419edo
| ← 418edo | 419edo | 420edo → |
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
| 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
| 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
- Cultural Appropriation? (2023)