5/4

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]\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
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~4.00659 bits
[sound info]
open this interval in xen-calc

English Wikipedia has an article on:

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, classic(al) major third, or ptolemaic 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 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 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
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

• 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