# 419edo

 ← 418edo 419edo 420edo →
Prime factorization 419 (prime)
Step size 2.86396¢
Fifth 245\419 (701.671¢)
Semitones (A1:m2) 39:32 (111.7¢ : 91.65¢)
Consistency limit 9
Distinct consistency limit 9

419 equal divisions of the octave (abbreviated 419edo or 419ed2), also called 419-tone equal temperament (419tet) or 419 equal temperament (419et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 419 equal parts of about 2.86 ¢ each. Each step represents a frequency ratio of 21/419, or the 419th root of 2.

## Theory

419edo is a decent 7-limit system, and is consistent to the 9-odd-limit. The equal temperament tempers out 32805/32768 (schisma) in the 5-limit; 235298/234375 (triwellisma), 420175/419904 (wizma) in the 7-limit. It supports and provides the optimal patent val for sextilififths, the 130 & 289 temperament, in the 7-limit.

Extending it to the 11-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the 419e val, it tempers out 3025/3024, 5632/5625, and 8019/8000. Using the patent val, it tempers out 441/440, 4000/3993, and 14700/14641 in the 11-limit. The patent val supports 11-limit sextilififths, though 289edo is better suited for that purpose.

The same can be said of the mapping for 13, with the 419e val tempering out 676/675, 1716/1715, 4225/4224, and 4459/4455, and the 419f val tempering out 729/728, 2200/2197, 2205/2197, 3584/3575, and 4459/4455.

### Odd harmonics

Approximation of prime harmonics in 419edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.28 +0.32 -0.81 +1.43 -1.39 +1.01 +0.34 -1.07 -1.41 +0.55
Relative (%) +0.0 -9.9 +11.2 -28.2 +49.8 -48.4 +35.3 +11.8 -37.2 -49.4 +19.2
Steps
(reduced)
419
(0)
664
(245)
973
(135)
1176
(338)
1450
(193)
1550
(293)
1713
(37)
1780
(104)
1895
(219)
2035
(359)
2076
(400)

### Subsets and supersets

419edo is the 81th prime edo.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-664 419 [419​ 664​]] +0.0897 0.0897 3.13
2.3.5 32805/32768, [41 43 -47 [419 ​664​ 973]] +0.0137 0.1301 4.54
2.3.5.7 32805/32768, 235298/234375, 420175/419904 [419 ​664​ 973 ​1176 ​]] +0.0821 0.1635 5.71
2.3.5.7.11 441/440, 4000/3993, 32805/32768, 420175/419904 [419​ 664 ​973 ​1176​ 1450]] (419) -0.0168 0.2460 8.59

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 29\419 83.05 21/20 Sextilififths (419f)
1 49\419 140.33 243/224 Tsaharuk (7-limit)
1 174\419 498.33 162/125 Helmholtz

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

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