1ed86.4c
| ← 0ed17069/16238 | 1ed17069/16238 | 2ed17069/16238 → |
13.888edo, also known as 1 equal division of 86.4¢ (1ed86.4c) or arithmetic pitch sequence of 86.4¢ (APS86.4¢), is an equal-step scale generated by dividing 2/1 into 13.888 equal parts of about 86.4 cents each, or every 9th step of 125edo.
Because 13.888 is not actually an integer, the scale that results does not actually include 2/1, so it is not a true edo. Instead it has a pseudo-octave almost 10 cents sharp, making it a “non-integer edo”.
13.888edo has been voted "monthly tuning" multiple times on the Monthly Tunings Facebook group, owing to its close approximations of many harmonics 3/1, 7/1, 9/1 and 11/1.
It is very close to zeta-stretched 14edo, a.k.a. 42zpi, which has a step size of 86.3¢.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +9.6 | -1.2 | +19.2 | -21.5 | +8.4 | +0.8 | +28.8 | -2.3 | -11.9 | -4.1 |
| Relative (%) | +11.1 | -1.3 | +22.2 | -24.9 | +9.8 | +0.9 | +33.3 | -2.7 | -13.8 | -4.8 | |
| Steps | 14 | 22 | 28 | 32 | 36 | 39 | 42 | 44 | 46 | 48 | |
| Harmonic | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +18.0 | -34.1 | +10.4 | -22.7 | +38.4 | +19.8 | +7.3 | +0.1 | -2.3 | -0.4 |
| Relative (%) | +20.9 | -39.5 | +12.0 | -26.2 | +44.4 | +23.0 | +8.4 | +0.1 | -2.7 | -0.4 | |
| Steps | 50 | 51 | 53 | 54 | 56 | 57 | 58 | 59 | 60 | 61 | |
14edo, 22edt, 32ed5, 39ed7 for comparison:
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | -16.2 | +0.0 | +42.3 | -16.2 | -26.0 | +0.0 | -32.5 | +42.3 | -37.0 | -16.2 |
| Relative (%) | +0.0 | -18.9 | +0.0 | +49.3 | -18.9 | -30.3 | +0.0 | -37.9 | +49.3 | -43.2 | -18.9 | |
| Steps (reduced) |
14 (0) |
22 (8) |
28 (0) |
33 (5) |
36 (8) |
39 (11) |
42 (0) |
44 (2) |
47 (5) |
48 (6) |
50 (8) | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +10.3 | +0.0 | +20.7 | -19.8 | +10.3 | +2.8 | +31.0 | +0.0 | -9.5 | -1.6 | +20.7 |
| Relative (%) | +12.0 | +0.0 | +23.9 | -22.9 | +12.0 | +3.3 | +35.9 | +0.0 | -11.0 | -1.8 | +23.9 | |
| Steps (reduced) |
14 (14) |
22 (0) |
28 (6) |
32 (10) |
36 (14) |
39 (17) |
42 (20) |
44 (0) |
46 (2) |
48 (4) |
50 (6) | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +19.0 | +13.6 | +38.0 | +0.0 | +32.6 | +27.0 | -30.0 | +27.3 | +19.0 | +28.2 | -35.4 |
| Relative (%) | +21.8 | +15.7 | +43.7 | +0.0 | +37.5 | +31.0 | -34.5 | +31.3 | +21.8 | +32.3 | -40.7 | |
| Steps (reduced) |
14 (14) |
22 (22) |
28 (28) |
32 (0) |
36 (4) |
39 (7) |
41 (9) |
44 (12) |
46 (14) |
48 (16) |
49 (17) | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +9.3 | -1.6 | +18.6 | -22.1 | +7.7 | +0.0 | +28.0 | -3.2 | -12.8 | -5.1 | +17.1 |
| Relative (%) | +10.8 | -1.8 | +21.6 | -25.6 | +8.9 | +0.0 | +32.4 | -3.7 | -14.8 | -5.9 | +19.7 | |
| Steps (reduced) |
14 (14) |
22 (22) |
28 (28) |
32 (32) |
36 (36) |
39 (0) |
42 (3) |
44 (5) |
46 (7) |
48 (9) |
50 (11) | |
Scala file
Tuning file for anything that supports Scala. Made with Scale Workshop.
! 13pt888edo.scl ! 13.888 equal divisions of the octave 14 ! 86.406 172.812 259.218 345.624 432.03 518.436 604.842 691.248 777.654 864.06 950.466 1036.872 1123.278 1209.684
Music
- The Butterfly’s Dream - Herman Miller (2022)