- Werckmeister temperament
Werckmeister temperament refers to any of the tuning systems described by
Andreas Werckmeisterin his writings [Andreas Werckmeister: Orgel-Probe (Frankfurt & Leipzig 1681), excerpts in Mark Lindley, "Stimmung und Temperatur", in "Hören, messen und rechnen in der frühen Neuzeit" pp. 109-331, Frieder Zaminer (ed.), vol. 6 of "Geschichte der Musiktheorie", Wissenschaftliche Buchgesellschaft (Darmstadt 1987).] [A. Werckmeister: Musicae mathematicae hodegus curiosus oder Richtiger Musicalischer Weg-Weiser (Quedlinburg 1686, Frankfurt & Leipzig 1687) ISBN 3-487-04080-8] [A. Werckmeister: Musicalische Temperatur (Quedlinburg 1691), reprint edited by Rudolf Rasch ISBN 90-70907-02-X] . The tuning systems are confusingly numbered in two different ways: the first refers to the order in which they were presented as "good temperaments" in Werckmeister's 1691 treatise, the second to their labelling on his monochord. The monochord labels start from III since just intonationis labelled I and quarter-comma meantoneis labelled II.
The tunings I (III), II (IV) and III (V) were presented graphically by a cycle of fifths and a list of
major thirds, giving the temperament of each in fractions of a comma. Werckmeister used the organbuilder's notation of ^ for a downwards tempered or narrowed interval and v for an upward tempered or widened one. (This appears counterintuitive - it is based on the use of a conical tuning tool which would reshape the ends of the pipes.) A pure fifths is simply a dash. Werckmeister was not explicit about whether the syntonic commaor Pythagorean commawas meant: the difference between them, the so-called schisma, is almost inaudible and he stated that it could be divided up among the fifths.
The last "Septenarius" tuning was not conceived in terms of fractions of a comma, despite some modern authors' attempts to approximate it by some such method. Instead, Werckmeister gave the string lengths on the monochord directly, and from that calculated how each fifth ought to be tempered.
Werckmeister I (III): "correct temperament" based on 1/4 comma divisions
This tuning uses mostly pure (perfect) fifths, as in
Pythagorean tuning, but each of the fifths C-G, G-D, D-A and B-F# is made smaller, i.e. tempered by 1/4 comma. Werckmeister designated this tuning as particularly suited for playing chromaticmusic ("ficte"), which may have led to its popularity as a tuning for J.S. Bach's music in recent years.
Werckmeister IV (VI): the Septenarius tunings
This tuning is based on a division of the
monochordlength into parts. The various notes are then defined by which 196-division one should place the bridge on in order to produce their pitches. The resulting scale has rationalfrequency relationships, so it is mathematically distinct from the irrationaltempered values above; however in practice, both involve pure and impure sounding fifths. Werckmeister also gave a version where the total length is divided into 147 parts, which is simply a transpositionof the intervals of the 196-tuning. He described the Septenarius as "an additional temperament which has nothing at all to do with the divisions of the comma, nevertheless in practice so correct that one can be really satisfied with it".
One apparent problem with these tunings is the value given to D (or A in the transposed version): Werckmeister writes it as 176. However this produces a musically bad effect because the fifth G-D would then be very flat (more than half a comma); the third Bb-D would be pure, but D-F# would be more than a comma too sharp - all of which contradict the rest of Werckmeister's writings on temperament. In the illustration of the monochord division, the number "176" is written one place too far to the right, where 175 should be. Therefore it is conceivable that the number 176 is a mistake for 175, which gives a musically much more consistent result. Both values are given in the table below.
In the tuning with D=175, the fifths C-G, G-D, D-A, B-F#, F#-C#, and Bb-F are tempered narrow, while the fifth G#-D# is tempered wider than pure; the other fifths are pure.
* [http://www.groenewald-berlin.de/Gliederung.html http://www.groenewald-berlin.de]
* [http://240edo.googlepages.com/equaldivisionsoflength(edl) 196-EDL & 1568-EDL and Septenarius tunings]
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