6/5
| Ratio | 6/5 |
| Factorization | 2 × 3 × 5-1 |
| Monzo | [1 1 -1⟩ |
| Size in cents | 315.64129¢ |
| Names | just minor third, classic(al) minor third, ptolemaic minor third |
| Color name | g3, gu 3rd |
| FJS name | [math]\text{m3}_{5}[/math] |
| Special properties | superparticular, reduced |
| Tenney height (log2 nd) | 4.90689 |
| Weil height (log2 max(n, d)) | 5.16993 |
| Wilson height (sopfr (nd)) | 10 |
| Harmonic entropy (Shannon, [math]\sqrt{nd}[/math]) |
~4.1012 bits |
[sound info] | |
| open this interval in xen-calc | |
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.
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 by edos
6/5 is very accurately approximated by 19edo (5\19), and hence the enneadecal temperament.
The following edos (up to 200) contain good approximations[2] of the interval 6/5. Errors are given by magnitude, the arrows in the table show if the edo representation is sharp (↑) or flat (↓).
| Edo | deg\edo | Absolute error (¢) |
Relative error (r¢) |
↕ | Equally acceptable multiples [3] |
|---|---|---|---|---|---|
| 15 | 4\15 | 4.3587 | 5.4484 | ↑ | |
| 19 | 5\19 | 0.1482 | 0.2346 | ↑ |
10\38, 15\57, 20\76, 25\95, 30\114, 35\133, 40\152, 45\171, 50\190 |
| 23 | 6\23 | 2.5978 | 4.9791 | ↓ | |
| 34 | 9\34 | 2.0058 | 5.683 | ↑ | |
| 42 | 11\42 | 1.3556 | 4.7445 | ↓ | |
| 53 | 14\53 | 1.3398 | 5.9176 | ↑ | |
| 61 | 16\61 | 0.8872 | 4.5099 | ↓ | |
| 72 | 19\72 | 1.0254 | 6.1523 | ↑ | |
| 80 | 21\80 | 0.6413 | 4.2752 | ↓ | |
| 91 | 24\91 | 0.8422 | 6.3869 | ↑ | |
| 99 | 26\99 | 0.4898 | 4.0406 | ↓ | |
| 110 | 29\110 | 0.7223 | 6.6215 | ↑ | |
| 118 | 31\118 | 0.387 | 3.806 | ↓ | |
| 129 | 34\129 | 0.6378 | 6.8562 | ↑ | |
| 137 | 36\137 | 0.3128 | 3.5714 | ↓ | |
| 156 | 41\156 | 0.2567 | 3.3367 | ↓ | |
| 175 | 46\175 | 0.2127 | 3.1021 | ↓ | |
| 194 | 51\194 | 0.1774 | 2.8675 | ↓ |
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