Twelfth root of two

The twelfth root of two or is an algebraic irrational number, approximately equal to 1.0594631. It is important in Western music theory, where it represents the frequency ratio 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 as consisting of different numbers of a single interval, the equal tempered semitone. A semitone itself is divided into 100 cents.

Numerical value

The twelfth root of two to 20 significant figures is.
The continued fraction begins, so a simple rational approximation is.

The equal-tempered chromatic scale

A musical interval is a ratio of frequencies and the equal-tempered chromatic scale divides the octave into twelve equal parts. Each note has a frequency that is 2 times that of the one below it.
Applying this value successively to the tones of a chromatic scale, starting from A above middle C with a frequency of 440 Hz, produces the following sequence of pitches:

Note	Standard interval name relating to A 440	Frequency	Multiplier	Coefficient	ratio
A	Unison	440.00	2		1	0
A/B	Minor second/Half step/Semitone	466.16	2		≈	+11.73
B	Major second/Full step/Whole tone	493.88	2		≈	−3.91
C	Minor third	523.25	2		≈	+15.64
C/D	Major third	554.37	2	[cube root of two#In music theory\|]	≈	−13.69
D	Perfect fourth	587.33	2		≈	−1.96
D/E	Augmented fourth/Diminished fifth/Tritone	622.25	2	[square root of two\|]	≈	+17.49
E	Perfect fifth	659.26	2		≈	+1.96
F	Minor sixth	698.46	2		≈	+13.69
F/G	Major sixth	739.99	2		≈	−15.64
G	Minor seventh	783.99	2		≈	+3.91
G/A	Major seventh	830.61	2		≈	−11.73
A	Octave	880.00	2		2	0

The final A is exactly twice the frequency of the lower A, 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.
The equal tempered Bohlen–Pierce scale uses the interval of the thirteenth root of three.
Stockhausen's Studie II makes use of the twenty-fifth root of five, a compound major third divided into 5×5 parts.
The delta scale is based on ≈.
The gamma scale is based on ≈.
The beta scale is based on ≈.
The alpha scale is based on ≈.

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. Modern recording studios utilize digital pitch shifting to achieve similar results, ranging from cents up to several half-steps. Reel-to-reel adjustments also affect the tempo of the recorded sound, while digital shifting does not.

History

Historically this number was proposed for the first time in relationship to musical tuning in 1580 by Simon Stevin. In 1581 Italian musician Vincenzo Galilei may be the first European to suggest twelve-tone equal temperament. The twelfth root of two was first calculated in 1584 by the Chinese mathematician and musician Zhu Zaiyu using an abacus to reach twenty four decimal places accurately, calculated circa 1605 by Flemish mathematician Simon Stevin, in 1636 by the French mathematician Marin Mersenne and in 1691 by German musician Andreas Werckmeister.