Twelfth root of two
The twelfth root of two or 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. 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. Fraction approximations in increasing order of accuracy are,, and., its numerical value has been computed to at least twenty billion decimal digits.
The equal-tempered chromatic scale
A musical interval is a ratio of frequencies and the equal-tempered chromatic scale divides the octave into twelve parts.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 | Just intonation ratio |
| A | Unison | 440.00 | 2 | 1 | |
| A/B | Minor second/Half step/Semitone | 466.16 | 2 | ≈ | |
| B | Major second/Full step/Whole tone | 493.88 | 2 | ≈ | |
| C | Minor third | 523.25 | 2 | ≈ | |
| C/D | Major third | 554.37 | 2 | cube root of two#In music theory| | ≈ |
| D | Perfect fourth | 587.33 | 2 | ≈ | |
| D/E | Augmented fourth/Diminished fifth/Tritone | 622.25 | 2 | square root of two| | ≈ |
| E | Perfect fifth | 659.26 | 2 | ≈ | |
| F | Minor sixth | 698.46 | 2 | ≈ | |
| F/G | Major sixth | 739.99 | 2 | ≈ | |
| G | Minor seventh | 783.99 | 2 | ≈ | |
| G/A | Major seventh | 830.61 | 2 | ≈ | |
| A | Octave | 880.00 | 2 | 2 |
The final A is exactly twice the frequency of the lower A, 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. 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 5x5 parts. The delta scale is based on ≈, the gamma scale is based on ≈, the beta scale is based on ≈, and 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.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.