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 most 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, so that impeccable performance in all tonalities is enabled, based on the extended just intonation (natural-harmonic tone system), 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 equal (equally beating) 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).

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.

Kenneth Robinson, attributes the invention of equal temperament to Zhu Zaiyu (Robinson 1980, vii) and provides textual quotations as evidence (Robinson and Needham 1962, 221). Fritz A. Kuttner is critical of his theory (Kuttner 1975, 163) and proposes that neither Zhu Zaiyu or Simon Stevin (1884) achieved equal temperament, and that neither of the two should be treated as inventors (Kuttner 1975, 200).


The term "well temperament" or "good temperament" (Barbour 1951, x, 221; Lindley 2001) 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, for example in Italy.

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). To say that they can only be used fleetingly is an exaggeration: they give a feeling of "home" in the keys with few accidentals and "remote" in keys to which one travelled for special effect and the exotic. This can contrast chords in which vibrations are concordant with others where the vibrations are not harmonically related. 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:

Some temperament schemes feature numbers of perfect, pure fifths and these give enhanced harmonic resonance to instruments and music on which they are played so that music moves into and out of focus between keys as vibrations lock together or not. Werckmeister features 8 perfect fifths, Kellner 7 and Vallotti 6.

The contemporary composer Douglas Leedy has written several works for harpsichord or organ in which the use of a well temperament is required.

See also


Further reading

External links

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