Well-tempered clavier.

A term used in particular by Bach (‘Das wohltemperirte Clavier’) to signify a tuning system suitable for all 24 keys. The fame of Bach’s 48 preludes and fugues, two in each of the 12 major and 12 minor keys (in fact only the first book, 1722, bore the title) has led to the mistaken assumption that wohltemperirt was a standard technical term in Bach’s day to designate a particular tuning; and on the basis of this assumption a difference of opinion has arisen as to whether it was equal temperament. Bach’s choice of title disallows any form of regular mean-tone temperament (which would have a wolf 5th) and calls for a tuning well adapted to all 24 keys; but equal temperament was not the only such scheme employed at that time. Moreover, while many theorists, including Meckenheuser (1727), Sorge (1748) and Marpurg (1756), referred to equal temperament as a good tuning or even as the best of the good tunings, other influential theorists from Werckmeister in the 1680s to Bach’s former pupil Kirnberger in 1776 held that a good temperament ‘makes a pleasing variety’ (Werckmeister, 1697) or does not ‘injure the variegation of the keys’ (Kirnberger, 1776–9).

There is no proof that Bach explicitly endorsed the latter view, but there is clear evidence that had he done so he would still have rejected the tunings advocated by his former pupil Kirnberger: they contain a pure major 3rd between C and E, whereas Bach had evidently taught Kirnberger to temper all the major 3rds larger than pure (see Marpurg, 1776, p.213). Thus not even the scheme referred to by scholars as ‘Kirnberger III’, in which the four 5ths C–G–D–A–E are each tempered by 1/4 comma, would have conformed to Bach’s practice, and the tuning primarily advocated by Kirnberger is an even more unlikely candidate as it obliges the two 5ths G–D–A to be diminished by the excessive amount of ½ comma each.

A selection of 18th-century temperaments which Bach would more probably have considered ‘good’ is shown in fig.1, which shows how much the various triadic concords are tempered according to each scheme, and how large the semitones are. Each hexachord diagram represents a spiral in which F, A and C are to the right of B, D and F; the unit of measure employed is the one most often used by 18th-century German theorists: 1/12 of the Pythagorean comma, or the amount by which each 5th is diminished in equal temperament (approximately 2 cents). To the right of each diagram the semitones are given in cents, and each row covers half the octave: by reading the numbers zig-zag between the two rows, one can see how the semitones vary according to their relation to the circle of 5ths. (In fig.1b, for example, the leading notes to C, G, D and A are 110, 106, 102 and 98 cents beneath them respectively.) Fig.1c is a theoretical model; Werckmeister suggested that in practice he might leave C–G–D–A as in 1/4-comma mean-tone temperament (tempered 2¾; units instead of 3). Fig.1e is a modern, approximate reconstruction to exemplify a style of tuning, not a specific theoretical model.

Bach would undoubtedly have been grateful to find any of these tunings (except perhaps the last one) on the organ. A hint of his more subtle preferences may be found in the remarks of his one-time pupil Lorenz Mizler (1737), his very well-informed friend G.A. Sorge (1748) and his son-in-law J.C. Altnickol (1753) to the effect that Johann Georg Neidhardt was a better theorist of tuning than Werckmeister. It is doubtful whether Bach had any one secret mathematical formula of his own; he was not so mathematically inclined in matters of theory (see Barbour, 1947). Werckmeister, Neidhardt and Sorge all pointed out that circumstances such as the social ambience, the presence or absence of transposing instruments, the use of Cammerton versus Chorton, or the chromaticism of the music could have a bearing upon the exact nuance of temperament to be preferred, and Bach was probably no less exacting; according to his son C.P.E. Bach, no one else could tune a keyboard instrument to his satisfaction. C.P.E. Bach’s own tuning advice allows for a few 5ths to be left untempered (implying a slightly unequal temperament), and he endorsed the mathematically vague instructions of Barthold Fritz (1757), which were ostensibly intended to render all keys ‘equally pure’, but actually tend to favour the diatonic 3rds.

