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]\text{m3}_{5}[/math]
Special properties	superparticular, reduced
Tenney height (log2 nd)	4.90689
Weil height (log2 max(n, d))	5.16993
Wilson height (sopfr(nd))	10
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~4.1012 bits
[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 by edos

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

The following edos (up to 200) contain good approximations of the interval 6/5. 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
15	4\15	4.3587	5.4484	↑
19	5\19	0.1482	0.2346	↑	10\38, 15\57, 20\76, 25\95, 30\114, 35\133, 40\152, 45\171, 50\190
23	6\23	2.5978	4.9791	↓
34	9\34	2.0058	5.683	↑
42	11\42	1.3556	4.7445	↓
53	14\53	1.3398	5.9176	↑
61	16\61	0.8872	4.5099	↓
72	19\72	1.0254	6.1523	↑
80	21\80	0.6413	4.2752	↓
91	24\91	0.8422	6.3869	↑
99	26\99	0.4898	4.0406	↓
110	29\110	0.7223	6.6215	↑
118	31\118	0.387	3.806	↓
129	34\129	0.6378	6.8562	↑
137	36\137	0.3128	3.5714	↓
156	41\156	0.2567	3.3367	↓
175	46\175	0.2127	3.1021	↓
194	51\194	0.1774	2.8675	↓

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