TUNING AND TEMPERAMENT

in music, the adjustment of one sound source, such as a voice or string, to produce a desired pitch in relation to a given pitch, and the modification of that tuning to lessen dissonance. The determination of pitch, the quality of sound that is described as high or low, is based upon the frequency of sound waves. Two concepts fundamental to the theory of tuning are those of frequency ratio and of consonance and dissonance. A given musical pitch is determined by the frequency of vibration of the sound wave that produces it, as a = 440 cycles per second. An interval, or distance between two pitches, can thus be mathematically described as the ratio of the frequency of the first pitch to the frequency of the second. Various frequency ratios can be reduced to the same basic relationship; for example, 440:220 and 30:15 and 750:375 can all be reduced to the ratio 2:1. When two tones are sounded together the subjective reaction may be anything from one of perfect consonance to one of extreme dissonance. Dissonance is produced by beats (interference between pulsations of sound waves), and it is found that maximum dissonance occurs when the rate of beats between the two tones is about 33 per second. Consonance results from the absence of beats, which occurs only when the ratio between the frequencies of the two tones is numerically simple. When the two tones are tuned to the same pitch, they are said to be in unison (ratio 1:1) and their consonance is absolute. Next in order of consonance comes the octave (2:1), the interval between c and c (encompassing eight notes of the piano keyboard); another highly consonant interval is the fifth (3:2, as from c to g). When a unison, octave, or fifth is slightly mistuned, the resulting combination is markedly dissonant and is judged out of tune. The slight mistunings that occur in systems of tempered tuning are necessary for reasons that will be discussed later in this article. in music, the adjustment of one sound source, such as a voice or string, to produce a desired pitch in relation to a given pitch, and the modification of that tuning to lessen dissonance. The determination of pitch, the quality of sound that is described as high or low, is based upon the frequency of vibration of sound waves. The sounding of a sustained pitch produces other vibrations called overtones, or partials, whose frequencies are the integral multiples of the frequency of the principal note. Fundamental to the concept of tuning is the mathematical relationship between intervals (the distance between two pitches). The earliest development of the system of simple mathematical ratios of intervals was that of the school of Pythagoras, in the 6th century BC. The starting point for all tuning systems in Western music is the interval of the octave. The diatonic scale (the basis of all Western music, as represented by the white keys on a keyboard) consists of seven intervals: five large (whole tones) and two small (semi-tones). Their frequency ratios are irrational numbers; these result in dissonance, subjectively described as harsh and unpleasant to the ear. The intervals whose frequency ratios are simple, i.e., the unison, octave, perfect fifth, perfect fourth, and major third, result in consonance, described as pleasing and harmonious. Dissonance results from the acoustical phenomenon of beats, the interference between sound waves of a close frequency, and consonance is the absence of beats. Temperament, or the modification of tuning in order to lessen dissonance, was first mentioned in Franchinus Gafurius' treatise Practica Musica (Milan, 1496). He advocated diminishing the fifths by a slight amount, spreading the D to A dissonance over several octaves. In mean-tone temperament, customary in the 16th century, the interval of two octaves and one third is tuned perfectly to the overtone series, and the difference is spread through the four fifths, which flattens them and sharpens the sixths by one fourth of a comma. Equal whole tones are created, thereby giving its name, the mean, or average, whole tone being exactly one-half of the pure third. This worked well within a limited number of keys, each of these being similar enough to the others to be harmonious but different enough to maintain highly individual characteristics. However, the use of enharmonic notes, often differing by almost a quarter of a tone, created problems. The move to equal temperament, which began in the 17th century, was brought about by the expansion of tonalities, chromaticism, and enharmonic modulation (changing key by using one note to serve as a pivot to another key). Here the octave is divided into 12 equal semitones of 100 cents each, and the only remaining pure interval is the octave. All of the others are tempered to some extent. The fifth is two cents flat, for example, and the major third is 14 cents sharp. Although equal temperament sacrifices purely tuned intervals, numerous gains result from its use. The complex harmonies and tonalities of 19th- and 20th-century music would be impossible without the concept of enharmonics, which is completely dependent upon equal temperament. The twelve tone music of Arnold Schoenberg, Alban Berg, and Anton Webern would not have developed without the complete equality of semitones, and though new techniques are gradually being used, this system still holds fast today. Additional reading Works on tuning and temperament include Hermann von Helmholtz, Die Lehre von den Tonempfindungen als physiologische Grundlage fr die Theorie der Musik (1862; Eng. trans. by Alexander J. Ellis, On the Sensations of Tone, 1875), still the classic treatment of the subject, with useful additional observations by the translator; J. Murray Barbour, Tuning and Temperament: A Historical Survey (1951), a comprehensive study of the theoretical aspects of the subject, with a good bibliography; Llewelyn S. Lloyd and Hugh Boyle, Intervals, Scales and Temperaments (1963), designed for the musician; Paul C. Greene, Violin Intonation, Journal of the Acoustical Society of America, 9:4344 (1937), disposes of the theory that violinists naturally play in just intonation; Roger E. Kirk, Tuning Preferences for Piano Unison Groups, ibid., 31:164448 (1959), shows that musicians prefer groups of unison strings on the piano to be mistuned; D.W. Martin and W.D. Ward, Subjective Evaluation of Musical Scale Temperament in Pianos, ibid., 33:582585 (1961), shows that musicians do not prefer just intonation to equal temperament; Fritz A. Kuttner and J. Murray Barbour, The Theory of Classical Greek Music; Meantone Temperament in Theory and Practice; and The Theory and Practice of Just Intonation, Musurgia Records, Theory Series A, No. 13, an opportunity to hear the three major tuning systems that preceded equal temperament, both in scales and chords and in actual music.