# Twelfth root of two

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Octaves (12 semitones) increase exponentially when measured on a linear frequency scale (Hz).
Octaves are equally spaced when measured on a logarithmic scale (cents).

The twelfth root of two or 122 is an algebraic irrational number. It is most important in Western music theory, where it represents the frequency ratio of a semitone ( ) in twelve-tone equal temperament. Historically this number was proposed for the first time in relationship to musical tuning in 1580 (drafted, rewritten 1610) by Simon Stevin.[1] It allows measurement and discussion of equal intervals (frequency ratios) as equally spaced. A semitone is 100 cents (1 cent = 12002).

## 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.[2]

## 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)
Approx.
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.

## Pitch adjustment

Since the frequency ratio of a semitone is close to 106%, 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).

DJ turntables can have an adjustment up to ±20%, but this is more often used for beat synchronization between songs than for pitch adjustment, which is mostly useful only in transitions between beatless and ambient parts. For beatmatching music of high melodic content the DJ would primarily try to look for songs that sound harmonic together when set to equal tempo.

## History

Calculated in 1636 by the French mathematician Marin Mersenne, and as the techniques for calculating logarithms developed, the original approach for calculation would eventually become trivial.

## References

1. ^ Christensen, Thomas, The Cambridge history of Western music theory (2002) - page 205
2. ^ Komsta, Lukasz. "Computations page".[unreliable source?]