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
which is equal to 1.0001 in binary, or 1 + 2−4.
The harmonic mean between unison and ditono is
which is equal to 1.0000111100001111000011110000111100001111... in binary.
The ratio 18:17 is the inversion of the eptadem maius (major seventh) (17:9), viz.
In Pythagorean tuning, the semitonium is equal to the ratio 256:243 (which is specifically called limma), i.e.
The Pythagorean diatonic scale has five toni, each of ratio 9:8, and two semitonia, each of ratio 256:243. Multiplying all of these together yields
which is diapason exactly.
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:
but notice that the semitonia are slightly unequal.
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:
and 1.00727 < 1.03494, so that the ratio 18:17 better approximates the ideal semitone.
It is possible to combine 18:17 and 17:16, so that there are ten 18:17 semitones and two 17:16 semitones:
which is extremely close to perfect diapason: the result is equal to 1199.4567 cents, less than one cent from a perfect octave. Also,
where 21/12 is exactly 100 cents: the semitone of equal temperament.
See also: unison, diapason, diapente, diatessaron, ditonus, semiditonus, tonus.
- Epistola de Ignoto Cantu (http://www.fh-augsburg.de/~harsch/gui_epi.html) by Guido of Arezzo. (Text in Latin.)