3ed4/3
| ← 2ed4/3 | 3ed4/3 | 4ed4/3 → |
(semiconvergent)
(semiconvergent)
3 equal divisions of 4/3 (abbreviated 3ed4/3) is a nonoctave tuning system that divides the interval of 4/3 into 3 equal parts of about 166 ¢ each. Each step represents a frequency ratio of (4/3)1/3, or the 3rd root of 4/3.
Theory
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -37.9 | -75.8 | -75.8 | +35.9 | +52.3 | -48.5 | +52.3 | +14.4 | -2.0 | -0.9 | +14.4 |
| Relative (%) | -22.8 | -45.7 | -45.7 | +21.6 | +31.5 | -29.2 | +31.5 | +8.7 | -1.2 | -0.6 | +8.7 | |
| Steps (reduced) |
7 (1) |
11 (2) |
14 (2) |
17 (2) |
19 (1) |
20 (2) |
22 (1) |
23 (2) |
24 (0) |
25 (1) |
26 (2) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +41.9 | +79.6 | -39.8 | +14.4 | +75.5 | -23.5 | +49.0 | -39.8 | +41.7 | -38.8 | +50.2 |
| Relative (%) | +25.2 | +47.9 | -24.0 | +8.7 | +45.5 | -14.1 | +29.5 | -24.0 | +25.1 | -23.4 | +30.3 | |
| Steps (reduced) |
27 (0) |
28 (1) |
28 (1) |
29 (2) |
30 (0) |
30 (0) |
31 (1) |
31 (1) |
32 (2) |
32 (2) |
33 (0) | |
Intervals
| # | Cents | Approximate ratios |
|---|---|---|
| 0 | 0.000 | 1/1 |
| 1 | 166.015 | 11/10 |
| 2 | 332.030 | |
| 3 | 498.045 | 4/3 |
| 4 | 664.060 | 22/15 |
| 5 | 830.075 | 13/8 |
| 6 | 996.090 | 16/9 |
| 7 | 1162.105 | 88/45 |
| 8 | 1328.120 | 13/6 |
| 9 | 1494.135 | 64/27 |
| 10 | 1660.150 | |
| 11 | 1826.165 | 13/9 |
| 12 | 1992.180 | |
| 13 | 2158.195 | |
| 14 | 2324.210 | |
| 15 | 2490.225 | 135/32 |
| 16 | 2656.240 | |
| 17 | 2822.255 | |
| 18 | 2988.270 | 45/8 |
| 19 | 3154.285 | |
| 20 | 3320.300 | 17/5 |
| 21 | 3486.315 | 15/2 |
| 22 | 3652.330 | |
| 23 | 3818.345 | 68/15 |
| 24 | 3984.360 | 10/1 |
Regular temperaments
3ed4/3 tuning is related to temperaments which temper out 4000/3993 (wizardharry temperament). The unit step of 3ed4/3 is approximately a cent sharp of 11/10. Tempering out 4000/3993 leads equating three 11/10s with 4/3, hence wizardharry temperaments split the fourth in three.
Tempering out both 55/54 and 100/99 (equating 10/9 with 11/10 and 12/11) leads to porcupine (2.3.5.11 subgroup) or sonic (full 11-limit). Sonic temperaments include porcupine, hystrix, porky, coendou, hedgehog, nautilus, ammonite, ceratitid, and opossum.
Other wizardharry temperaments include octoid, harry, tritikleismic, wizard, septisuperfourth, unthirds, supers, alphaquarter, quincy, stearnscape, pogo, marvolo, cotritone, echidna, marvo, mystery, zarvo, escaped, thuja, and escapade.