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) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 5/4.
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;).
{{Interval edo approximation|interval = 5/4| max_edo=200}}
 
{| 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; ||
|}
 
<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]] – sound sample that illustrates 5/4 as the interval between sung overtones
* [[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 13:15, 3 November 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 norm (log2 nd) 4.32193
Weil norm (log2 max(n, d)) 4.64386
Wilson norm (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) contain good approximations[2] of the interval 5/4.

Edo approximations for 5/4 (386.31 ¢)
≤ 200edo, relative error ≤ 10%
Edo Step size Cents (¢) Absolute error (¢) Relative error (%)
3 1\3 400.00 +13.69 +3.42
6 2\6 400.00 +13.69 +6.84
22 7\22 381.82 -4.50 -8.24
25 8\25 384.00 -2.31 -4.82
28 9\28 385.71 -0.60 -1.40
31 10\31 387.10 +0.78 +2.02
34 11\34 388.24 +1.92 +5.44
37 12\37 389.19 +2.88 +8.87
50 16\50 384.00 -2.31 -9.64
53 17\53 384.91 -1.41 -6.22
56 18\56 385.71 -0.60 -2.80
59 19\59 386.44 +0.13 +0.62
62 20\62 387.10 +0.78 +4.05
65 21\65 387.69 +1.38 +7.47
81 26\81 385.19 -1.13 -7.62
84 27\84 385.71 -0.60 -4.20
87 28\87 386.21 -0.11 -0.77
90 29\90 386.67 +0.35 +2.65
93 30\93 387.10 +0.78 +6.07
96 31\96 387.50 +1.19 +9.49
109 35\109 385.32 -0.99 -9.02
112 36\112 385.71 -0.60 -5.59
115 37\115 386.09 -0.23 -2.17
118 38\118 386.44 +0.13 +1.25
121 39\121 386.78 +0.46 +4.67
124 40\124 387.10 +0.78 +8.09
140 45\140 385.71 -0.60 -6.99
143 46\143 386.01 -0.30 -3.57
146 47\146 386.30 -0.01 -0.15
149 48\149 386.58 +0.26 +3.27
152 49\152 386.84 +0.53 +6.69
168 54\168 385.71 -0.60 -8.39
171 55\171 385.96 -0.35 -4.97
174 56\174 386.21 -0.11 -1.55
177 57\177 386.44 +0.13 +1.87
180 58\180 386.67 +0.35 +5.29
183 59\183 386.89 +0.57 +8.72
196 63\196 385.71 -0.60 -9.79
199 64\199 385.93 -0.38 -6.37

See also

Notes

  1. ↑ For reference, see 5-limit.
  2. ↑ error magnitude below 7, both, absolute (in ¢) and relative (in r¢)
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