Well temperament (also circular or circulating temperament) is a type of tempered tuning described in 20th-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier. The phrase wohl temperiert also occurs in the works of Bach's predecessor, the organ tuner and music theorist Andreas Werckmeister.
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"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 most major or minor keys and it will not sound perceptibly out of tune. 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).
The wolf was not a problem if music was played in a small number of keys (or to be more precise, transposed modes) with few accidentals, but it prevented players from transposing and modulating freely. Some instrument-makers sought to remedy the problem by introducing more than twelve notes per octave, producing enharmonic keyboards which could provide, for example, a D♯ and an E♭ with different pitches so that the thirds B–D♯ and E♭–G could both be euphonious.
However, Werckmeister realised that these "subsemitonia", as he called them, were unnecessary, and even counterproductive in music with chromatic progressions and extensive modulations. He described a series of tunings where enharmonic notes had the same pitch: in other words, the same note was used as both (say) E♭ and D♯, thereby "bringing the keyboard into the form of a circle". This refers to the fact that the notes or keys may be arranged in a circle of fifths and it is possible to modulate from one key to another unrestrictedly.
According to Sinologist Robert K. G. Temple, the well temperament was first invented by the Chinese prince of the Ming dynasty Chu Tsai-Yü in 1584 and came in contact with Western culture during exchange fairs organized by the Cantonese viceroy in that time. How exactly it travelled to Europe is not documented, but it is likely that Jesuits in China brought the knowledge over to Europe (Temple 2007, ). Other scholars, however, state that Prince Chu accurately calculated not a well temperament but equal temperament—though only in his second treatise, Lii Lu Ching I, written in 1595–96 and probably first published in 1606 did he achieve the full chromatic complement of 12 notes (Kutter 1975, 166–67). However, at just the same time, the Flemish mathematician Simon Stevin (1548–1620) wrote an essay containing the correct mathematical formulation of equal temperament for the first time in Western musical theory, though his manuscript remained unpublished until long after the author's death (Kutter 1975, 167–68; Stevin 1884).
The term "well temperament" usually means some sort of irregular temperament in which the tempered fifths are of different sizes but no key has very impure intervals. Historical irregular temperaments usually have the narrowest fifths between the diatonic notes ("naturals") producing purer thirds, and wider fifths among the chromatic notes ("sharps and flats"). Each key then has a slightly different intonation, hence different keys have distinct characters. Such "key-color" was an essential part of much 18th- and 19th-century music and was described in treatises of the period.
The first circular temperament was described by the organist Arnolt Schlick in the early 16th century, but "well temperaments" did not become widely used until the baroque period. They persisted through the classical period, and even survived into the late 19th century in some areas.
There are many well temperament schemes, some nearer meantone temperament, others nearer equal temperament. Although such tunings have no wolf fifth, keys with many sharps or flats still do not sound very well in tune (due to their thirds), and can only be used fleetingly. Some theorists have sought to define "well temperament" more narrowly to exclude fifths wider than pure, which rules out many such schemes.
Some well-known well temperaments go by the following names:
The contemporary composer Douglas Leedy has written several works for harpsichord or organ in which the use of a well temperament is required.
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