12edt
| ← 11edt | 12edt | 13edt → |
12 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 12edt or 12ed3), is a nonoctave tuning system that divides the interval of 3/1 into 12 equal parts of about 158 ¢ each. Each step represents a frequency ratio of 31/12, or the 12th root of 3.
12edt corresponds to 7.571 edo, and can be used as a generator chain for hemikleismic temperament. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +68.0 | +0.0 | +66.6 | -40.4 | -30.4 | -2.6 | +8.4 | -25.6 | -39.4 | +34.8 | +77.8 |
| Relative (%) | +42.9 | +0.0 | +42.0 | -25.5 | -19.2 | -1.7 | +5.3 | -16.2 | -24.9 | +21.9 | +49.1 | |
| Steps (reduced) |
8 (8) |
12 (0) |
18 (6) |
21 (9) |
26 (2) |
28 (4) |
31 (7) |
32 (8) |
34 (10) |
37 (1) |
38 (2) | |
Theory
In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup—ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen–Pierce) and 12edt handles the 2.3.5.13.17.19—and, it is a multiple of 4edt which is the simplest BP equal temperament.
Macrodiatonic and macromeantone
12edt can be viewed as a version of 12edo with octaves stretched out to tritaves, so it contains an extremely stretched diatonic scale or macrodiatonic 5L 2s⟨3/1⟩ scale. This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable, since, for example, the generator is now the size of a major seventh instead of a perfect fifth. The stretched perfect fifth can be approximated by 17/9 and the stretched major third by 13/9. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 subgroup, equating 4 17/9 to 13/9 tritave-reduced, rather than 4 3/2 to 5/4 octave-reduced (although this is not a completely exact stretching of meantone, unlike some macromeantones like meansquared which repeats at 4/1).
Another example of a macrodiatonic scale is hyperpyth which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.
Interval table
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 158.5 | 108.3 | 21/19, 23/21 |
| 2 | 317 | 216.7 | 6/5, 13/11, 17/14, 23/19 |
| 3 | 475.5 | 325 | 17/13 |
| 4 | 634 | 433.3 | 13/9, 19/13 |
| 5 | 792.5 | 541.7 | 11/7, 14/9 |
| 6 | 951 | 650 | 19/11 |
| 7 | 1109.5 | 758.3 | 17/9, 21/11 |
| 8 | 1268 | 866.7 | 19/9, 23/11 |
| 9 | 1426.5 | 975 | |
| 10 | 1585 | 1083.3 | 5/2 |
| 11 | 1743.5 | 1191.7 | 19/7 |
| 12 | 1902 | 1300 | 3/1 |
Scala file
! C:\Cakewalk\scales\tritave-in-12.scl ! 3/1 in 12 12 ! 158.49625 316.99250 475.48875 633.98500 792.48125 950.97750 1109.47375 1267.97000 1426.46625 1584.96250 1743.45875 3/1