Werckmeister temperaments are the tuning systems described by Andreas Werckmeister in his writings. 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 intonation is labelled I and quarter-comma meantone is labelled II.
Contents
- Werckmeister I III correct temperament based on 14 comma divisions
- Werckmeister II IV another temperament included in the Orgelprobe divided up through 13 comma
- Werckmeister III V an additional temperament divided up through 14 comma
- Werckmeister IV VI the Septenarius tunings
- References
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 fifth is simply a dash. Werckmeister was not explicit about whether the syntonic comma or Pythagorean comma was 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 chromatic music ("ficte"), which may have led to its popularity as a tuning for J.S. Bach's music in recent years.
Play major tonic chord
Modern authors have calculated exact mathematical values for the frequency relationships and intervals using the Pythagorean comma:
Werckmeister II (IV): another temperament included in the Orgelprobe, divided up through 1/3 comma
In Werckmeister II the fifths C-G, D-A, E-B, F♯-C♯, and B♭-F are tempered narrow by 1/3 comma, and the fifths G♯-D♯ and E♭-B♭ are widened by 1/3 comma. The other fifths are pure. Werckmeister designed this tuning for playing mainly diatonic music (i.e. rarely using the "black notes"). Most of its intervals are close to sixth-comma meantone. Werckmeister also gave a table of monochord lengths for this tuning, setting C=120 units, a practical approximation to the exact theoretical values. Following the monochord numbers the G and D are somewhat lower than their theoretical values but other notes are somewhat higher.
Werckmeister III (V): an additional temperament divided up through 1/4 comma
In Werckmeister III the fifths D-A, A-E, F♯-C♯, C♯-G♯, and F-C are narrowed by 1/4, and the fifth G♯-D♯ is widened by 1/4 comma. The other fifths are pure. This temperament is closer to equal temperament than the previous two.
Werckmeister IV (VI): the Septenarius tunings
This tuning is based on a division of the monochord length into
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 B♭-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 B♭-F are tempered narrow, while the fifth G♯-D♯ is tempered wider than pure; the other fifths are pure.