1619edo
Theory
1619edo is excellent in the 13-limit, where it tempers out 4225/4224, 4375/4374, 6656/6655, 78125/78078, and 117649/117612. It also notably tempers out quartisma (117440512/117406179) and 123201/123200.
1619edo is the 256th Prime edo. It can be seen as a prime counterpart to 270edo in its excellent ability to act as a very fine closed 13-limit system, and it has an advantage over 270edo in being prime, since every generator produces a unique MOS.
Temperaments
It supports vidar, which has the comma basis 4225/4224, 4375/4374, and 6656/6655, and other unnamed expansions of the ragismic temperament such as the 270 & 441 & 1619, tempering out 4225/4224, 4375/4374, 655473/655360, or the 72 & 270 & 494 & 1619 temperament tempering out 6656/6655 and 2912000/2910897.
1619edo supports the rank 5 temperament tempering out the jacobin comma, 6656/6655 and its fifth-order maximal evenness scale, 24 & 72 & 270 & 494 & 1619, is represented by every 3rd step of the 72 & 270 & 494 & 1619 temperament.
1619edo tunes keenanisma very finely, to 6 steps, and can use it as a microchroma. In addition, it can use the keenanisma as a generator in the keenanose temperament, 270 & 1619, in which it highlights the relationship between 270 keenanismas and the octave. It also achieves this since 270 * 6 = 1620, and 1619 is 1 short of that and also excellent in the 13-limit. 1619edo has 7/6 on 360th step, a highly divisible number, 27/25 on 180th, and 33/32 on 72nd as a consequence of tempering out the commas. This means that 72ed33/32 is virtually equivalent to 1619edo. When it comes to using 33/32 as the generator, 1619edo supports the ravine temperament, which tempers out 196625/196608, 200000/199927, 2912000/2910897, and 3764768/3764475.
Since 33/32 is close to 1\45, 7\6 is close to 1\9, and 385/384 is close to 1\270, 1619edo can be thought of as 1620edo where one step was extracted and all others were moved into a more harmonically just position.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.040 | -0.149 | -0.080 | +0.134 | -0.009 | +0.295 | -0.293 | +0.262 | -0.053 |
| Relative (%) | +0.0 | -5.4 | -20.2 | -10.8 | +18.0 | -1.2 | +39.8 | -39.5 | +35.3 | -7.1 | |
| Steps (reduced) |
1619 (0) |
2566 (947) |
3759 (521) |
4545 (1307) |
5601 (744) |
5991 (1134) |
6618 (142) |
6877 (401) |
7324 (848) |
7865 (1389) | |
Table of intervals
Intervals named in accordance to their most just 13-limit counterpart using the names accepted on the wiki.
| Step | Cents | Ratio | Name |
|---|---|---|---|
| 0 | 0.000 | 1/1 | prime, unison |
| 6 | 4.447 | 385/384 | keenanisma |
| 72 | 53.366 | 33/32 | al-Farabi quarter-tone |
| 360 | 266.831 | 7/6 | septimal subminor third |
| 1619 | 1200.000 | 2/1 | perfect octave |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-2566 1619⟩ | [⟨1619 2566]] | +0.0127 | 0.0127 | 1.71 |
| 2.3.5 | [-69 45 -1⟩, [-82 -1 36⟩ | [⟨1619 2566 3759]] | +0.0299 | 0.0265 | 3.57 |
| 2.3.5.7 | 4375/4374, 52734375/52706752, [-67 14 6 11⟩ | [⟨1619 2566 3759 4545]] | +0.0295 | 0.0229 | 3.09 |
| 2.3.5.7.11 | 4375/4374, 117649/117612, 759375/758912, [24 -6 0 1 -5⟩ | [⟨1619 2566 3759 4545 5601]] | +0.0159 | 0.0341 | 4.60 |
| 2.3.5.7.11.13 | 4225/4224, 4375/4374, 6656/6655, 78125/78078, 117649/117612 | [⟨1619 2566 3759 4545 5601 5991]] | +0.0136 | 0.0315 | 4.26 |
Rank-2 temperaments by generator
| Periods
per octave |
Generator
(reduced) |
Cents
(reduced) |
Associated
ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 6\1619 | 4.447 | 385/384 | Keenanose |
| 1 | 72\1619 | 53.366 | 33/32 | Ravine |