4/3: Difference between revisions

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{{interwiki

| de =

| en =

| es =

| ja =

| 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''', 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'''. 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 3/2. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (↑) or flat (↓).

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"

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|}

== 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 ==

* [[3/2]] – its [[octave complement]]

* [[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 =
+| 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''', 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'''. 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 3/2. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&uarr;) or flat (&darr;).
+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"
@@ Line 64: / Line 78: @@
 |}
 <references/>
+== 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 ==
 * [[3/2]] – its [[octave complement]]
 * [[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]]