1848edo
| ← 1847edo | 1848edo | 1849edo → |
1848 equal divisions of the octave (abbreviated 1848edo or 1848ed2), also called 1848-tone equal temperament (1848tet) or 1848 equal temperament (1848et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1848 equal parts of about 0.649 ¢ each. Each step represents a frequency ratio of 21/1848, or the 1848th root of 2.
Theory
1848edo is an extremely strong 11-limit division, having the lowest 11-limit relative error than any division until 6079.
In the 5-limit it tempers out the minortone comma, [-16 35 -17⟩ and the atom, [161 -84 -12⟩ and thus tunes the atomic temperament, for which it also provides the optimal patent val in the 11-limit. In the 7-limit it tempers out the landscape comma, 250047/250000, so it supports domain and akjayland. In the 11-limit it tempers out 9801/9800, 151263/151250, 1771561/1771470, 3294225/3294172, and the spoob.
It is distinctly consistent through the 15-odd-limit (though just barely), and tempers out the 13-limit commas 4225/4224 and 6656/6655. Higher-limit prime harmonics represented by 1848edo with less than 10% error are 37, 61, and 83, of which 61 is accurate to 0.002 edosteps (and is inherited from 231edo). The harmonics represented by less than 20% error are 19, 47, 59, 67, 89, and the 2.3.5.7.11.19 subgroup is the simplest and most natural choice for using 1848edo with higher limits. In the 2.3.5.7.11.19, it tempers out 5776/5775.
1848edo is unique in that it consistently tunes both 81/80 and 64/63 to an integer fraction of the octave, 1\56 and 1\44 respectively. As a corollary, it supports barium and ruthenium temperaments, which have periods 56 and 44 respectively. While every edo that is a multiple of 616 shares the property of directly mapping 81/80 and 64/63 to fractions of the octave, 1848edo is unique due to its strength in simple harmonics and it actually shows how 81/80 and 64/63 are produced. In 2.3.5.7.11.19, it also tempers 96/95 to 1\66, thus making it a valuable system where important raising or lowering commas are represented by intervals that fit evenly within the octave.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.007 | +0.050 | +0.005 | -0.019 | -0.268 | +0.239 | -0.110 | +0.297 | +0.293 | -0.230 |
| Relative (%) | +0.0 | -1.1 | +7.7 | +0.8 | -3.0 | -41.3 | +36.9 | -17.0 | +45.8 | +45.1 | -35.5 | |
| Steps (reduced) |
1848 (0) |
2929 (1081) |
4291 (595) |
5188 (1492) |
6393 (849) |
6838 (1294) |
7554 (162) |
7850 (458) |
8360 (968) |
8978 (1586) |
9155 (1763) | |
Subsets and supersets
Since 1848 factors into 23 × 3 × 7 × 11, 1848edo has subset edos 2, 3, 4, 6, 7, 8, 11, 12, 14, 21, 22, 24, 28, 33, 42, 44, 56, 66, 77, 84, 88, 132, 154, 168, 231, 264, 308, 462, 616, 924.
5544edo, which divides the edostep into three, provides a good correction for 13- and the 17-limit.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-2929 1848⟩ | [⟨1848 2929]] | 0.002192 | 0.002192 | 0.34 |
| 2.3.5 | [-16 35 -17⟩, [129 -14 -46⟩ | [⟨1848 2929 4291]] | −0.005705 | 0.011311 | 1.74 |
| 2.3.5.7 | 250047/250000, [-4 17 1 -9⟩, [43 -1 -13 -4⟩ | [⟨1848 2929 4291 5188]] | −0.004748 | 0.009935 | 1.53 |
| 2.3.5.7.11 | 9801/9800, 151263/151250, 1771561/1771470, 67110351/67108864 | [⟨1848 2929 4291 5188 6393]] | −0.002686 | 0.009797 | 1.51 |
| 2.3.5.7.11.13 | 4225/4224, 6656/6655, 9801/9800, 151263/151250, 1771561/1771470 | [⟨1848 2929 4291 5188 6393 6838]] | +0.009828 | 0.029378 | 4.52 |
| 2.3.5.7.11.19 | 5776/5775, 9801/9800, 10241/10240, 250047/250000, 233744896/233735625 | [⟨1848 2929 4291 5188 6393 7850]] | +0.002094 | 0.013936 | 2.15 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 281\1848 | 182.467 | 10/9 | Minortone |
| 1 | 523\1848 | 339.610 | [36 -24 1⟩ | Empress |
| 3 | 281\1848 | 182.467 | 10/9 | Domain |
| 12 | 767\1848 (3\1848) |
498.052 (1.948) |
4/3 (32805/32768) |
Atomic |
| 21 | 901\1848 (21\1848) |
585.065 (13.636) |
91875/65536 (126/125) |
Akjayland |
| 22 | 767\1848 (11\1848) |
498.052 (7.143) |
4/3 ([16 -13 2⟩) |
Major arcana |
| 44 | 767\1848 (11\1848) |
498.052 (7.143) |
4/3 (18375/18304) |
Ruthenium |
| 56 | 767\1848 (8\1848) |
498.052 (5.195) |
4/3 (126/125) |
Barium |
| 77 | 581\1848 (42\1848) |
377.273 (27.273) |
975/784 (?) |
Iridium |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct