419edo
| ← 418edo | 419edo | 420edo → |
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.
| 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
| 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
- Cultural Appropriation? by Francium