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. This word also appears in the title of J.S. Bach's famous composition "Das wohltemperierte Klavier", The Well-Tempered Clavier.

Origins
As the term was used in the 17th century, "Well tempered" meant that the twelve notes per octave of the standard keyboard were tuned in such a way that it was possible to play music in all major or minor keys that were commonly in use, and it would not sound perceptibly out of tune. Well temperament is called "Wohltemperiert", in the German language. This wording was first used by Werckmeister in the subtitle of "Orgelprobe" 1681: "Unterricht, Wie durch Anweiß und Hülffe des Monochordi ein Clavier und zu stimmen sei, damit man nach heutiger Manier alle modos fictos in einer erträglichen und angehmen harmoni vernehme". The facsimile of the cover on the right is copied from page 18 of: “A Passable and Good Temperament; A New Methodology for Studying Tuning and Temperament in Organ Music”, from Johan Norback. Studies from the Department of Musicology, Göteborg University, no. 70, 2002, ISBN 91 85974-66-8, ISSN 1650-9285. The words "wohl" and "temperiren became combined as "Wohltemperiert". The "Orgelprobe" 1681, and other Werckmeister publications were the source of inspiration for Prof. H. Kelletat, for the elaboration of the "Wholtemperiert" definition given below, and published in "Zur Musikalischen Temperatur", page 9 (ISBN 3-87537-156-9):

"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 interval". 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 (syntonic comma) or tweflth-comma meantone (pythagorean comma) 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 attributes the invention of equal temperament to Zhu Zaiyu and provides textual quotations as evidence. Fritz A. Kuttner is critical of this theory and proposes that neither Zhu Zaiyu nor Simon Stevin  achieved equal temperament, and that neither of the two should be treated as inventors.

Forms
The term "well temperament" or "good 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, 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). 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:


 * Werckmeister temperament (invented by Andreas Werckmeister)
 * French Temperament Ordinaire
 * Neidhardt
 * Kirnberger
 * Kellner
 * Vallotti (invented by Francesco Antonio Vallotti)
 * Young

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.