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 (419edo), or 419-tone equal temperament (419tet), 419 equal temperament (419et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 419 equal parts of about 2.86 ¢ each. .

Theory

419et tempers out 32805/32768, 29360128/29296875, 1959552/1953125, 420175/419904 and 2100875/2097152 in the 7-limit and 35156250/35153041, 25165824/25109315, 26214400/26198073, 4000/3993, 1366875/1362944, 759375/758912, 21437500/21434787, 2359296/2358125, 472392/471625, 369140625/369098752, 200704/200475, 441/440, 43046721/43025920, 17537553/17500000, 422576/421875, 160083/160000, 199297406/199290375, 244515348/244140625 and 67110351/67108864 in the 11-limit. It supports the sextilififths temperament. 419edo is the 81th prime edo.

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)

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, 420175/419904, 1959552/1953125 419 ​664​ 973 ​1176​] +0.0821 0.1635 5.71
2.3.5.7.11 441/440, 200704/200475, 160083/160000, 472392/471625 419​ 664 ​973 ​1176​ 1450] -0.0168 0.2460 8.59
2.3.5.7.11.13 441/440, 2080/2079, 1001/1000, 78975/78848, 472392/471625 419​ 664 ​973 ​1176​ 1450 1550] +0.0484 0.2678 9,35

Rank-2 temperaments

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

Scales

Music