4/3: Difference between revisions
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'''4/3''' is the [[frequency ratio]] of the '''just perfect fourth''', which is easily one of the more heavily discussed intervals outside of xenharmony- in fact, some of these usages have gone on to inspire other music theories within xenharmonic contexts, such as certain ideas about [[tetrachord]]s. As its inversion is the perfect fifth, [[3/2]], 4/3 is the [[octave reduced]] form of the third [[subharmonic]]. In the [[Wikipedia: Medieval music #Early polyphony: organum|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. | '''4/3''' is the [[frequency ratio]] of the '''just perfect fourth''', which is easily one of the more heavily discussed intervals outside of xenharmony- in fact, some of these usages have gone on to inspire other music theories within xenharmonic contexts, such as certain ideas about [[tetrachord]]s. As its inversion is the perfect fifth, [[3/2]], 4/3 is the [[octave reduced]] form of the third [[subharmonic]]. In the [[Wikipedia: Medieval music #Early polyphony: organum|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. | ||
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 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, though because [[16/9]] clashes with the [[octave]] through crowding, these types of chords are dissonances. Regardless, the usage of 4/3 as a framework for chords is intimately connected with the use of [[tritave]]s in the same capacity- at least in [[octave-equivalent]] systems. | 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 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, though because [[16/9]] clashes with the [[octave]] through crowding, these types of chords are dissonances. Regardless, the usage of 4/3 as a framework for chords is intimately connected with the use of [[tritave]]s in the same capacity- at least in [[Octave #Octave equivalence|octave-equivalent]] systems. | ||
== Approximations by EDOs == | == Approximations by EDOs == | ||
Revision as of 13:58, 24 December 2024
| Interval information |
reduced,
reduced subharmonic
[sound info]
4/3 is the frequency ratio of the just perfect fourth, which is easily one of the more heavily discussed intervals outside of xenharmony- in fact, some of these usages have gone on to inspire other music theories within xenharmonic contexts, such as certain ideas about tetrachords. As its inversion is the perfect fifth, 3/2, 4/3 is the octave reduced form of the third subharmonic. 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.
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 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, though because 16/9 clashes with the octave through crowding, these types of chords are dissonances. 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.
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 | ↓ |