5/3
| Interval information |
classic(al) major sixth,
ptolemaic major sixth
[sound info]
In 5-limit just intonation, 5/3 is the just major sixth, classic(al) major sixth, or ptolemaic major sixth[1] of about 884.4¢. It represents the difference between the 5th and 3rd harmonics, and appears in just chords such as 3:4:5 (a 2nd inversion major triad). Its inversion is 6/5, the 5-limit minor third. It differs from the Pythagorean major sixth of 27/16 (about 905.9¢) by the syntonic comma of 81/80 (about 21.5¢). This means that in systems which temper out the syntonic comma, such as 12edo and meantone systems, 5/3 and 27/16 are conflated.
5/3 has a more mellow sound than 27/16, owing to its simpler beating pattern as well as its smaller size.
Approximation
5/3 is very accurately approximated by 19edo (14\19), and hence the enneadecal temperament.
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 4 | 3\4 | 900.00 | +15.64 | +5.21 |
| 15 | 11\15 | 880.00 | -4.36 | -5.45 |
| 19 | 14\19 | 884.21 | -0.15 | -0.23 |
| 23 | 17\23 | 886.96 | +2.60 | +4.98 |
| 34 | 25\34 | 882.35 | -2.01 | -5.68 |
| 38 | 28\38 | 884.21 | -0.15 | -0.47 |
| 42 | 31\42 | 885.71 | +1.36 | +4.74 |
| 46 | 34\46 | 886.96 | +2.60 | +9.96 |
| 53 | 39\53 | 883.02 | -1.34 | -5.92 |
| 57 | 42\57 | 884.21 | -0.15 | -0.70 |
| 61 | 45\61 | 885.25 | +0.89 | +4.51 |
| 65 | 48\65 | 886.15 | +1.80 | +9.72 |
| 72 | 53\72 | 883.33 | -1.03 | -6.15 |
| 76 | 56\76 | 884.21 | -0.15 | -0.94 |
| 80 | 59\80 | 885.00 | +0.64 | +4.28 |