5/4: Difference between revisions

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{{interwiki

| de = Naturterz

| en = 5/4

| es =

| ja =

| ro = 5/4 (ro)

}}

{{Infobox Interval

~~| Icon = [[File:5_4_glyph.png|alt=5 4 glyph.png|113x123px|5 4 glyph.png]] JI Glyph~~

| Name = just major third, classic(al) major third, ptolemaic major third

~~| Ratio = 5/4~~

| Color name = y3, yo 3rd

~~| Monzo = -2 0 1~~

~~| Cents = 386.31371~~

| Name = just major third

| Sound = jid_5_4_pluck_adu_dr220.mp3

}}

In [[~~Just Intonation~~]], '''5/4''' is the frequency ratio between the 5th and 4th ~~harmonics~~. Measuring about 386.3[[cent|¢]], it is about 13.7¢ away from [[12edo]]'s major third of ~~400¢~~. It has a distinctive "sweet" sound, and has been described as more "laid back" than its 12edo counterpart. Providing a novel consonance after 3, it is the basis for [[5-limit]] harmony. It is distinguished from the [[Pythagorean]] major third of [[81/64]] by the syntonic comma of [[81/80]], which measures about 21.5¢. 81/64 and 5/4 are both just intonation "major thirds," 81/64 having a more active and discordant quality, 5/4 sounding more "restful".

In [[5-limit]] [[just intonation]], '''5/4''' is the [[frequency ratio]] between the 5th and 4th [[harmonic]]s. It has been called the '''just major third''', '''classic(al) major third''', or '''ptolemaic major third'''<ref>For reference, see [[5-limit]].</ref> to distinguish it from other intervals in that neighborhood. Measuring about 386.3 [[cent|¢]], it is about 13.7{{c}} away from [[12edo]]'s major third of 400{{c}}. It has a distinctive "sweet" sound, and has been described as more "laid back" than its 12edo counterpart. Providing a novel consonance after 3, it is the basis for [[5-limit]] harmony. It is distinguished from the [[Pythagorean]] major third of [[81/64]] by the syntonic comma of [[81/80]], which measures about 21.5{{c}}, and from the Pythagorean diminished fourth of [[8192/6561]] by the [[schisma]], which measures about 1.95{{c}}. 81/64 and 5/4 are both just intonation "major thirds", 81/64 having a more active and discordant quality, 5/4 sounding more "restful".

In the context of the harmonic series, 5/4 can be heard between the 4th and 5th member of the series, demonstrated here melodically in singing into a resonant [http://udderbot.wikispaces.com/home udderbot] (from the fundamental up to 5 and then noodling between 5 and 4).

[[File:5-4.mp3]] ~~ ~~ [[~~:File:~~5~~-4.mp3|~~5-4.~~mp3]]~~

In the context of the harmonic series, 5/4 can be heard between the 4th and 5th member of the series, demonstrated in [[:File: 5-4.mp3]] melodically in singing into a resonant [[udderbot]] (from the fundamental up to 5 and then noodling between 5 and 4).

~~5/4 converted~~ to ~~cents~~ (¢)~~: 1200 * log~~ (5/4) ~~/ log~~ (2) ~~= 386.314..~~.

== Approximations by edos ==

Following [[edo]]s (up to 200, and also 643) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 5/4. Errors are given by magnitude, the arrows in the table show if the edo representation is sharp (↑) or flat (↓).

5/4 ~~the interval~~, ~~like all ''intervals''~~, ~~refers to a ''relation'' between two pitches~~. ~~It has been called the ''just major third'' to distinguish it from the other intervals in that neighborhood~~.

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"

|-

! [[Edo]]

! class="unsortable" | deg\edo

! Absolute error ([[Cent|¢]])

! Relative error ([[Relative cent|r¢]])

