5/4: Difference between revisions

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{{interwiki
| de = Naturterz
| en = 5/4
| es =
| ja =
| ro = 5/4 (ro)
}}
{{Infobox Interval
{{Infobox Interval
| JI glyph = [[File:5_4_glyph.png|x48px]]
| Name = just major third, classic(al) major third, ptolemaic major third
| Ratio = 5/4
| Monzo = -2 0 1
| Cents = 386.31371
| Name = classic major third
| Color name = y3, yo 3rd
| Color name = y3, yo 3rd
| FJS name = M3<sup>5</sup>
| Sound = jid_5_4_pluck_adu_dr220.mp3
| Sound = jid_5_4_pluck_adu_dr220.mp3
}}
}}
{{Wikipedia|Major third}}


In [[5-limit]] [[Just Intonation]], '''5/4''' is the [[frequency ratio]] between the 5th and 4th harmonics. It has been called the '''just major third''' or '''classic major third''' to distinguish it from other intervals in that neighborhood. 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 [[udderbot]] (from the fundamental up to 5 and then noodling between 5 and 4).
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).


== Approximations by EDOs ==
== 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;).
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 5/4. 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"
{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
|-
! [[EDO]]
! [[Edo]]
! class="unsortable" | deg\edo
! class="unsortable" | deg\edo
! Absolute <br> error ([[Cent|¢]])
! Absolute <br> error ([[Cent|¢]])
! Relative <br> error ([[Relative cent|r¢]])
! Relative <br> error ([[Relative cent|r¢]])
! &#8597;
! &#8597;
! class="unsortable" | Equally acceptable multiples <ref>Super EDOs up to 200 within the same error tolerance</ref>
! 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; ||
|  [[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]],
|  [[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]],
|  [[31edo|31]]  || 10\31  || 0.7831 || 2.0229 || &uarr; || [[62edo|20\62]], [[93edo|30\93]]
|-
|-
|  [[34edo|34]]  || 11\34  || 1.9216 || 5.4445 || &uarr; ||
|  [[34edo|34]]  || 11\34  || 1.9216 || 5.4445 || &uarr; ||
Line 37: Line 39:
|  [[53edo|53]]  || 17\53  || 1.4081 || 6.2189 || &darr; ||
|  [[53edo|53]]  || 17\53  || 1.4081 || 6.2189 || &darr; ||
|-
|-
|  [[59edo|59]]  || 19\59  || 0.1270 || 0.6242 || &uarr; || [[118edo|38\118]], [[177edo|57\177]],
|  [[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]],
|  [[87edo|87]]  || 28\87  || 0.1068 || 0.7744 || &darr; || [[174edo|56\174]]
|-
|-
|  [[90edo|90]]  || 29\90  || 0.3530 || 2.6471 || &uarr; || [[180edo|58\180]],
|  [[90edo|90]]  || 29\90  || 0.3530 || 2.6471 || &uarr; || [[180edo|58\180]]
|-
|-
| [[115edo|115]] || 37\115 || 0.2268 || 2.1731 || &darr; ||
| [[115edo|115]] || 37\115 || 0.2268 || 2.1731 || &darr; ||
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|-
|-
| [[199edo|199]] || 64\199 || 0.3841 || 6.3691 || &darr; ||
| [[199edo|199]] || 64\199 || 0.3841 || 6.3691 || &darr; ||
|-
| [[643edo|643]] || 207\643 || 0.0004 || 0.0235 || &uarr; ||
|}
|}
<references/>


== See also ==
== See also ==
* [[8/5]] – its [[octave complement]]
* [[8/5]] – its [[octave complement]]
* [[6/5]] – its [[fifth complement]]
* [[6/5]] – its [[fifth complement]]
* [[5/2]] – the interval plus one [[octave]] sounds even more [[consonant]]
* [[16/15]] – its [[fourth complement]]
* [[Gallery of Just Intervals]]
* [[5/2]] – the interval up one [[octave]] which sounds even more [[consonant]]
* [[Wikipedia:Major third|Major third - Wikipedia]]
* [[Ed5/4]]
* [[:File:5-4.mp3]] – another sound sample
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]
 
== Notes ==
<references/>


[[Category:5-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Interval]]
[[Category:Third]]
[[Category:Third]]
[[Category:Major third]]
[[Category:Major third]]
[[Category:Superparticular]]
[[Category:Overtone]]
[[Category:Over-2]]
<!-- interwiki -->
[[de:Naturterz]]

Latest revision as of 13:37, 16 April 2025

Interval information
Ratio 5/4
Factorization 2-2 × 5
Monzo [-2 0 1⟩
Size in cents 386.3137¢
Names just major third,
classic(al) major third,
ptolemaic major third
Color name y3, yo 3rd
FJS name [math]\displaystyle{ \text{M3}^{5} }[/math]
Special properties superparticular,
reduced,
reduced harmonic
Tenney height (log2 nd) 4.32193
Weil height (log2 max(n, d)) 4.64386
Wilson height (sopfr(nd)) 9

[sound info]
Open this interval in xen-calc
English Wikipedia has an article on:

In 5-limit just intonation, 5/4 is the frequency ratio between the 5th and 4th harmonics. It has been called the just major third, classic(al) major third, or ptolemaic major third[1] to distinguish it from other intervals in that neighborhood. Measuring about 386.3 ¢, 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 ¢, and from the Pythagorean diminished fourth of 8192/6561 by the schisma, which measures about 1.95 ¢. 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 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).

Approximations by edos

Following edos (up to 200, and also 643) contain good approximations[2] of the interval 5/4. Errors are given by magnitude, the arrows in the table show if the edo representation is sharp (↑) or flat (↓).

Edo deg\edo Absolute
error (¢) 		Relative
error (r¢) 		↕ Equally acceptable multiples [3]
25 8\25 2.3137 4.8202 ↓
28 9\28 0.5994 1.3987 ↓ 18\56, 27\84, 36\112, 45\140
31 10\31 0.7831 2.0229 ↑ 20\62, 30\93
34 11\34 1.9216 5.4445 ↑
53 17\53 1.4081 6.2189 ↓
59 19\59 0.1270 0.6242 ↑ 38\118, 57\177
87 28\87 0.1068 0.7744 ↓ 56\174
90 29\90 0.3530 2.6471 ↑ 58\180
115 37\115 0.2268 2.1731 ↓
121 39\121 0.4631 4.6701 ↑
143 46\143 0.2997 3.5718 ↓
146 47\146 0.0123 0.1502 ↓
149 48\149 0.2635 3.2714 ↑
152 49\152 0.5284 6.6930 ↑
171 55\171 0.3488 4.9704 ↓
199 64\199 0.3841 6.3691 ↓
643 207\643 0.0004 0.0235 ↑

See also

Notes

  1. ↑ For reference, see 5-limit.
  2. ↑ error magnitude below 7, both, absolute (in ¢) and relative (in r¢)
  3. ↑ Super-edos up to 200 within the same error tolerance
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