4/3: Difference between revisions

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{{interwiki

| de =

| en =

| es =

| ja = 4/3

| ko =

| ro = 4/3 (ro)

}}

{{Infobox Interval

~~| Ratio = 4/3~~

~~| Monzo = 2 -1~~

~~| Cents = 498.04500~~

| Name = just perfect fourth

| Color name = w4, wa 4th

~~| FJS name = P4~~

| Sound = jid_4_3_pluck_adu_dr220.mp3

}}

'''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 (↑) or flat (↓).

{| 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¢]])

! ↕

! 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 || ↑ || [[24edo|10\24]], [[36edo|15\36]]

|-

| [[17edo|17]] || 7\17 || 3.9274 || 5.5637 || ↓ ||

|-

| [[29edo|29]] || 12\29 || 1.4933 || 3.6087 || ↓ ||

|-

| [[41edo|41]] || 17\41 || 0.4840 || 1.6537 || ↓ || [[82edo|34\82]], [[123edo|51\123]], [[164edo|68\164]]

|-

| [[53edo|53]] || 22\53 || 0.0682 || 0.3013 || ↑ || [[106edo|44\106]], [[159edo|66\159]]

|-

| [[65edo|65]] || 27\65 || 0.4165 || 2.2563 || ↑ || [[130edo|54\130]], [[195edo|81\195]]

|-

| [[70edo|70]] || 29\70 || 0.9021 || 5.2625 || ↓ ||

|-

| [[77edo|77]] || 32\77 || 0.6563 || 4.2113 || ↑ ||

|-

| [[89edo|89]] || 37\89 || 0.8314 || 6.1663 || ↑ ||

|-

| [[94edo|94]] || 39\94 || 0.1727 || 1.3525 || ↓ || [[188edo|78\188]]

|-

| [[111edo|111]] || 46\111 || 0.7477 || 6.9162 || ↓ ||

|-

| [[118edo|118]] || 49\118 || 0.2601 || 2.5575 || ↑ ||

|-

| [[135edo|135]] || 56\135 || 0.2672 || 3.0062 || ↓ ||

|-

| [[142edo|142]] || 59\142 || 0.5466 || 6.4675 || ↑ ||

|-

| [[147edo|147]] || 61\147 || 0.0858 || 1.0512 || ↓ ||

|-

| [[171edo|171]] || 71\171 || 0.2006 || 2.8588 || ↑ ||

|-

| [[176edo|176]] || 73\176 || 0.3177 || 4.6600 || ↓ ||

|-

| [[183edo|183]] || 76\183 || 0.3157 || 4.8138 || ↑ ||

|-

| [[200edo|200]] || 83\200 || 0.0450 || 0.7500 || ↓ ||

|-

|}

~~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 ==

Line 18:

Line 86:

* [[9/8]] – its [[fifth complement]]

* [[Fourth complement]]

* [[Ed4/3]]

* [[Gallery of just intervals]]

~~[[Category:3-limit]]~~

~~[[Category:Interval]]~~

~~[[Category:Ratio]]~~

[[Category:Fourth]]

~~[[Category:Pythagorean]]~~

[[Category:Over-3 intervals]]

~~[[Category:Superparticular]]~~

[[Category:Tritave-reduced harmonics]]

~~[[Category:Subharmonic]]~~

[[Category:Over-3]]

[[Category:~~Stub~~]]

@@ Line 1: / Line 1: @@
+{{interwiki
+| de =
+| en =
+| es =
+| ja = 4/3
+| ko =
+| ro = 4/3 (ro)
+}}
 {{Infobox Interval
-| Ratio = 4/3
-| Monzo = 2 -1
-| Cents = 498.04500
 | Name = just perfect fourth
 | Color name = w4, wa 4th
-| FJS name = P4
 | Sound = jid_4_3_pluck_adu_dr220.mp3
 }}
 {{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 ==
+/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 ==
@@ Line 18: / Line 86: @@
 * [[9/8]] – its [[fifth complement]]
 * [[Fourth complement]]
+* [[Ed4/3]]
 * [[Gallery of just intervals]]
-[[Category:3-limit]]
-[[Category:Interval]]
-[[Category:Ratio]]
 [[Category:Fourth]]
-[[Category:Pythagorean]]
+[[Category:Over-3 intervals]]
-[[Category:Superparticular]]
+[[Category:Tritave-reduced harmonics]]
-[[Category:Subharmonic]]
-[[Category:Over-3]]
-[[Category:Stub]]