Twelfth root of two

The Twelfth root of two is a quantity representing the frequency ratio between any two consecutive notes of a modern chromatic scale in equal temperament. The quantity was popularized as a solution to the problem of a 'growling' sound made by early claviers, harpsichords and the forerunners of the modern piano in playing certain intervals which were "out of tune" in systems of unequal temperament.

Errors in temperament resulting from the use of harmonic tuning caused dissonance in the harmonic content produced by multi-stringed musical instruments when the harmonic overtones of lower strings were used to tune other strings. This method was ideal for four-stringed instruments such as violins, cellos and basses, but proved to be problematic for the harp and the family of keyboard instruments developed during the European Renaissance. What was needed was a way to render uniform resonance throughout the entire range of notes and a more complete understanding of the mathematical relationships between the members of scale and chord families. Combining the five tones of the pentatonic scale from the Orient (represented by the black keys) with the seven tones of the heptatonic scale from the Occident (represented by the white keys) resulted in the quintessential compromise creating an ideal approximation based on a nominal uniform ratio between each of the twelve tones of the chromatic scale:

or 1.05946309436.

See Also

Equal temperament, Piano key frequencies, Well-Tempered Clavier, Musical tuning

References

• From http://www.physlink.com/Education/AskExperts/ae165.cfm:

"Mathematically, ... each successive pitch is related to the previous pitch by a factor of the twelfth root of 2. That is, the ratio between the frequencies of any two successive pitches in either standard is 1.05946309436."