4/3
| Interval information |
reduced,
reduced subharmonic
(Shannon, [math]\sqrt{nd}[/math])
[sound info]
4/3 is the frequency ratio of the just perfect fourth. As its inversion is the perfect fifth, 3/2, 4/3 is the octave reduced form of the third subharmonic. 4/3 is one of the most common intervals one finds in the world's musical traditions, past and present.
Among many other uses, 4/3 forms the basis of tetrachords in many musical traditions, such as Ancient Greek music, as well as in modern just intonation and xenharmony.
History
In the florid organum of Medieval music, 4/3 was reliably considered a consonance, and indeed was frequently emphasized. Once major thirds with a tuning approximating 5/4 began to be treated as consonances, however, the perception of 4/3 was altered to where it was at times considered a dissonance. However, as of late, the perfect fourth is once again being reevaluated as a consonance.
Chord construction
Much like 3/2, 4/3 is valuable as a framework for constructing chords. However, while 3/2 provides the framework for 5-limit triads involving intervals like 5/4 and 6/5, 4/3 provides a possible framework for 7-limit triads involving intervals like 7/6 and 8/7, though such triads are ambisonances (that is, they're both consonant and dissonant at the same time) at best.
Because up to two instances of 4/3 can fit within the span of an octave, it is very easy to create xenharmonic chords using 4/3 as a framework. Regardless, the usage of 4/3 as a framework for chords is intimately connected with the use of tritaves in the same capacity- at least in octave-equivalent systems- due to the same pitch classes being involved in both 6:7:8 and 4:7:12 where 7 is kept as the same note, thus rendering the two chords as different voicings of the same underlying harmonic unit.
Approximations by EDOs
The following EDOs (up to 200) contain good approximations[1] of the interval 4/3. 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 [2] |
|---|---|---|---|---|---|
| 12 | 5\12 | 1.9550 | 1.9550 | ↑ | 10\24, 15\36 |
| 17 | 7\17 | 3.9274 | 5.5637 | ↓ | |
| 29 | 12\29 | 1.4933 | 3.6087 | ↓ | |
| 41 | 17\41 | 0.4840 | 1.6537 | ↓ | 34\82, 51\123, 68\164 |
| 53 | 22\53 | 0.0682 | 0.3013 | ↑ | 44\106, 66\159 |
| 65 | 27\65 | 0.4165 | 2.2563 | ↑ | 54\130, 81\195 |
| 70 | 29\70 | 0.9021 | 5.2625 | ↓ | |
| 77 | 32\77 | 0.6563 | 4.2113 | ↑ | |
| 89 | 37\89 | 0.8314 | 6.1663 | ↑ | |
| 94 | 39\94 | 0.1727 | 1.3525 | ↓ | 78\188 |
| 111 | 46\111 | 0.7477 | 6.9162 | ↓ | |
| 118 | 49\118 | 0.2601 | 2.5575 | ↑ | |
| 135 | 56\135 | 0.2672 | 3.0062 | ↓ | |
| 142 | 59\142 | 0.5466 | 6.4675 | ↑ | |
| 147 | 61\147 | 0.0858 | 1.0512 | ↓ | |
| 171 | 71\171 | 0.2006 | 2.8588 | ↑ | |
| 176 | 73\176 | 0.3177 | 4.6600 | ↓ | |
| 183 | 76\183 | 0.3157 | 4.8138 | ↑ | |
| 200 | 83\200 | 0.0450 | 0.7500 | ↓ |