6/5

Interval information

Ratio	6/5
Factorization	2 × 3 × 5-1
Monzo	[1 1 -1⟩
Size in cents	315.6413¢
Names	just minor third, classic(al) minor third, ptolemaic minor third
Color name	g3, gu 3rd
FJS name	[math]\displaystyle{ \text{m3}_{5} }[/math]
Special properties	superparticular, reduced
Tenney norm (log2 nd)	4.90689
Weil norm (log2 max(n, d))	5.16993
Wilson norm (sopfr(nd))	10
[sound info]
Open this interval in xen-calc

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Minor third

In 5-limit just intonation, 6/5 is the just minor third, classic(al) minor third, or ptolemaic minor third, measuring about 315.6¢. It is sharp of the Pythagorean minor third of 32/27 (about 294.1¢) as well as the 300¢ minor third of 4edo, 12edo and all other 4n-edos. It arises in the harmonic series between the 5th and 6th harmonics and appears in the 5-limit otonal triad of 4:5:6. A 5-limit minor triad in just intonation can be written 10:12:15, with 6/5 falling between 10 and 12, 5/4 falling between 12 and 15, and 3/2 falling between 10 and 15.

In higher-limit JI, 6/5 is only one of many minor thirds. A popular one in the 7-limit is 7/6 (about 266.9¢), the septimal subminor third, which is 36/35 (about 48.8¢) flat of 6/5. Another in the 13-limit is 13/11 (about 289.2¢), which is 66/65 (about 26.4¢) flat of 6/5. Both of these are more complex intervals than 6/5 and have their own character to them.

Approximation

6/5 is very accurately approximated by 19edo (5\19), and hence the enneadecal temperament.

Edo approximations for 6/5 (315.64 ¢)
≤ 80edo, relative error ≤ 10%

Edo	Step size	Cents (¢)	Absolute error (¢)	Relative error (%)
4	1\4	300.00	-15.64	-5.21
15	4\15	320.00	+4.36	+5.45
19	5\19	315.79	+0.15	+0.23
23	6\23	313.04	-2.60	-4.98
34	9\34	317.65	+2.01	+5.68
38	10\38	315.79	+0.15	+0.47
42	11\42	314.29	-1.36	-4.74
46	12\46	313.04	-2.60	-9.96
53	14\53	316.98	+1.34	+5.92
57	15\57	315.79	+0.15	+0.70
61	16\61	314.75	-0.89	-4.51
65	17\65	313.85	-1.80	-9.72
72	19\72	316.67	+1.03	+6.15
76	20\76	315.79	+0.15	+0.94
80	21\80	315.00	-0.64	-4.28

See also

• 5/3 – its octave complement
• 5/4 – its fifth complement
• 10/9 – its fourth complement
• Gallery of just intervals
• List of superparticular intervals
• File:Ji-6-5-csound-foscil-220hz.mp3 – another sound example