# Twelfth root of two

The twelfth root of two or ${\sqrt[{12}]{2}}$ (or equivalently $2^{1/12}$ ) 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 = ${\sqrt[{1200}]{2}}=2^{1/1200}$ ).

## The equal-tempered chromatic scale

A musical interval is a ratio of frequencies and the equal-tempered chromatic scale divides the octave (which has a ratio of 2:1) into twelve 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 169
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.

### Other Tuning Scales

Other tuning scales use slightly different interval ratios:

• 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 5×5 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.
• 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).