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The twelfth root of two or (or equivalently /) is an algebraic irrational number, approximately equal to 1.0594631.It is most important in Western music theory, where it represents the frequency ratio (musical interval) of a semitone (Play ⓘ) in twelve-tone equal temperament.
12 tone equal temperament chromatic scale on C, one full octave ascending, notated only with sharps. Play ascending and descending ⓘ. An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequencies of any adjacent pair of notes is the same.
Twelve-tone equal temperament (12 TET) is obtained by making all semitones the same size, with each equal to one-twelfth of an octave; i.e. with ratios 12 √ 2 : 1. Relative to Pythagorean tuning , it narrows the perfect fifths by about 2 cents or 1 / 12 th of a Pythagorean comma to give a frequency ratio of 2 7 / 12 : 1 {\displaystyle ...
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are determined by choosing a sequence of fifths [2] which are "pure" or perfect, with ratio :. This is chosen because it is the next harmonic of a vibrating string, after the octave (which is the ratio 2 : 1 {\displaystyle 2:1} ), and hence is the ...
12-tone equal temperament chromatic scale on C, one full octave ascending, notated only with sharps. Play ascending and descending ⓘ. 12 equal temperament (12-ET) [a] is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 (≈ 1.05946).
In musical tuning theory, a Pythagorean interval is a musical interval with a frequency ratio equal to a power of two divided by a power of three, or vice versa. [1] For instance, the perfect fifth with ratio 3/2 (equivalent to 3 1 / 2 1) and the perfect fourth with ratio 4/3 (equivalent to 2 2 / 3 1) are Pythagorean intervals.
An octave—two notes that have a frequency ratio of 2:1—spans twelve semitones and therefore 1200 cents. The ratio of frequencies one cent apart is precisely equal to 2 1 ⁄ 1200 = 1200 √ 2, the 1200th root of 2, which is approximately 1.000 577 7895. Thus, raising a frequency by one cent corresponds to multiplying the original frequency ...
The value of 5 1 ⁄ 8 ·35 1 ⁄ 3 is very close to 4, which is why a 7-limit interval 6144:6125 (which is the difference between the 5-limit diesis 128:125 and the septimal diesis 49:48), equal to 5.362 cents, appears very close to the quarter-comma ( 81 / 80 ) 1 ⁄ 4 of 5.377 cents.