# Twelfth root of two

The twelfth root of two or ${\sqrt[]}$ (or equivalently $2^$ ) is an algebraic irrational number. It is most important in Western music theory, where it represents the frequency ratio (musical interval) of a semitone () in twelve-tone equal temperament. This number was proposed for the first time in relationship to musical tuning in the sixteenth and seventeenth centuries. It allows measurement and comparison of different intervals (frequency ratios) as consisting of different numbers of a single interval, the equal tempered semitone (for example, a minor third is 3 semitones, a major third is 4 semitones, and perfect fifth is 7 semitones).[a] A semitone itself is divided into 100 cents (1 cent = ${\sqrt[]}=2^$ ).

## Numerical value

The twelfth root of two to 20 significant figures is 1.0594630943592952646. Fraction approximations in order of accuracy are 18/17, 196/185, and 18904/17843.

As of December 2013, its numerical value has been computed to at least twenty billion decimal digits.

## The equal-tempered chromatic scale

Since a musical interval is a ratio of frequencies, the equal-tempered chromatic scale divides the octave (which has a ratio of 2:1) into twelve equal parts.

Applying this value successively to the tones of a chromatic scale, starting from A above middle C (known as A4) with a frequency of 440 Hz, produces the following sequence of pitches:

Note Standard interval name(s)
relating to A 440
Frequency
(Hz)
Multiplier Coefficient
(to six places)
Just intonation
ratio
A Unison 440.00 2012 1.000000 1
A/B Minor second/Half step/Semitone 466.16 2112 1.059463 ≈ ​1615
B Major second/Full step/Whole tone 493.88 2212 1.122462 ≈ ​98
C Minor third 523.25 2312 1.189207 ≈ ​65
C/D Major third 554.37 2412 1.259921 ≈ ​54
D Perfect fourth 587.33 2512 1.334839 ≈ ​43
D/E Augmented fourth/Diminished fifth/Tritone 622.25 2612 1.414213 ≈ ​75
E Perfect fifth 659.26 2712 1.498307 ≈ ​32
F Minor sixth 698.46 2812 1.587401 ≈ ​85
F/G Major sixth 739.99 2912 1.681792 ≈ ​53
G Minor seventh 783.99 21012 1.781797 ≈ ​95
G/A Major seventh 830.61 21112 1.887748 ≈ ​158
A Octave 880.00 21212 2.000000 2

The final A (A5: 880 Hz) is exactly twice the frequency of the lower A (A4: 440 Hz), that is, one octave higher.

The just or Pythagorean perfect fifth is 3/2, and the difference between the equal tempered perfect fifth and the just is a grad, the twelfth root of the Pythagorean comma (12531441/524288). The equal tempered Bohlen–Pierce scale uses the interval of the thirteenth root of three (133). Stockhausen's Studie II (1954) makes use of the twenty-fifth root of five (255), a compound major third divided into 5x5 parts. The delta scale is based on ≈503/2, the gamma scale is based on ≈203/2, the beta scale is based on ≈113/2, and the alpha scale is based on ≈93/2.

Since the frequency ratio of a semitone is close to 106% ($1.05946\times 100=105.946$ ), increasing or decreasing the playback speed of a recording by 6% will shift the pitch up or down by about one semitone, or "half-step". Upscale reel-to-reel magnetic tape recorders typically have pitch adjustments of up to ±6%, generally used to match the playback or recording pitch to other music sources having slightly different tunings (or possibly recorded on equipment that was not running at quite the right speed). Modern recording studios utilize digital pitch shifting to achieve similar results, ranging from cents up to several half-steps (note that reel-to-reel adjustments also affect the tempo of the recorded sound, while digital shifting does not).