3 × 67, 402edo has subset edos 2, 3, 6, 67, 134, and 201. 804edo, which doubles it, gives a good correction to the harmonics 5 and 7.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-637 402⟩ [⟨402 637]] 0.1459 0.1459 4.89
2.3.5 2109375/2097152, [25 -48 22⟩ [⟨402 637 933]] 0.2752 0.2182 7.31

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 91\402 271.64 75/64 Orson (402)
1 115\402 343.28 8000/6561 Raider (402)
2 70\402 208.96 44/39 Abigail (402)
2 167\402
(34\402) 		498.51
(101.49) 		4/3
(200/189) 		Bischismic (402c, 7-limit)

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

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