5/4: Difference between revisions

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another table of EDO approximations
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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).
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).
== 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. 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"
|-
! [[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; ||
|}
<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]]

Revision as of 12:42, 25 October 2020

Interval information
Ratio 5/4
Factorization 2-2 × 5
Monzo [-2 0 1⟩
Size in cents 386.3137¢
Name classic 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

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¢, 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 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).

Approximations by EDOs

Following EDOs (up to 200) contain good approximations[1] 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 [2]
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 ↓
  1. ↑ error magnitude below 7, both, absolute (in ¢) and relative (in r¢)
  2. ↑ Super EDOs up to 200 within the same error tolerance

See also

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