Editing Semitone (section)

===Equal temperament===

[[12-tone equal temperament]] is a form of meantone tuning in which the diatonic and chromatic semitones are exactly the same, because its circle of fifths has no break. Each semitone is equal to one twelfth of an octave. This is a ratio of [[Twelfth root of two|2<sup>1/12</sup>]] (approximately 1.05946), or 100 cents, and is 11.7 cents narrower than the 16:15 ratio (its most common form in [[just intonation]], [[#Just intonation|discussed below]]).

All diatonic intervals can be expressed as an equivalent number of semitones. For instance a [[major sixth]] equals nine semitones.

There are many approximations, [[Rational number|rational]] or otherwise, to the equal-tempered semitone. To cite a few:
:*<math>18 / 17 \approx 99.0 \text{ cents,}</math><br />suggested by [[Vincenzo Galilei]] and used by [[luthier]]s of the [[Renaissance music|Renaissance]],

:*<math>\sqrt[4]{\frac{2}{3-\sqrt{2}}} \approx 100.4 \text{ cents,}</math><br />suggested by [[Marin Mersenne]] as a [[Constructible number|constructible]] and more accurate alternative,

:*<math>(139 / 138 )^8  \approx 99.9995 \text{ cents,}</math><br />used by [[Julián Carrillo]] as part of a sixteenth-tone system.

For more examples, see Pythagorean and Just systems of tuning below.