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{| class="wikitable"
{{interwiki
|-
| de = Naturterz
| | [[File:5_4_glyph.png|alt=5 4 glyph.png|113x123px|5 4 glyph.png]]
| en = 5/4
|-
| es =
| | JI Glyph for 5/4
| ja =
|}
| ro = 5/4 (ro)
}}
{{Infobox Interval
| Name = just major third, classic(al) major third, ptolemaic major third
| Color name = y3, yo 3rd
| Sound = jid_5_4_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major third}}


'''5/4'''
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".
|-2 0 1&gt;


386.31371 cents
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).


[[File:jid_5_4_pluck_adu_dr220.mp3]] [[:File:jid_5_4_pluck_adu_dr220.mp3|sound sample]]
== 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;).


In [[Just_intonation|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|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|5-limit]] harmony. It is distinguished from the [[Pythagorean|Pythagorean]] major third of [[81/64|81/64]] by the syntonic comma of [[81/80|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".
{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
 
|-
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).
! [[Edo]]
 
! class="unsortable" | deg\edo
[[File:5-4.mp3]]
! Absolute <br> error ([[Cent|¢]])
 
! Relative <br> error ([[Relative cent|r¢]])
[[:File:5-4.mp3|5-4.mp3]]
! &#8597;
 
! class="unsortable" | Equally acceptable multiples <ref>Super-edos up to 200 within the same error tolerance</ref>
5/4 converted to cents (¢): 1200 * log (5/4) / log (2) = 386.314...
|-
|  [[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; ||
|}


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.
== 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]]


==5/4 quotes==
== Notes ==
got any?
<references/>


See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]      [[Category:5-limit]]
[[Category:Third]]
[[Category:just_interval]]
[[Category:Major third]]
[[Category:major_third]]
[[Category:ratio]]
[[Category:superparticular]]
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