Pure-tuned harmonium developed by the German physicist and music theorist Arthur Joachim von Oettingen (b Dorpat, Livonia, 28 March 1836; d Bensheim, Germany, 5 Sept 1920) and built by Schiedmayer in Stuttgart. An example from 1914 is in the Musikinstrumenten Museum, Staatliches Institut für Musikforschung, Berlin. Designed to sound pure 3rds, 4ths, and 5ths, it is based on an octave division into 53 (in some versions, 72) tones and has a complex multilayered but symmetrical keyboard similar to that of Bosanquet’s enharmonic harmonium. Oettingen studied astronomy and physics at the University of Dorpat and continued his education in Paris and Berlin. He was appointed a professor in Dorpat in 1863 and moved in 1893 to Leipzig, where he worked until 1919. He advocated the theory of ‘harmonic dualism’, later elaborated by Hugo Riemann, and introduced the interval measurement called the ‘millioctave’, based on division of the octave into 1000 tones. See ...
Tunings of the scale in which some or all of the concords are made slightly impure in order that few or none will be left distastefully so. Equal temperament, in which the octave is divided into 12 uniform semitones, is the standard Western temperament today except among specialists in early music. This article traces the history of temperaments in performing practice and in relation to the main lines of development in the history of harmony; for additional technical and historical details see Tuning, Pythagorean intonation, Just intonation, Microtone, Mean-tone, Well-tempered clavier, Equal temperament and Interval, especially Table 1.
Since the 15th century, tempered tuning has characterized keyboard music and in Western culture the art music of fretted instruments such as the lute. Its prevalence is due mainly to the fact that the concords of triadic music – octaves, 5ths and 3rds – are in many cases incommensurate in their pure forms. Three pure major 3rds (e.g. A♭–C–E–G♯) fall short of a pure octave by approximately one fifth of a whole tone (lesser diesis); four pure minor 3rds (G♯–B–D–F–A♭) exceed an octave by half as much again (greater diesis); the circle of twelve 5ths, if the 5ths are pure, does not quite cumulate in a perfect unison; and, most important of all in the context of Renaissance and Baroque music, the whole tone produced by subtracting a pure minor 3rd from a pure 4th (C–F–D) is about 11% smaller than the whole tone produced by subtracting a pure 4th from a pure 5th (C–G–D). These discrepancies are summarized in ...