814edo
|
This page presents a novelty topic.
It may contain ideas which are less likely to find practical applications in music, or numbers or structures that are arbitrary or exceedingly small, large, or complex. Novelty topics are often developed by a single person or a small group. As such, this page may also contain idiosyncratic terms, notation, or conceptual frameworks. |
|
This page is a stub. You can help the Xenharmonic Wiki by expanding it. |
Theory
814edo is distinctly consistent to the 17-odd-limit and is a strong 17-limit system. The equal temperament is enfactored in the 5-limit, tempering out the schisma as does 407et. In the 7-limit it tempers out 2401/2400 so that it supports and gives a good tuning for sesquiquartififths. In the 11-limit it tempers out 9801/9800, in the 13-limit 4225/4224 and 6656/6655, and in the 17-limit 1701/1700, 2058/2057, 2601/2600, 4914/4913 and 5832/5831. The 171 & 643 temperament gives an extension of sesquiquartififths to the 17-limit for which 814edo provides the optimal patent val.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.235 | -0.073 | -0.276 | +0.033 | -0.233 | -0.287 | +0.276 | -0.265 | -0.585 | +0.419 |
| Relative (%) | +0.0 | -15.9 | -4.9 | -18.7 | +2.3 | -15.8 | -19.5 | +18.7 | -17.9 | -39.7 | +28.4 | |
| Steps (reduced) |
814 (0) |
1290 (476) |
1890 (262) |
2285 (657) |
2816 (374) |
3012 (570) |
3327 (71) |
3458 (202) |
3682 (426) |
3954 (698) |
4033 (777) | |
Subsets and supersets
Since 814 factors into 2 × 11 × 37, 814edo has subset edos 2, 11, 22, 37, 74, and 407.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7 | 2401/2400, 32805/32768, [25 20 -22 -2⟩ | [⟨814 1290 1890 2285]] | +0.0695 | 0.0577 | 3.91 |
| 2.3.5.7.11 | 2401/2400, 9801/9800, 32805/32768, 20155392/20131375 | [⟨814 1290 1890 2285 2816]] | +0.0536 | 0.0605 | 4.11 |
| 2.3.5.7.11.13 | 2401/2400, 4225/4224, 6656/6655, 9801/9800, 32805/32768 | [⟨814 1290 1890 2285 2816 3012]] | +0.0552 | 0.0554 | 3.76 |
| 2.3.5.7.11.13.17 | 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4225/4224, 6656/6655 | [⟨814 1290 1890 2285 2816 3012 3327]] | +0.0573 | 0.0528 | 3.50 |
- 814et is notable in the 17- and 23-limit with lower absolute errors than any previous equal temperaments, beating 764 in the 17-limit and 742i in the 23-limit, and is only bettered by 935 in either subgroup.
Rank-2 temperaments
Note: 5-limit temperaments supported by 407edo are not included.
| Periods per Octave |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 119\814 | 175.43 | 448/405 | Sesquiquartififths |