5/4
| Ratio | 5/4 |
| Factorization | 2-2 × 5 |
| Monzo | [-2 0 1⟩ |
| Size in cents | 386.31371¢ |
| Names | just major third, classic(al) major third, ptolemaic major third |
| Color name | y3, yo 3rd |
| FJS name | [math]\text{M3}^{5}[/math] |
| Special properties | superparticular, reduced |
| Tenney height (log2 nd) | 4.32193 |
| Weil height (log2 max(n, d)) | 4.64386 |
| Wilson height (sopfr (nd)) | 9 |
| Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~4.45116 bits |
[sound info] | |
| open this interval in xen-calc | |
In 5-limit just intonation, 5/4 is the frequency ratio between the 5th and 4th harmonics. It has been called the just major third, classic(al) major third, or ptolemaic major third[1] to distinguish it from other intervals in that neighborhood. Measuring about 386.3 ¢, it is about 13.7 ¢ away from 12edo's major third of 400 ¢. It has a distinctive "sweet" sound, and has been described as more "laid back" than its 12edo counterpart. Providing a novel consonance after 3, it is the basis for 5-limit harmony. It is distinguished from the Pythagorean major third of 81/64 by the syntonic comma of 81/80, which measures about 21.5 ¢. 81/64 and 5/4 are both just intonation "major thirds", 81/64 having a more active and discordant quality, 5/4 sounding more "restful".
In the context of the harmonic series, 5/4 can be heard between the 4th and 5th member of the series, demonstrated in File: 5-4.mp3 melodically in singing into a resonant udderbot (from the fundamental up to 5 and then noodling between 5 and 4).
Approximations by edos
Following edos (up to 200, and also 643) contain good approximations[2] of the interval 5/4. 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] |
|---|---|---|---|---|---|
| 25 | 8\25 | 2.3137 | 4.8202 | ↓ | |
| 28 | 9\28 | 0.5994 | 1.3987 | ↓ | 18\56, 27\84, 36\112, 45\140 |
| 31 | 10\31 | 0.7831 | 2.0229 | ↑ | 20\62, 30\93 |
| 34 | 11\34 | 1.9216 | 5.4445 | ↑ | |
| 53 | 17\53 | 1.4081 | 6.2189 | ↓ | |
| 59 | 19\59 | 0.1270 | 0.6242 | ↑ | 38\118, 57\177 |
| 87 | 28\87 | 0.1068 | 0.7744 | ↓ | 56\174 |
| 90 | 29\90 | 0.3530 | 2.6471 | ↑ | 58\180 |
| 115 | 37\115 | 0.2268 | 2.1731 | ↓ | |
| 121 | 39\121 | 0.4631 | 4.6701 | ↑ | |
| 143 | 46\143 | 0.2997 | 3.5718 | ↓ | |
| 146 | 47\146 | 0.0123 | 0.1502 | ↓ | |
| 149 | 48\149 | 0.2635 | 3.2714 | ↑ | |
| 152 | 49\152 | 0.5284 | 6.6930 | ↑ | |
| 171 | 55\171 | 0.3488 | 4.9704 | ↓ | |
| 199 | 64\199 | 0.3841 | 6.3691 | ↓ | |
| 643 | 207\643 | 0.0004 | 0.0235 | ↑ |
See also
- 8/5 – its octave complement
- 6/5 – its fifth complement
- 16/15 – its fourth complement
- 5/2 – the interval plus one octave sounds even more consonant
- Gallery of just intervals
- List of superparticular intervals