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 2^{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* (*http://www.fh-augsburg.de/~harsch/gui_epi.html*) by Guido of Arezzo. (Text in Latin.) |