4ed4/3
| ← 3ed4/3 | 4ed4/3 | 5ed4/3 → |
(semiconvergent)
4 equal divisions of 4/3 (abbreviated 4ed4/3) is a nonoctave tuning system that divides the interval of 4/3 into 4 equal parts of about 125 ¢ each. Each step represents a frequency ratio of (4/3)1/4, or the 4th root of 4/3.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1, 21/20 |
| 1 | 124.5 | 11/10, 13/12, 14/13, 18/17, 19/18, 20/19, 21/19 |
| 2 | 249 | 7/6, 10/9, 17/14, 17/15, 19/17 |
| 3 | 373.5 | 6/5, 9/7, 11/9, 13/11, 17/13, 20/17, 21/17 |
| 4 | 498 | 13/10, 14/11, 15/11, 18/13, 19/14, 19/15 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +45.1 | -34.3 | -34.3 | -47.1 | +10.8 | -7.0 | +10.8 | +55.9 | -2.0 | -42.4 | +55.9 |
| Relative (%) | +36.2 | -27.5 | -27.5 | -37.8 | +8.7 | -5.6 | +8.7 | +44.9 | -1.6 | -34.1 | +44.9 | |
| Steps (reduced) |
10 (2) |
15 (3) |
19 (3) |
22 (2) |
25 (1) |
27 (3) |
29 (1) |
31 (3) |
32 (0) |
33 (1) |
35 (3) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +41.9 | +38.1 | +43.2 | +55.9 | -49.0 | -23.5 | +7.4 | +43.2 | -41.3 | +2.7 | +50.2 |
| Relative (%) | +33.6 | +30.6 | +34.7 | +44.9 | -39.4 | -18.8 | +6.0 | +34.7 | -33.2 | +2.1 | +40.3 | |
| Steps (reduced) |
36 (0) |
37 (1) |
38 (2) |
39 (3) |
39 (3) |
40 (0) |
41 (1) |
42 (2) |
42 (2) |
43 (3) |
44 (0) | |
Similar tunings
125cET
125-cent equal temperament (125cET, also known as 1ed125c or APS125¢), is an equal nonoctave scale generated by making a continuous chain of intervals of exactly 125¢. It is a stretched tuning of 4ed4/3, with its step size wider by about 0.5 ¢.
Intervals
| Degree | Cents | Associated ratios | Interval name |
|---|---|---|---|
| 0 | 0 | 1/1 | unison |
| 1 | 125 | 14/13, 15/14 | 2/3-tone, trienthird |
| 2 | 250 | 15/13, 22/19 | semifourth |
| 3 | 375 | 5/4 | narrow perde segah, marvelous major third, near just major third |
| 4 | 500 | 4/3 | perfect fourth |
| 5 | 625 | 10/7, 23/16 | pental diminished fifth, classic diminshed fifth |
| 6 | 750 | 17/11, 20/13 | septendecimal subminor sixth |
| 7 | 875 | 5/3 | major sixth |
| 8 | 1000 | 16/9 | Pythagorean minor seventh |
| 9 | 1125 | 21/11, 23/12 | classic (5-limit) diminished octave |
| 10 | 1250 | 33/16 |
11ed11/5
11 equal divisions of 11/5 (abbreviated 11ed11/5) is a nonoctave tuning system that divides the interval of 11/5 into 11 equal parts of about 124 ¢ each. Each step represents a frequency ratio of (11/5)1/11, or the 11th root of 11/5. It is a compressed tuning of 4ed4/3, with its step size narrower by about 0.4 ¢.
Intervals
| Step | Interval (¢) | JI approximated | Simplified ratios |
|---|---|---|---|
| 1 | 124.09 | 29/27 | |
| 2 | 248.18 | 22/19 | |
| 3 | 372.27 | 31/25 | |
| 4 | 496.37 | 8/6, 29/22 | 4/3 |
| 5 | 620.46 | 52/36 | 13/9 |
| 6 | 744.55 | 80/52 | 20/13 |
| 7 | 868.46 | 10/6, 36/22 | 5/3, 18/11 |
| 8 | 992.73 | 39/22, 55/31, 64/36 | 16/9 |
| 9 | 1116.82 | 19/10 | |
| 10 | 1240.91 | 39/19 | |
| 11 | 1365.00 | 11/5 |
The subgroup interpretation used is 11/5.6.8.10.19.25.27.29.31.39.45.52. Other interpretations are possible. Don't forget that fractions can multiply, e.g. 10*11/5=22, 25*11/5=55.
|
Todo: cleanup
Reorganize interval tables? |