# minor-3rd / minor third / min3 / m3

[Joe Monzo]

The smaller of the two diatonic intervals which encompass 3 degrees of the of the diatonic scale. The minor-3rd is composed of 1 tone (i.e., whole-tone) and 1 diatonic semitone, and is one chromatic semitone smaller than the major-3rd. Here is one example:

```       minor-3rd
|         |
C     D     E   F     G     A     B   C
t    s
M2   m2

minor-3rd   =   t  + s    =   (t-s) + 2s
=   M2 + m2   =   +1    + 2(m2)
```

Thus, the minor-3rd contains 1 chromatic semitone and 2 diatonic semitones (or equivalently, 1 augmented-prime and 2 minor-2nds). In 12-edo, the minor-3rd encompasses 3 equal semitones. It is the characteristic interval of the minor chord, minor mode, and minor scale, thus giving its name to all of them.

Narrowing the minor-3rd by one more chromatic semitone results in the diminished-3rd.

. . . . . . . . .
[John Chalmers, Divisions of the Tetrachord]

An interval in the range of 300 cents (¢). Common minor thirds in Just Intonation are 6/5 (315¢), 19/16 (298¢) and the Pythagorean 32/27 (294¢).

This interval is also called the "trihemitone" and "augmented second" in certain contexts.

. . . . . . . . .
[Joe Monzo]

More precise values for the ratios in the Chalmers definition above are:

```ratio           ~cents
6/5    315.641287  ~= 315 5/8
19/16   297.5130161 ~= 297 1/2
32/27   294.1349974 ~= 294 1/7
```

The 7-limit ratio 7/6 is also known as the "subminor 3rd". It is ~266.8709056 (~= 266 7/8) cents. Another type of "subminor 3rd" is the 5-limit ratio 75/64, which is ~274.5824286 (~= 274 7/12) cents.

Successively closer small-integer rational approximations to the 12-edo minor-3rd are:

```ratio    ~cents

13/11  289.2097194
19/16  297.5130161
25/21  301.8465204
44/37  299.9739036, less than 1/38 cent narrower than 2(3/12)
```
. . . . . . . . .

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