6/5
In 5-limit just intonation, 6/5 is the just minor third, classic(al) minor third, or ptolemaic minor third[1], measuring about 315.6¢. It is sharp of the Pythagorean minor third of 32/27 (about 294.1¢) as well as the 300¢ minor third of 4edo, 12edo and all other 4n-edos. It arises in the harmonic series between the 5th and 6th harmonics and appears in the 5-limit otonal triad of 4:5:6. A 5-limit minor triad in just intonation can be written 10:12:15, with 6/5 falling between 10 and 12, 5/4 falling between 12 and 15, and 3/2 falling between 10 and 15.
| Interval information |
classic(al) minor third,
ptolemaic minor third
reduced
[sound info]
In higher-limit JI, 6/5 is only one of many minor thirds. A popular one in the 7-limit is 7/6 (about 266.9¢), the septimal subminor third, which is 36/35 (about 48.8¢) flat of 6/5. Another in the 13-limit is 13/11 (about 289.2¢), which is 66/65 (about 26.4¢) flat of 6/5. Both of these are more complex intervals than 6/5 and have their own character to them.
Approximation
6/5 is very accurately approximated by 19edo (5\19), and hence the enneadecal temperament.
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 4 | 1\4 | 300.00 | -15.64 | -5.21 |
| 15 | 4\15 | 320.00 | +4.36 | +5.45 |
| 19 | 5\19 | 315.79 | +0.15 | +0.23 |
| 23 | 6\23 | 313.04 | -2.60 | -4.98 |
| 34 | 9\34 | 317.65 | +2.01 | +5.68 |
| 38 | 10\38 | 315.79 | +0.15 | +0.47 |
| 42 | 11\42 | 314.29 | -1.36 | -4.74 |
| 46 | 12\46 | 313.04 | -2.60 | -9.96 |
| 53 | 14\53 | 316.98 | +1.34 | +5.92 |
| 57 | 15\57 | 315.79 | +0.15 | +0.70 |
| 61 | 16\61 | 314.75 | -0.89 | -4.51 |
| 65 | 17\65 | 313.85 | -1.80 | -9.72 |
| 72 | 19\72 | 316.67 | +1.03 | +6.15 |
| 76 | 20\76 | 315.79 | +0.15 | +0.94 |
| 80 | 21\80 | 315.00 | -0.64 | -4.28 |
See also
- 5/3 – its octave complement
- 5/4 – its fifth complement
- 10/9 – its fourth complement
- Gallery of just intervals
- List of superparticular intervals
- File:Ji-6-5-csound-foscil-220hz.mp3 – another sound example