214edo
| ← 213edo | 214edo | 215edo → |
214 equal divisions of the octave (abbreviated 214edo or 214ed2), also called 214-tone equal temperament (214tet) or 214 equal temperament (214et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 214 equal parts of about 5.61 ¢ each. Each step represents a frequency ratio of 21/214, or the 214th root of 2.
Theory
214edo is (uniquely) consistent through the 7-odd-limit. The patent val for 214edo is ⟨214 339 497 601 740 792], which tempers out the following commas: 78732/78125 (sensipent comma) and [-51 19 9⟩ (untriton comma) in the 5-limit; 6144/6125 (porwell comma), 16875/16807 (mirkwai comma), 321489/320000 (varunisma), and [22 -1 -10 1⟩ (quasiorwellisma) in the 7-limit; 540/539, 1375/1372, 5632/5625, in the 11-limit; 351/350, 847/845, 1001/1000, 1188/1183, 1573/1568, and 4096/4095 in the 13-limit. It can be viewed as a 2.3.5.13.19.23 subgroup temperament, as its approximations for lower prime limits are very poor but this makes 214edo an exceptionally xenharmonic tuning.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.02 | +0.60 | +1.27 | -1.79 | +0.59 | +1.59 | -0.32 | -0.24 | +2.20 | -1.11 |
| Relative (%) | +0.0 | -18.2 | +10.7 | +22.6 | -31.8 | +10.6 | +28.3 | -5.6 | -4.2 | +39.2 | -19.8 | |
| Steps (reduced) |
214 (0) |
339 (125) |
497 (69) |
601 (173) |
740 (98) |
792 (150) |
875 (19) |
909 (53) |
968 (112) |
1040 (184) |
1060 (204) | |
Subsets and supersets
Since 214 factors into 2 × 107, 214edo contains 2edo and 107edo as its subsets.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-339 214⟩ | [⟨214 339]] | +0.3219 | 0.3220 | 5.74 |
| 2.3.5 | 78732/78125, [-49 28 2⟩ | [⟨214 339 497]] | +0.1281 | 0.3797 | 6.77 |
| 2.3.5.7 | 6144/6125, 16875/16807, 78732/78125 | [⟨214 339 497 601]] | −0.0169 | 0.4137 | 7.38 |
| 2.3.5.7.11 | 540/539, 1375/1372, 5632/5625, 72171/71680 | [⟨214 339 497 601 740]] | +0.0897 | 0.4270 | 7.61 |
| 2.3.5.7.11.13 | 351/350, 540/539, 847/845, 1375/1372, 4096/4095 | [⟨214 339 497 601 740 792]] | +0.0480 | 0.4008 | 7.15 |
| 2.3.5.7.11.13.17 | 351/350, 540/539, 715/714, 847/845, 936/935, 4096/4095 | [⟨214 339 497 601 740 792 875]] | −0.0144 | 0.4012 | 7.15 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 27\214 | 151.40 | 12/11 | Browser |
| 1 | 69\214 | 386.92 | 5/4 | Grendel |
| 1 | 79\214 | 442.99 | 162/125 | Sensipent |
| 1 | 105\214 | 588.79 | 7/5 | Aufo |
| 2 | 28\214 | 157.01 | 35/32 | Bison (214e) |
| 2 | 29\214 | 162.62 | 1125/1024 | Kwazy |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct