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 402-tone equal temperament (402tet), 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 of 402edo represents a frequency ratio of 21/402, or the 402nd root of 2.

Theory

402et is only consistent to the 5-limit, tempering out the semicomma. There are three possible mappings in the 7-limit:

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

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 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, 1500625/1492992 and 2460375/2458624 in the 7-limit. It supports the landscape temperament.

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 -42 +44 -31 +31 +42 +43 -16 +33 +29 -47
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

402 factors into 2 × 3 × 67, with subset edos 2, 3, 6, 67, 134, and 201. 804edo, which doubles it, gives a good correction to the harmonic 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
(reduced)*
Cents
(reduced)*
Associated
Ratio*
Temperaments
1 91\402 271.64 75/64 Orson
1 115\402 343.28 8000/6561 Raider

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