! ↕

! class="unsortable" | Equally acceptable multiples <ref>Super-edos up to 200 within the same error tolerance</ref>

|-

| [[25edo|25]] || 8\25 || 2.3137 || 4.8202 || ↓ ||

|-

| [[28edo|28]] || 9\28 || 0.5994 || 1.3987 || ↓ || [[56edo|18\56]], [[84edo|27\84]], [[112edo|36\112]], [[140edo|45\140]]

|-

| [[31edo|31]] || 10\31 || 0.7831 || 2.0229 || ↑ || [[62edo|20\62]], [[93edo|30\93]]

|-

| [[34edo|34]] || 11\34 || 1.9216 || 5.4445 || ↑ ||

|-

| [[53edo|53]] || 17\53 || 1.4081 || 6.2189 || ↓ ||

|-

| [[59edo|59]] || 19\59 || 0.1270 || 0.6242 || ↑ || [[118edo|38\118]], [[177edo|57\177]]

|-

| [[87edo|87]] || 28\87 || 0.1068 || 0.7744 || ↓ || [[174edo|56\174]]

|-

| [[90edo|90]] || 29\90 || 0.3530 || 2.6471 || ↑ || [[180edo|58\180]]

|-

| [[115edo|115]] || 37\115 || 0.2268 || 2.1731 || ↓ ||

|-

| [[121edo|121]] || 39\121 || 0.4631 || 4.6701 || ↑ ||

|-

| [[143edo|143]] || 46\143 || 0.2997 || 3.5718 || ↓ ||

|-

| [[146edo|146]] || 47\146 || 0.0123 || 0.1502 || ↓ ||

|-

| [[149edo|149]] || 48\149 || 0.2635 || 3.2714 || ↑ ||

|-

| [[152edo|152]] || 49\152 || 0.5284 || 6.6930 || ↑ ||

|-

| [[171edo|171]] || 55\171 || 0.3488 || 4.9704 || ↓ ||

|-

| [[199edo|199]] || 64\199 || 0.3841 || 6.3691 || ↓ ||

|-

| [[643edo|643]] || 207\643 || 0.0004 || 0.0235 || ↑ ||

|}

~~:''~~ See also [[Gallery of ~~Just Intervals~~]]''

== See also ==

* [[8/5]] – its [[octave complement]]

* [[6/5]] – its [[fifth complement]]

* [[16/15]] – its [[fourth complement]]

* [[5/2]] – the interval up one [[octave]] which sounds even more [[consonant]]

* [[Ed5/4]]

* [[Gallery of just intervals]]

* [[List of superparticular intervals]]