Table 1 shows by what fraction of a comma the 5ths are tuned smaller than pure according to various theoretical well-tempered systems. Several of them distribute 1/4 of the Pythagorean comma among the three 5ths C–G–D–A; 1/6 among E–B–F–C and 1/3 among A–E–B–F–C–G. As fig.1 shows, the tempering of any 3rd or 6th will vary inversely with the sum of the tempering among the 5ths or 4ths that it comprises in the chain. (If the sum of tempering among them is an entire syntonic comma (11 units) the 3rd or 6th will be pure; if they are all pure, the 3rd or 6th will be tempered by an entire syntonic comma; etc.) Of Werckmeister’s schemes, only the one singled out by Christiaan Huygens, Sorge and Marpurg in their discussions of Werckmeister is included in Table 1. The systems of Kellner and Barnes are modern proposals for the music of Bach, and share with Werckmeister’s scheme a musically unfelicitous mixing of tempered and untempered 5ths among the diatonic notes. (Werckmeister, 1697, p.32, did this in order to convert from 1/4-comma mean-tone temperament by returning only some of the notes and yet make C–E larger than pure.) Lambert’s 1/7-comma arrangement might be regarded as a kind of synthesis of the four schemes preceding it in Table 1. The remaining systems are more elaborate in that they provide for more than one size of tempered 5th. Of Neidhardt’s schemes, those which he recommended for a large or small town have been included, and Bach probably favoured something along these lines. The first volume of Das wohltemperirte Clavier was composed before the publication of any of these schemes, however, and may rather have been inspired by a non-mathematical book on the musical significance of unequal temperament published by Johann Mattheson in 1720; Bach visited Hamburg at the end of the same year, and probably met Mattheson.

For the 3rds involving a sharp or flat, an averaging of the rigidly mathematical theoretical schemes of ‘good’ unequal temperament that Bach might have known about would be better suited to the subtle demands of his music than would any one of those schemes on its own. This is because the unit of measurement used by even the most meticulous of the German theorists of the day, 1/12 Pythagorean comma, is not fine enough to represent the nuances as subtly as a good tuner can control them by ear. The theoretical problem could have been overcome by dividing the unit, but none of the theorists did that.

One point which emerges from a comprehensive study of Bach’s organ music, however, is that the most heavily tempered major 3rd was the one above C/D. Bach would, for instance, more readily treat F minor as transposed Phrygian than transposed Dorian, and so would characteristically use F–A in a more straightforward way than C–E as a vertical sonority. This can be observed in such chorale settings as Aus tiefer Not (bwv 687), Erbarm dich (bwv 721) and Herzlich tut mich (bwv 727). On the other hand, the ‘St Anne’ Prelude (bwv 552) shows that the nuances of tempering on at least some organs available to Bach were subtle enough that he could compose virile, allegro organ music in E major. The climactic (albeit inverted) D triad towards the end of the Passacaglia in C minor (bwv 582) occurs at such a stressful moment that some harshness in the intonation is, in this particular context, expressively apt. In a quieter way the tender effect of the relatively low intonation, in a ‘good’ temperament, of D (and, though to a lesser extent, of A) when used as the 7th in a dominant 7th chord is tellingly exploited in such well-known chorale-preludes as O Mensch bewein’ (bwv 622) and Schmücke dich (bwv 654). To tune really well for Bach, one should think in such terms and test with some characteristic examples of his fine use of the various notes of the chromatic scale.

The modern schemes included in Table 1 are only two of several neo-Baroque keyboard temperaments invented since the mid-1970s. These have often been improved upon after being tried out for a while. For instance, The Frobenius organ at the Queen’s College, Oxford, was tuned in the 1980s to the scheme shown in fig.2a. In 1992 the tuning was improved by raising slightly the chromatic notes and F, thereby letting D–F be tempered more than F–A, preventing E–G from being tempered distincly more than B–D, and likewise gaining a better balance between the temperings of A–C and E–G. The result is shown in fig.2b.

For additional discussion of the musical characteristics and historical importance of well-tempered tunings, see Temperaments, §§7–8 and 10. Instructions for tuning some of them have been published by Lindley (1977), Jorgenson (1977, pp.304, 323) and Blood (1979), and some have been reduced by Barbour to tables showing the distance (in cents) of the 11 notes of the chromatic scale above C.

For bibliography see Temperaments.

MARK LINDLEY