Well temperament

Well temperament (also good temperament, circular or circulating temperament) is a type of tempered tuning described in 20th-century music theory. The term is modeled on the German word wohltemperiert that appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier. The phrase wohl temperiert first appeared in Musikalische Temperatur (1691) by Bach's predecessor Andreas Werckmeister (1645–1706), an organ tuner and music theorist.

"Well tempered" means that the twelve notes per octave of the standard keyboard are tuned in such a way that it is possible to play music in all major or minor keys and it will not sound perceptibly out of tune.

It was defined by Werckmeister, in "Orgelprobe, 1681":

Wohltemperierung heißt mathematisch-akustische und praktisch-musikalischen Einrichtung von Tonmaterial innerhalb der zwölfstufigen Oktavskala zum einwandfreien Gebrauch in allen Tonarten auf der Grundlage des natürlich-harmonischen Systems mit Bestreben möglichster Reinerhaltung der diatonische Intervalle. Sie tritt auf als proportionsgebundene, sparsam temperierende Lockerung und Dehnung des mitteltönigen Systems, als ungleichschwebende Semitonik und als gleichschwebende Temperatur.

Well temperament means a mathematical-acoustic and musical-practical organisation of the tone system within the twelve steps of an octave, with the goal of impeccable performance in all tonalities, based on the natural-harmonic tone system [i.e., extended just intonation], while striving to keep the diatonic intervals as pure as possible. This temperament acts, while tied to given pitch ratios, as a thriftily tempered smoothing and extension of the meantone, as unequally beating half tones and as equally beating [i.e., equal] temperament.

In most tuning systems used before 1700, one or more intervals on the twelve-note keyboard were so far from any pure interval that they were unusable in harmony and were called a "wolf". Until about 1650 the most common keyboard temperament was quarter-comma meantone, in which the fifths were narrowed to the extent that they were just usable, and would thereby produce justly tuned thirds. The syntonic comma was distributed between four intervals, with most of the comma accommodated in the sol♯ to mi♭ diminished sixth, which expands to nearly a minor sixth. It is this interval that is usually called the "wolf", because it is so far out of consonance. The term "mean tone", the basis for meantone temperament, refers to the mathematical averaging of thirds, in which the middle note (for example the D between C and E) is in the "mean" position between the notes making the third. Another example of this is equal temperament (which is actually eleventh-comma meantone if seen in the perspective as to how to divide the comma between the fifths).

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