402edo

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← 401edo402edo403edo →
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; 4375/4374, 7381125/7340032 and 3200000/3176523 in the 7-limit. It supports abigail.

Using the 402d val, it tempers out 250047/250000, 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. It supports 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 it is distinct