12edt
| ← 11edt | 12edt | 13edt → |
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding 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.
Interval table
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 158.496 | 21/19, 23/21 |
| 2 | 316.993 | 6/5, 13/11, 17/14, 23/19 |
| 3 | 475.489 | 17/13 |
| 4 | 633.985 | 13/9, 19/13 |
| 5 | 792.481 | 11/7, 14/9 |
| 6 | 950.978 | 19/11 |
| 7 | 1109.474 | 17/9, 21/11 |
| 8 | 1267.97 | 19/9, 23/11 |
| 9 | 1426.466 | |
| 10 | 1584.963 | 5/2 |
| 11 | 1743.459 | 19/7 |
| 12 | 1901.955 | 3/1 |
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) | |
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
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 scale (5L 2s<3/1>). This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable-for example, the "perfect fifth" is inflated to the size of a major seventh. 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.