tuning systems, methods for assigning pitches to the twelve Western pitch names that constitute the octave. The term usually refers to this procedure in the tuning of keyboard instruments. The need for a tuning system hinges on the conflict of pitch relationships in the natural overtone series and the exigencies of musical compositional systems, specifically those utilizing the familiar diatonic scale. Chronologically, the conflict occurred in the early Renaissance when composers had an increasing desire to modulate from one key to another. Implicit in the concept of modulation is the condition of identity of intervals between corresponding scale degrees in different modes or keys. A keyboard instrument tuned to a function of any natural interval except the octave will not satisfy that condition. The Pythagorean system, derived from a scale supposedly invented by Pythagoras (c.550 BC), was generated by acoustically perfect fifths. It exhibited an audible difference between the interval of a semitone and the interval resulting from the subtraction of the semitone from the whole tone. The mean-tone system generated the scale with fifths just flat enough to eliminate this difference, producing a scale containing acoustically perfect thirds. Discrepancy between chromatic notes (semitones) rendered this system unsuitable for successive modulations. Equal temperament tuning, which replaced mean-tone tuning in the 18th cent. and is universally accepted for Western music today, partitions the octave into twelve equal semitones. All intervals except the octave are acoustically out of tune, but by a tolerable degree, making complex modulations and atonality possible.