398edo

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← 397edo398edo399edo →
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 398-tone equal temperament (398tet), 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 of 398edo 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 +18 -13 -33 +15 +22 +19 +32 -38 -48 +23
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