5/4: Difference between revisions

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{{Wikipedia|Major third}}
{{Wikipedia|Major third}}


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 ¢ 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 ¢ 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". It is a [[schisma]] sharp of 8192/6561, another 3-limit interval in the major third range often called a diminished fourth.


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).
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).

Revision as of 08:44, 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 ¢. 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". It is a schisma sharp of 8192/6561, another 3-limit interval in the major third range often called a diminished fourth.

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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