# 6/5

 Ratio 6/5 Factorization 2 × 3 × 5-1 Monzo [1 1 -1⟩ Size in cents 315.64129¢ Names just minor third,classic(al) minor third,ptolemaic minor third Color name g3, gu 3rd FJS name $\text{m3}_{5}$ 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, $\sqrt{nd}$) ~4.1012 bits https://en.xen.wiki/w/File:Jid_6_5_pluck_adu_dr220.mp3[sound info] open this interval in xen-calc
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In 5-limit just intonation, 6/5 is the just minor third, classic(al) minor third, or ptolemaic minor third[1], 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[2] 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 ()
Equally acceptable multiples [3]
15 4\15 4.3587 5.4484
19 5\19 0.1482 0.2346
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