~~[[Category:5-limit]]~~

== Notes ==

~~[[Category:just_interval]]~~

~~[[Category:major_third]]~~

~~[[Category:ratio]]~~

~~[[Category:superparticular]]~~

~~~~

[[Category:Third]]

[[de:~~Naturterz~~]]

[[Category:Major third]]

@@ Line 1: / Line 1: @@
+{{interwiki
+| de = Naturterz
+| en = 5/4
+| es =
+| ja =
+| ro = 5/4 (ro)
+}}
 {{Infobox Interval
-| Icon = [[File:5_4_glyph.png|alt=5 4 glyph.png|113x123px|5 4 glyph.png]] <small><br/>JI Glyph</small>
+| Name = just major third, classic(al) major third, ptolemaic major third
-| Ratio = 5/4
+| Color name = y3, yo 3rd
-| Monzo = -2 0 1
-| Cents = 386.31371
-| Name = just major third
 | Sound = jid_5_4_pluck_adu_dr220.mp3
 }}
+{{Wikipedia|Major third}}
-In [[Just Intonation]], '''5/4''' is the frequency ratio between the 5th and 4th harmonics. Measuring about 386.3[[cent|¢]], it is about 13.7¢ away from [[12edo]]'s major third of 400¢. It has a distinctive "sweet" sound, and has been described as more "laid back" than its 12edo counterpart. Providing a novel consonance after 3, it is the basis for [[5-limit]] harmony. It is distinguished from the [[Pythagorean]] major third of [[81/64]] by the syntonic comma of [[81/80]], which measures about 21.5¢. 81/64 and 5/4 are both just intonation "major thirds," 81/64 having a more active and discordant quality, 5/4 sounding more "restful".
+In [[5-limit]] [[just intonation]], '''5/4''' is the [[frequency ratio]] between the 5th and 4th [[harmonic]]s. It has been called the '''just major third''', '''classic(al) major third''', or '''ptolemaic major third'''<ref>For reference, see [[5-limit]].</ref> to distinguish it from other intervals in that neighborhood. Measuring about 386.3 [[cent|¢]], it is about 13.7{{c}} away from [[12edo]]'s major third of 400{{c}}. It has a distinctive "sweet" sound, and has been described as more "laid back" than its 12edo counterpart. Providing a novel consonance after 3, it is the basis for [[5-limit]] harmony. It is distinguished from the [[Pythagorean]] major third of [[81/64]] by the syntonic comma of [[81/80]], which measures about 21.5{{c}}, and from the Pythagorean diminished fourth of [[8192/6561]] by the [[schisma]], which measures about 1.95{{c}}. 81/64 and 5/4 are both just intonation "major thirds", 81/64 having a more active and discordant quality, 5/4 sounding more "restful".
-In the context of the harmonic series, 5/4 can be heard between the 4th and 5th member of the series, demonstrated here melodically in singing into a resonant [http://udderbot.wikispaces.com/home udderbot] (from the fundamental up to 5 and then noodling between 5 and 4).
-[[File:5-4.mp3]] <br/> [[:File:5-4.mp3|5-4.mp3]]
+In the context of the harmonic series, 5/4 can be heard between the 4th and 5th member of the series, demonstrated in [[:File: 5-4.mp3]] melodically in singing into a resonant [[udderbot]] (from the fundamental up to 5 and then noodling between 5 and 4).
-/4 converted to cents (¢): 1200 * log (5/4) / log (2) = 386.314...
+== Approximations by edos ==
+Following [[edo]]s (up to 200, and also 643) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 5/4. Errors are given by magnitude, the arrows in the table show if the edo representation is sharp (&uarr;) or flat (&darr;).
-/4 the interval, like all ''intervals'', refers to a ''relation'' between two pitches. It has been called the ''just major third'' to distinguish it from the other intervals in that neighborhood.
+{| 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>
+|-
+|  [[25edo|25]]  ||  8\25  || 2.3137 || 4.8202 || &darr; ||
+|-
+|  [[28edo|28]]  ||  9\28  || 0.5994 || 1.3987 || &darr; || [[56edo|18\56]], [[84edo|27\84]], [[112edo|36\112]], [[140edo|45\140]]
+|-
+|  [[31edo|31]]  || 10\31  || 0.7831 || 2.0229 || &uarr; || [[62edo|20\62]], [[93edo|30\93]]
+|-
+|  [[34edo|34]]  || 11\34  || 1.9216 || 5.4445 || &uarr; ||
+|-
+|  [[53edo|53]]  || 17\53  || 1.4081 || 6.2189 || &darr; ||
+|-
+|  [[59edo|59]]  || 19\59  || 0.1270 || 0.6242 || &uarr; || [[118edo|38\118]], [[177edo|57\177]]
+|-
+|  [[87edo|87]]  || 28\87  || 0.1068 || 0.7744 || &darr; || [[174edo|56\174]]
+|-
+|  [[90edo|90]]  || 29\90  || 0.3530 || 2.6471 || &uarr; || [[180edo|58\180]]
+|-
+| [[115edo|115]] || 37\115 || 0.2268 || 2.1731 || &darr; ||
+|-
+| [[121edo|121]] || 39\121 || 0.4631 || 4.6701 || &uarr; ||
+|-
+| [[143edo|143]] || 46\143 || 0.2997 || 3.5718 || &darr; ||
+|-
+| [[146edo|146]] || 47\146 || 0.0123 || 0.1502 || &darr; ||
+|-
+| [[149edo|149]] || 48\149 || 0.2635 || 3.2714 || &uarr; ||
+|-
+| [[152edo|152]] || 49\152 || 0.5284 || 6.6930 || &uarr; ||
+|-
+| [[171edo|171]] || 55\171 || 0.3488 || 4.9704 || &darr; ||
+|-
+| [[199edo|199]] || 64\199 || 0.3841 || 6.3691 || &darr; ||
+|-
+| [[643edo|643]] || 207\643 || 0.0004 || 0.0235 || &uarr; ||
+|}
-:'' See also [[Gallery of Just Intervals]]''
+== See also ==
+* [[8/5]] – its [[octave complement]]
+* [[6/5]] – its [[fifth complement]]
+* [[16/15]] – its [[fourth complement]]
+* [[5/2]] – the interval up one [[octave]] which sounds even more [[consonant]]
+* [[Ed5/4]]
+* [[Gallery of just intervals]]
+* [[List of superparticular intervals]]
-[[Category:5-limit]]
+== Notes ==
-[[Category:just_interval]]
+<references/>
-[[Category:major_third]]
-[[Category:ratio]]
-[[Category:superparticular]]
-<!-- interwiki -->
+[[Category:Third]]
-[[de:Naturterz]]
+[[Category:Major third]]