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'''. 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 (↓).

~~'''~~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.~~

{| 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~~, as ~~the~~ [[~~Wikipedia: Perfect fourth|corresponding Wikipedia article~~]] ~~makes abundantly clear. 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 17:

Line 86:

* [[9/8]] – its [[fifth complement]]

* [[Fourth complement]]

* [[Ed4/3]]

* [[Gallery of just intervals]]

* [[Wikipedia: Perfect fourth]]

~~[[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'''. 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;).
-'''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.
+{| 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, as the [[Wikipedia: Perfect fourth|corresponding Wikipedia article]] makes abundantly clear.  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 17: / Line 86: @@
 * [[9/8]] – its [[fifth complement]]
 * [[Fourth complement]]
+* [[Ed4/3]]
 * [[Gallery of just intervals]]
-* [[Wikipedia: Perfect fourth]]
-[[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]]