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. Sine waves of various frequencies; the lower waves have higher frequencies than those above. ... In algebra, a ratio is the relationship between two quantities. ... The chromatic scale is any musical scale that contains more than one consecutive half-step (in other words two adjacent pairs of scale degrees or members which are separated by a semitone). ... Equal temperament is a scheme of musical tuning in which the octave is divided into a series of equal steps (equal frequency ratios). ... Clavier is a municipality located in the Belgian province of Liège. ... A harpsichord is the general term for a family of European keyboard instruments, including the large instrument nowadays called a harpsichord, but also the smaller virginals, the muselar virginals and the spinet. ... This article is about the modern musical instrument. ...

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: Just intonation is any musical tuning in which the frequencies of notes are related by whole number ratios. ... In poetry, dissonance is the deliberate avoidance of patterns of repeated vowel sounds (see assonance). ... A musical instrument is a device that has been constructed or modified with the purpose of making music. ... An overtone is a sinusoidal component of a waveform, of greater frequency than its fundamental frequency. ... The violin is a stringed musical instrument that has four strings tuned a perfect fifth apart. ... A cropped image to show the relative size of a cello to a human (Uncropped Version) The cello (also violoncello or cello) is a stringed instrument and a member of the violin family. ... Bass or Basses can refer to many different things. ... The harp is a chordophone whose strings are positioned perpendicular to the soundboard. ... A keyboard instrument is a musical instrument played with a musical keyboard. ... By Region: Italian Renaissance Northern Renaissance -French Renaissance -German Renaissance -English Renaissance The Renaissance was an influential cultural movement which brought about a period of scientific revolution and artistic transformation, at the dawn of modern European history. ... This article is about resonance in physics. ... In music, a pentatonic scale is a scale with five notes per octave. ... The term the Orient literally refers simply to the rising of the sun, being derived from the Latin word oriens. ... In music a heptatonic scale is a scale (music) with seven (hepta) degrees. ... Occident has a number of meanings. ...

or 1.05946309436.

See also

Equal temperament, Piano key frequencies, Well-Tempered Clavier, Musical tuning Equal temperament is a scheme of musical tuning in which the octave is divided into a series of equal steps (equal frequency ratios). ... This is a virtual piano with 88 keys tuned to A440, showing the frequencies, in cycles per second (Hz), of each note. ... Title-page of Das wohtemperierte Klavier A flat major (As-dur) fugue from the second part of Das wohtemperierte Klavier (manuscript) The Well-Tempered Clavier (Das wohltemperierte Klavier in German -- Klavier means piano, but the English word clavier (which means keyboard) looks more like the German title) consists of two... This page is about musical systems of tuning, for the musical process of tuning see 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."