16edt
16 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 16edt or 16ed3), is a nonoctave tuning system that divides the interval of 3/1 into 16 equal parts of about 119 ¢ each. Each step represents a frequency ratio of 31/16, or the 16th root of 3.
Properties
As the double of 8EDT, this division of the tritave is harmonically fraternal to 10EDO. Its unit step is ~1.128 ¢ flat of 1\10EDO. Unlike 10EDO, it does not really have a 7 or 13 because it is not using its approximation of 2 as equivalent though the accumulated flatness of a stack of its unit step leads to an excellent 13:21 and a decent 7:13. When twos are admitted, it turns into a tritave-repeating version of Blackwood temperament.
Harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | -52.3 | -40.4 | +0.0 | +9.2 | -42.3 | -52.3 | -31.2 | +14.0 | -40.4 | +39.8 |
| Relative (%) | +0.0 | -44.0 | -34.0 | +0.0 | +7.7 | -35.5 | -44.0 | -26.2 | +11.8 | -34.0 | +33.5 | |
| Steps (reduced) |
16 (0) |
23 (7) |
28 (12) |
32 (0) |
35 (3) |
37 (5) |
39 (7) |
41 (9) |
43 (11) |
44 (12) |
46 (14) | |
| Harmonic | 25 | 27 | 29 | 31 | 33 | 35 | 37 | 39 | 41 | 43 | 45 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +14.4 | +0.0 | -4.8 | -1.4 | +9.2 | +26.2 | +48.9 | -42.3 | -10.0 | +26.5 | -52.3 |
| Relative (%) | +12.1 | +0.0 | -4.1 | -1.2 | +7.7 | +22.1 | +41.1 | -35.5 | -8.4 | +22.3 | -44.0 | |
| Steps (reduced) |
47 (15) |
48 (0) |
49 (1) |
50 (2) |
51 (3) |
52 (4) |
53 (5) |
53 (5) |
54 (6) |
55 (7) |
55 (7) | |
Intervals
| Degree | Size in | |
|---|---|---|
| Cents | Hekts | |
| 1 | 118.87219 | 81.25 |
| 2 | 237.74438 | 162.5 |
| 3 | 356.61656 | 243.75 |
| 4 | 475.48875 | 325 |
| 5 | 594.36094 | 406.25 |
| 6 | 713.23312 | 487.5 |
| 7 | 832.10531 | 568.75 |
| 8 | 950.9775 | 650 |
| 9 | 1069.84969 | 731.25 |
| 10 | 1188.72188 | 812.5 |
| 11 | 1307.59406 | 893.75 |
| 12 | 1426.46625 | 975 |
| 13 | 1545.33844 | 1056.25 |
| 14 | 1664.21063 | 1137.5 |
| 15 | 1783.08281 | 1218.75 |
| 16 | 1901.955 | 1300 |
Related temperament
16EDT is also be thought of as a generator of the subsedia temperament, which is a cluster temperament with 10 clusters of notes in an octave.
Subsedia (10 & 111)
7-limit
Comma list: 16875/16807, 65536/64827
Mapping: [⟨1 0 5 4], ⟨0 16 -27 -12]]
POTE generator: ~15/14 = 118.965
Optimal ET sequence: 10, 101, 111, 121, 232d
Badness: 0.157658
11-limit
Comma list: 540/539, 1375/1372, 65536/64827
Mapping: [⟨1 0 5 4 -1], ⟨0 16 -27 -12 45]]
POTE generator: ~15/14 = 118.968
Optimal ET sequence: 10, 101, 111, 121, 232d
Badness: 0.066838
13-limit
Comma list: 352/351, 540/539, 676/675, 1375/1372
Mapping: [⟨1 0 5 4 -1 4], ⟨0 16 -27 -12 45 -3]]
POTE generator: ~15/14 = 118.968
Optimal ET sequence: 10, 101, 111, 121, 232d
Badness: 0.031635
17-limit
Comma list: 256/255, 352/351, 442/441, 540/539, 715/714
Mapping: [⟨1 0 5 4 -1 4 3], ⟨0 16 -27 -12 45 -3 11]]
POTE generator: ~15/14 = 118.968
Optimal ET sequence: 10, 101, 111, 121, 232dg
Badness: 0.019707