1848edo: Difference between revisions
+notable fact about 11-limit atomic |
→Rank-2 temperaments: correct me if I'm wrong, I calculated this using finding out what 11\1684 would correspond to |
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| Line 91: | Line 91: | ||
| 767\1848<br>(11\1848) | | 767\1848<br>(11\1848) | ||
| 498.052<br>(7.143) | | 498.052<br>(7.143) | ||
| 4/3<br>( | | 4/3<br>(18375/18304) | ||
| [[Ruthenium]] | | [[Ruthenium]] | ||
|- | |- | ||
Revision as of 14:33, 13 November 2022
| ← 1847edo | 1848edo | 1849edo → |
Theory
1848edo is a super strong 11-limit division, having the lowest 11-limit relative error than any division until 6079. It tempers out the 11-limit commas 9801/9800, 151263/151250, 1771561/1771470 and 3294225/3294172. In the 5-limit it is an atomic system, tempering out the atom, [161 -84 -12⟩ and also the minortone comma, [-16 35 -17⟩. It also tempers out the 7-limit landscape comma, 250047/250000, so it supports domain and akjayland. It is distinctly consistent through the 15-odd-limit, and tempers out the 13-limit commas 4225/4224 and 6656/6655.
It provides the optimal patent val for 11-limit atomic.
1848 factors as 23 × 3 × 7 × 11. It is a superabundant number in the no-fives subgroup, that is, if only numbers not divisible by 5 are counted. Its divisors are 1, 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.
Fractional-octave temperaments
1848edo is unique in that it consistently tunes both 81/80 and 64/63 to an integer fraction of the octave, 1/56th and 1/44th 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 this property, 1848edo is unique due to its strength in simple harmonics and it actually shows how 81/80 and 64/63 are produced.
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) | |
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 |
Rank-2 temperaments
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 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 |
| 44 | 767\1848 (11\1848) |
498.052 (7.143) |
4/3 (18375/18304) |
Ruthenium |
| 56 | 767\1848 (21\1848) |
498.052 (13.636) |
4/3 (126/125) |
Barium |