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Encyclopedia > 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

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.


External link

  • Epistola de Ignoto Cantu (http://www.fh-augsburg.de/~harsch/gui_epi.html) by Guido of Arezzo. (Text in Latin.)

  Results from FactBites:
 
JACSP2B TEXT (16746 words)
Alias consimiles habere potest semitonium maius notificationes, ut quod est illa particula in qua tonus diesim vincit, ditonus semiditonum, tritonus diatessaron, tetratonus diapente, et, secundum Guidonem, quae, iuncta minori semitonio, tonum implet; et multas tales quae verae sunt.
Tertio, quia minus semitonium in pluribus locis in compositione reperitur gammatis quam maius semitonium, exclusa falsa musica.
Cum enim voces ipsius in spatio sunt vel linea, ut apparet ipsarum distinctio, plura requisita sunt signa vel littera alio et alio modo figurata; distinguuntur etiam in quibusdam aliis quae tangentur.
  More results at FactBites »

 
 

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