# Semitonium

In harmony, the semitonium is the ratio 17:16 — or 18:17 — between a pair of frequencies or, equivalently, the ratio 16:17 — or 17:18 — between a pair of wavelengths (or lengths of a monochord). It is the mean between unison and ditono.The arithmetic mean between unison and ditono is

^{−4}.

The harmonic mean between unison and ditono is

The ratio 18:17 is the inversion of the *eptadem maius* (major seventh) (17:9), viz.

*limma*), i.e.

- .

The semitonium is also called minor second, or semitone.

A tone is equal to a pair of semitones. That is, a tonus can be composed by joining together a pair of semitonia:

- ,

Of the two ratios given above for the semitonium, the ratio 18:17 is closer to the minor second of equal temperament. The reason is that, given that an octave should equal twelve semitones, then both

(17/16)^{12} and (18/17)^{12} should be close to 2, but (18/17)^{12} is closer:

It is possible to combine 18:17 and 17:16, so that there are ten 18:17 semitones and two 17:16 semitones:

^{1/12}is exactly 100 cents: the semitone of equal temperament.

**See also:** unison, diapason, diapente, diatessaron, ditonus, semiditonus, tonus.

## External link

*Epistola de Ignoto Cantu*by Guido of Arezzo. (Text in Latin.)