Editing Well temperament (section)

==Origins==
[[File:Orgelprobe 1681 3 gemerkt 2.jpg|thumb|upright=1.5|300px|Cover of ''Orgelprobe'' (1681)|alt=Cover of "Orgelprobe" 1681]]
As used in the 17th century, the term "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 and minor|major]] or [[Major and minor|minor]] [[key (music)|keys]] that were commonly in use, without sounding perceptibly out of tune.{{sfn|Duffin|2007||p=37}}

One of the first attestations of the concept of "well tempered" is found in a treatise in German by the music theorist [[Andreas Werckmeister]].<ref name="Ledbetter2002">{{cite book |last1=Ledbetter |first1=David |title=Bach's Well-tempered clavier : the 48 preludes and fugues |date=2002 |publisher=Yale University Press |location=New Haven |isbn=978-0-300-12898-7 |url=https://books.google.com/books?id=02rmCC1MgvoC}}</ref>{{rp|37}} In the subtitle of his ''Orgelprobe'', from 1681, he writes:<ref name="Norrback2002">{{cite journal |last1=Norrback |first1=Johan |title=A Passable and Good Temperament. A New Methodology for Studying Tuning and Temperament in Organ Music |journal=Studies from the Department of Musicology |date=27 September 2002 |volume=70 |hdl=2077/15641 |url=http://hdl.handle.net/2077/15641 |access-date=22 December 2020}}</ref>
{{blockquote|{{lang|de|Unterricht, Wie durch Anweiß und Hülffe des Monochordi ein Clavier '''wohl zu temperiren''' und zu stimmen sei, damit man nach heutiger Manier alle modos fictos in einer erträglichen und angenehmen harmoni vernehme.|italic=no}}}}

The words {{lang|de|wohl}} and {{lang|de|temperieren}} were subsequently combined into {{lang|de|Wohltemperiert}}. A modern definition of "well temperament", from [[Herbert Kelletat]], is given below:{{sfn|Kelletat|1981–82/94|p=9}}
{{clear}}
:{| style="padding: 0 1em"
|style="vertical-align: top;"|{{lang|de|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.|italic=no}}
|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 [[Just intonation|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 [[perfect fifth|fifths]] were narrowed so as to maximize the number of pure [[major third]]s. [[Syntonic comma]]s were distributed across most sequences of four narrowed fifths, with one remaining correction accommodated usually in the diminished sixth [[G♯ (musical note)|G{{music|sharp}}]] to [[E♭ (musical note)|E{{music|flat}}]],{{sfn|Lindley|1990}} which expands to almost a justly tuned [[minor sixth]]. It is this interval that is usually called the "[[wolf interval|wolf]]", because it is so far from [[Consonance and dissonance|consonance]].<ref name="Cyr1992">{{cite book |last1=Cyr |first1=Mary |title=Performing baroque music |date=1992 |publisher=Scolar Press |location=Aldershot, England |isbn=978-0-85967-960-2}}</ref>{{rp|65}}

The wolf was not a problem if music was played in a small number of keys (or to be more precise, transposed [[mode (music)|modes]]) with few [[accidental (music)|accidentals]], but it prevented players from [[transposition (music)|transposing]] and [[modulation (music)|modulating]] freely. Some instrument-makers sought to remedy the problem by introducing more than twelve notes per octave, producing [[enharmonic keyboard]]s which could provide, for example, a D{{music|sharp}} and an E{{music|flat}} with different pitches so that the thirds B–D{{music|sharp}} and E{{music|flat}}–G could both be euphonious. These solutions could include [[split sharp|split keys]] and multiple [[manual (music)|manuals]]; one such solution, the [[archicembalo]], was mentioned by [[Nicola Vicentino]] in 1555.<ref>{{cite journal |last1=Kaufmann |first1=Henry W. |title=More on the Tuning of the Archicembalo |journal=Journal of the American Musicological Society |date=1 April 1970 |volume=23 |issue=1 |pages=84–94 |doi=10.2307/830349|jstor=830349 }}</ref>

However, Werckmeister realised that split keys, or "subsemitonia" as he called them, were unnecessary, and even counterproductive in music with [[chromatic scale|chromatic]] progressions and extensive [[modulation (music)|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{{music|flat}} and D{{music|sharp}}, 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 without restriction.{{r|Ledbetter2002|p=37}} This is also the source of the terms "circular temperament" or "circulating temperament".<ref>{{cite book |title=Harpsichord and clavichord : an encyclopedia |date=2007 |publisher=Routledge |location=New York |isbn=978-1-135-94978-5 |page=68}}</ref><ref name="Campbell2004">{{cite book |last1=Campbell |first1=Murray |title=Musical instruments : history, technology, and performance of instruments of western music |date=2004 |publisher=Oxford University Press |location=Oxford |isbn=978-0-19-816504-0 |page=341}}</ref>

Although [[equal temperament]] is discussed by Werckmeister in his treatises,<ref name="Bartel2015">{{cite journal |last1=Bartel |first1=Dietrich |title=Andreas Werckmeister's final tuning: the path to equal temperament: 1 |journal=Early Music |date=25 June 2015 |volume=43 |issue=3 |pages=503–512 |doi=10.1093/em/cav047 |access-date=22 December 2020 |url=https://academic.oup.com/em/article-abstract/43/3/503/414311}}</ref> it is distinguished from non-equal well temperaments.{{r|Cyr1992|p=66}}