15edt
| ← 14edt | 15edt | 16edt → |
15 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 15edt or 15ed3), is a nonoctave tuning system that divides the interval of 3/1 into 15 equal parts of about 127 ¢ each. Each step represents a frequency ratio of 31/15, or the 15th root of 3.
Theory
15edt corresponds to 9.4639…edo. It has harmonics 5 and 13 closely in tune, but does not do so well for 11, which is quite sharp. The main appeal of 15edt is that it allows for strong tritave equivalency, while supporting more conventional harmony. It achieves this with fantastic approximation of the 4th harmonic, and terrible approximation of the octave. In other words; 3:4:5 is available, but 4:5:6 is not. Like the octave, the 7th harmonic is about halfway between steps, so 6:7:8 is well approximated, but not 4:5:7. It also tempers out the syntonic comma, 81/80, in the 3.4.5 subgroup, as the major third is three perfect fourths below a tritave. As a 3.5.13-subgroup system, it tempers out 2197/2187 and 3159/3125, and if these commas are added, 15edt is related to the 2.3.5.13-subgroup temperament 19 & 123, which has a mapping [⟨1 0 0 0], ⟨0 15 22 35]], where the generator, an approximate 27/25, has a POTE tuning of 126.773, very close to 15edt.
Using the patent val, it tempers out 375/343 and 6561/6125 in the 7-limit; 81/77, 125/121, and 363/343 in the 11-limit; 65/63, 169/165, 585/539, and 1287/1225 in the 13-limit; 51/49, 121/119, 125/119, 189/187, and 195/187 in the 17-limit (no-twos subgroup). With the patent 4, it tempers out 36/35, 64/63, and 375/343 in the 3.4.5.7 subgroup; 45/44, 80/77, 81/77, and 363/343 in the 3.4.5.7.11 subgroup; 52/49, 65/63, 65/64, 143/140, and 169/165 in the 3.4.5.7.11.13 subgroup; 51/49, 52/51, 85/84, and 121/119 in the 3.4.5.7.11.13.17 subgroup ( that 15edt treated this way is essentially a retuning of 19ed4). The 2*15 subgroup of 15edt is 3.4.5.14.22.13.34, on which b15 tempers out the same commas as the patent val for 30edt.
15edt is also associated with tempering out the mowgli comma, [0 22 -15⟩ in the 5-limit, which fixes 5/3 to 7\15edt; in an octave context, this temperament is supported by 19edo but has an optimal patent val of 303edo.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -58.83 | +0.00 | +9.14 | +3.22 | -58.83 | +54.69 | -49.68 | +0.00 | -55.61 | +32.98 | +9.14 |
| Relative (%) | -46.4 | +0.0 | +7.2 | +2.5 | -46.4 | +43.1 | -39.2 | +0.0 | -43.9 | +26.0 | +7.2 | |
| Steps (reduced) |
9 (9) |
15 (0) |
19 (4) |
22 (7) |
24 (9) |
27 (12) |
28 (13) |
30 (0) |
31 (1) |
33 (3) |
34 (4) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.63 | -4.13 | +3.22 | +18.29 | +40.13 | -58.83 | -25.63 | +12.36 | +54.69 | -25.84 | +24.00 | -49.68 |
| Relative (%) | -2.1 | -3.3 | +2.5 | +14.4 | +31.6 | -46.4 | -20.2 | +9.8 | +43.1 | -20.4 | +18.9 | -39.2 | |
| Steps (reduced) |
35 (5) |
36 (6) |
37 (7) |
38 (8) |
39 (9) |
39 (9) |
40 (10) |
41 (11) |
42 (12) |
42 (12) |
43 (13) |
43 (13) | |
Intervals
| # | Cents | Hekts | Approximate ratios | Polaris nonatonic notation |
|---|---|---|---|---|
| 0 | 0.0 | 0.0 | 1/1 | H |
| 1 | 126.8 | 86.7 | 14/13, 15/14, 16/15, 29/27 | Ib |
| 2 | 253.6 | 173.3 | 15/13 | vH#, ^Ib |
| 3 | 380.4 | 260.0 | 5/4 | H# |
| 4 | 507.2 | 346.7 | 4/3 | I |
| 5 | 634.0 | 433.3 | 13/9 | J |
| 6 | 760.8 | 520.0 | 14/9 | K |
| 7 | 887.6 | 606.7 | 5/3 | L |
| 8 | 1014.4 | 793.3 | 9/5 | Mb |
| 9 | 1141.2 | 780.0 | 27/14 | vL#, ^Mb |
| 10 | 1268.0 | 866.7 | 27/13 | L# |
| 11 | 1394.8 | 953.3 | 9/4 | M |
| 12 | 1521.6 | 1040.0 | 12/5 | N |
| 13 | 1648.4 | 1126.7 | 13/5 | O |
| 14 | 1775.2 | 1213.3 | 14/5 | P |
| 15 | 1902.0 | 1300.0 | 3/1 | H |
15edt contains 4 intervals from 5edt and 2 intervals from 3edt, meaning that it contains 6 redundant intervals and 8 new intervals. The new intervals introduced include good approximations to 15/14, 15/13, 4/3, 5/3 and their tritave inverses. This allows for new chord possibilities such as 1:3:4:5:9:12:13:14:15:16…
15edt also contains a 5L 5s mos similar to Blackwood Decatonic, which I[who?] call Ebony. This mos has a period of 1/5 of the tritave and the generator is a single step. The major scale is sLsLsLsLsL, and the minor scale is LsLsLsLsLs.
15edt approximates the 5th and 13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L 3s mos "augmented scale", in which three 13/9 intervals close to a tritave, and another three are set 5/3 away.
JI approximation
Z function
Below is a plot of the no-twos Z function in the vicinity of 15edt:
Audio examples
A short composition by Unque.
Music
- ox-idation (2012)