# 417edo

 ← 416edo 417edo 418edo →
Prime factorization 3 × 139
Step size 2.8777¢
Fifth 244\417 (702.158¢)
Semitones (A1:m2) 40:31 (115.1¢ : 89.21¢)
Consistency limit 5
Distinct consistency limit 5

417 equal divisions of the octave (abbreviated 417edo or 417ed2), also called 417-tone equal temperament (417tet) or 417 equal temperament (417et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 417 equal parts of about 2.88 ¢ each. Each step represents a frequency ratio of 21/417, or the 417th root of 2.

## Theory

417et is only consistent to the 5-odd-limit. Using the patent val, it tempers out 589824/588245, 43046721/43025920, 33554432/33480783 and 65625/65536 in the 7-limit; 78121827/77948684, 20155392/20131375, 10333575/10307264, 1019215872/1019046875, 46656/46585, 1366875/1362944, 78675968/78594219, 536870912/535869675, 7168000/7144929, 496125/495616, 514714375/514434888, 2359296/2358125, 540/539, 1265625/1261568, 17561600/17537553, 180224/180075, 1375/1372, 645922816/645700815, 3025/3024, 9453125/9437184 and 1362944/1361367 in the 11-limit. It supports familia and 5-limit fortune.

### Prime harmonics

Approximation of prime harmonics in 417edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.20 -0.70 +0.96 +1.20 -0.24 -1.36 -1.11 -0.94 +0.64 +0.29
Relative (%) +0.0 +7.1 -24.4 +33.3 +41.7 -8.3 -47.2 -38.6 -32.5 +22.2 +10.0
Steps
(reduced)
417
(0)
661
(244)
968
(134)
1171
(337)
1443
(192)
1543
(292)
1704
(36)
1771
(103)
1886
(218)
2026
(358)
2066
(398)

### Subsets and supersets

417 factors into 3 × 139, with 3edo and 139edo as its subset edos.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [661 -417 [417 661]] -0.0641 0.0641 2.23
2.3.5 1600000/1594323, [-80 8 29 [417 661 968]] +0.0580 0.1806 6.28
2.3.5.7 16875/16807, 65625/65536, 1600000/1594323 [417 661 968 1171]] -0.0418 0.2331 8.10
2.3.5.7.11 540/539, 3025/3024, 496125/495616, 7168000/7144929 [417 661 968 1171 1443]] -0.1029 0.2416 8.40

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator
(reduced)*
Cents
(reduced)*
Associated
Ratio*
Temperaments
1 77\417 221.58 8388608/7381125 Fortune
1 118\417 339.57 243/200 Amity
1 121\417 348.20 60/49 Eris
3 39\417 112.23 16/15 Tertiosec

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