419edo

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← 418edo419edo420edo →
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 419-tone equal temperament (419tet), 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 of 419edo 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 -10 +11 -28 +50 -48 +35 +12 -37 -49 +19
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

Scales

Music

Francium