Musical Tuning Systems

Musical Tuning Systems, theoretical or practical systems for determining the correct tuning for the intervals of a scale. Because some leeway exists within which the ear recognizes two notes as a given interval, cultural concepts of “correct” pitch and intervals vary; for example, intervals of 200 cents, 204 cents, and 182 cents are all heard as whole steps (1,200 cents equals one octave).

In strongly concordant intervals, the sound-wave frequencies of the upper and lower notes form simple mathematical ratios, such as 2:1 (octave), 5:4 (major third), and 3:2 (fifth). This last ratio, the “pure” or “natural” fifth, is the basis of Pythagorean tuning, used in ancient Greece, ancient China, medieval Islamic countries, and medieval Europe. Tuning a series of fifths, beginning on F, produces the seven notes of the C-major scale, F C G D A E B, then the five notes F# C# G# D# A#, and finally E# and B# (theoretically identical with F and C, hence the term circle of fifths for this series). The Pythagorean B#, however, is slightly higher than the initial C, making the system unusable on fretted and keyboard instruments; moreover, the thirds, sharper than the natural third, are strongly dissonant. The system works best for unharmonized melodies, sung or played on a violin or other instrument of adjustable pitch.

In the tuning system known as just intonation, some intervals are derived from pure fifths and others from pure thirds. Mostly a theoretical system, it produces euphonious chords but has serious disadvantages, including an important out-of-tune fifth (D-A).

From early times, these ideal scales were tempered, or slightly adjusted, when fretted and keyboard instruments were to be used. In mean-tone temperament, popular during the Baroque era (roughly 1600-1750), several series of four minutely flattened fifths result in pure major thirds in the most commonly used parts of the C-major scale. One seldom-used interval (D#-G#), called the wolf, is always out of tune. Except for special effects, modulation (change of key) is limited to keys closely related to C major. As late 18th-century musical styles developed, musicians became more interested in equal temperament, a system that was adopted only gradually—by 1800 in Germany and by 1850 in England. In equal temperament the octave is divided into twelve precisely equidistant intervals. This means that all fifths are slightly flattened equally, so that B# is identical with C as the circle of fifths is completed. The major third is somewhat sharp, but within acceptable limits. The most important feature of equal temperament is that all keys, even the most distant from C, become interchangeable, so that any modulation is feasible. This is fundamental to the ever-increasing chromaticism and harmonic complexity of music composed from the mid-19th century onwards.