# 397edo

 ← 396edo 397edo 398edo →
Prime factorization 397 (prime)
Step size 3.02267¢
Fifth 232\397 (701.259¢)
Semitones (A1:m2) 36:31 (108.8¢ : 93.7¢)
Consistency limit 5
Distinct consistency limit 5

397 equal divisions of the octave (abbreviated 397edo or 397ed2), also called 397-tone equal temperament (397tet) or 397 equal temperament (397et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 397 equal parts of about 3.02 ¢ each. Each step represents a frequency ratio of 21/397, or the 397th root of 2.

## Theory

397edo is only consistent to the 5-odd-limit, with three mappings possible for the 7-limit:

• 397 629 922 1115] (patent val)
• 397 629 922 1114] (397d val)
• 397 629 921 1114] (397cd val)

Using the patent val, it tempers out 129140163/128000000 and [55 -1 -23 in the 5-limit; 6144/6125, 16875/16807 and 129140163/128000000 in the 7-limit; supporting grendel.

Using the 397d val, it tempers out 129140163/128000000 and [55 -1 -23 in the 5-limit; 3136/3125, 420175/419904 and 33756345/33554432 in the 7-limit; supporting sengagen.

Using the 397cd val, it tempers out 390625000/387420489 and [-55 23 8 in the 5-limit; 2401/2400, 390625/387072 and 14348907/14336000 in the 7-limit; supporting cotritone.

### Prime harmonics

Approximation of prime harmonics in 397edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.70 +0.59 +1.45 -1.19 -0.23 +0.84 -1.29 +0.44 +1.15 +0.56
Relative (%) +0.0 -23.0 +19.5 +48.0 -39.4 -7.5 +27.7 -42.7 +14.6 +38.2 +18.4
Steps
(reduced)
397
(0)
629
(232)
922
(128)
1115
(321)
1373
(182)
1469
(278)
1623
(35)
1686
(98)
1796
(208)
1929
(341)
1967
(379)

### Subsets and supersets

397edo is the 78th prime edo. 1588edo, which quadruples 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 [-629 397 [397 629]] 0.2194 0.2195 7.26
2.3.5 129140163/128000000, [55 -1 -23 [397 629 922]] 0.0618 0.2859 9.46

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 8\397 24.18 686/675 Sengagen (397d)
1 37\397 111.84 16/15 Vavoom
1 128\397 386.90 5/4 Grendel (397)
1 171\397 516.88 27/20 Gravity
1 193\397 583.38 7/5 Cotritone (397cd)

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

Francium