# 398edo

 ← 397edo 398edo 399edo →
Prime factorization 2 × 199
Step size 3.01508¢
Fifth 233\398 (702.513¢)
Semitones (A1:m2) 39:29 (117.6¢ : 87.44¢)
Consistency limit 5
Distinct consistency limit 5

398 equal divisions of the octave (abbreviated 398edo or 398ed2), also called 398-tone equal temperament (398tet) or 398 equal temperament (398et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 398 equal parts of about 3.02 ¢ each. Each step represents a frequency ratio of 21/398, or the 398th root of 2.

## Theory

398edo is only consistent to the 5-odd-limit, though it has a reasonable approximation to the full 13-limit using the patent val, which tempers out 10976/10935, 65625/65536, 1500625/1492992, 102760448/102515625, 102942875/102036672, and 200120949/200000000 in the 7-limit; 3025/3024, 4000/3993, 6250/6237, 59290/59049, 117649/117128, and 131072/130977 in the 11-limit; and 625/624, 1575/1573, 2080/2079, 2200/2197, 4096/4095, and 4225/4224 in the 13-limit. It supports yarman I, bisupermajor and semiquindromeda.

### Prime harmonics

Approximation of prime harmonics in 398edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.56 -0.38 -0.99 +0.44 +0.68 +0.57 +0.98 -1.14 -1.44 +0.69
Relative (%) +0.0 +18.5 -12.7 -32.7 +14.6 +22.5 +19.0 +32.5 -37.8 -47.6 +23.0
Steps
(reduced)
398
(0)
631
(233)
924
(128)
1117
(321)
1377
(183)
1473
(279)
1627
(35)
1691
(99)
1800
(208)
1933
(341)
1972
(380)

### Subsets and supersets

Since 398 factors into 2 × 199, 398edo has 2edo and 199edo as its subsets.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [631 -398 [398 631]] -0.1759 0.1759 5.83
2.3.5 390625000/387420489, [-53 10 16 [398 631 924]] -0.0622 0.2157 7.15
2.3.5.7 10976/10935, 65625/65536, 200120949/200000000 [398 631 924 1117]] +0.0412 0.2588 8.58
2.3.5.7.11 3025/3024, 4000/3993, 10976/10935, 65625/65536 [398 631 924 1117 1377]] +0.0075 0.2411 8.00
2.3.5.7.11.13 625/624, 1575/1573, 2080/2079, 2200/2197, 10976/10935 [398 631 924 1117 1377 1473]] -0.0243 0.2313 7.67

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 5\398 15.08 126/125 Yarman I
1 183\398 551.76 11/8 Emkay
2 54\398 162.81 11/10 Bisupermajor

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