4/3: Difference between revisions

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{{interwiki
| de =
| en =
| es =
| ja =
| ko =
| ro = 4/3 (ro)
}}
{{Infobox Interval
{{Infobox Interval
| Ratio = 4/3
| Monzo = 2 -1
| Cents = 498.04500
| Name = just perfect fourth
| Name = just perfect fourth
| Color name = w4, wa 4th
| Color name = w4, wa 4th
| FJS name = P4
| Sound = jid_4_3_pluck_adu_dr220.mp3
| Sound = jid_4_3_pluck_adu_dr220.mp3
}}
}}
{{Wikipedia|Perfect fourth}}
{{Wikipedia|Perfect fourth}}


'''4/3''' is the [[frequency ratio]] of the '''just perfect fourth'''. Its inversion is the perfect fifth, [[3/2]]. 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.
'''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 [[Approaches to Musical Tuning|musical traditions]], past and present.
 
Among many other uses, 4/3 forms the basis of [[tetrachord]]s in many musical traditions, such as [[Ancient Greek music]], as well as in modern [[just intonation]] and [[xenharmonic|xenharmony]].
 
== History ==
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.
 
== Chord construction ==
Much like 3/2, 4/3 is valuable as a framework for constructing [[chord]]s. 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 [[Condissonance|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 [[tritave]]s in the same capacity- at least in [[Octave #Octave equivalence|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 [[EDO]]s (up to 200) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 4/3. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&uarr;) or flat (&darr;).
 
{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute <br> error ([[Cent|¢]])
! Relative <br> error ([[Relative cent|r¢]])
! &#8597;
! class="unsortable" | Equally acceptable multiples <ref>Super EDOs up to 200 within the same error tolerance</ref>
|-
|  [[12edo|12]]  ||  5\12  || 1.9550 || 1.9550 || &uarr; || [[24edo|10\24]], [[36edo|15\36]]
|-
|  [[17edo|17]]  ||  7\17  || 3.9274 || 5.5637 || &darr; ||
|-
|  [[29edo|29]]  ||  12\29  || 1.4933 || 3.6087 || &darr; ||
|-
|  [[41edo|41]]  ||  17\41  || 0.4840 || 1.6537 || &darr; || [[82edo|34\82]], [[123edo|51\123]], [[164edo|68\164]]
|-
|  [[53edo|53]]  ||  22\53  || 0.0682 || 0.3013 || &uarr; || [[106edo|44\106]], [[159edo|66\159]]
|-
|  [[65edo|65]]  ||  27\65  || 0.4165 || 2.2563 || &uarr; || [[130edo|54\130]], [[195edo|81\195]]
|-
|  [[70edo|70]]  ||  29\70  || 0.9021 || 5.2625 || &darr; ||
|-
|  [[77edo|77]]  ||  32\77  || 0.6563 || 4.2113 || &uarr; ||
|-
|  [[89edo|89]]  ||  37\89  || 0.8314 || 6.1663 || &uarr; ||
|-
|  [[94edo|94]]  ||  39\94  || 0.1727 || 1.3525 || &darr; || [[188edo|78\188]]
|-
| [[111edo|111]] ||  46\111 || 0.7477 || 6.9162 || &darr; ||
|-
| [[118edo|118]] ||  49\118 || 0.2601 || 2.5575 || &uarr; ||
|-
| [[135edo|135]] ||  56\135 || 0.2672 || 3.0062 || &darr; ||
|-
| [[142edo|142]] ||  59\142 || 0.5466 || 6.4675 || &uarr; ||
|-
| [[147edo|147]] ||  61\147 || 0.0858 || 1.0512 || &darr; ||
|-
| [[171edo|171]] || 71\171 || 0.2006 || 2.8588 || &uarr; ||
|-
| [[176edo|176]] || 73\176 || 0.3177 || 4.6600 || &darr; ||
|-
| [[183edo|183]] || 76\183 || 0.3157 || 4.8138 || &uarr; ||
|-
| [[200edo|200]] || 83\200 || 0.0450 || 0.7500 || &darr; ||
|-
|}
<references/>


The 4/3 interval is easily heavily discussed outside of xenharmony. In fact, some of the usages discussed there have gone on to inspire other music theories in xenharmonic contexts, and indeed continue to inform certain ideas about [[tetrachord]]s.
== Temperaments ==
4/3 can be used as an alternative generator for temperaments generated by an octave and a fifth of 3/2, such as [[meantone]], [[superpyth]], and [[schismic]]. See [[3/2 #In regular temperament theory]] for details.


== See also ==
== See also ==
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* [[9/8]] – its [[fifth complement]]
* [[9/8]] – its [[fifth complement]]
* [[Fourth complement]]
* [[Fourth complement]]
* [[Ed4/3]]
* [[Gallery of just intervals]]
* [[Gallery of just intervals]]


[[Category:3-limit]]
[[Category:Interval]]
[[Category:Ratio]]
[[Category:Fourth]]
[[Category:Fourth]]
[[Category:Pythagorean]]
[[Category:Over-3 intervals]]
[[Category:Superparticular]]
[[Category:Tritave-reduced harmonics]]
[[Category:Subharmonic]]
[[Category:Over-3]]
[[Category:Stub]]

Latest revision as of 11:09, 27 January 2026

Interval information
Ratio 4/3
Factorization 22 × 3-1
Monzo [2 -1⟩
Size in cents 498.045¢
Name just perfect fourth
Color name w4, wa 4th
FJS name [math]\displaystyle{ \text{P4} }[/math]
Special properties square superparticular,
reduced,
reduced subharmonic
Tenney norm (log2 nd) 3.58496
Weil norm (log2 max(n, d)) 4
Wilson norm (sopfr(nd)) 7

[sound info]
Open this interval in xen-calc
English Wikipedia has an article on:

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 ↓
  1. ↑ error magnitude below 7, both, absolute (in ¢) and relative (in r¢)
  2. ↑ Super EDOs up to 200 within the same error tolerance

Temperaments

4/3 can be used as an alternative generator for temperaments generated by an octave and a fifth of 3/2, such as meantone, superpyth, and schismic. See 3/2 #In regular temperament theory for details.

See also

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