Das Wohltemperirte Clavier

 Pitch, Tuning and Temperament Design

© Copyright 2005, John Charles Francis BSc (Hons.), MSc, PhD

CH-3072 Ostermundigen, Switzerland

Francis@datacomm.ch

1 July 2005

Background

A previous article by the author[1] published February 1^st, 2005, detailed the results of a systematic mathematical analysis of the diagram on the cover sheet of Das Wohltemperirte Clavier. Performed in terms of beats per second, the analysis considered diagram orientation (left-to-right, right-to-left), tuning direction (towards sharps, toward flats), starting position (12 positions) and beat rates for the interval closing the circle of fifths (0, 1, and 2). In total 144 possibilities were analysed, and for each, a system of twelve linear equations in twelve unknowns was constructed and solved to analytically obtain the pitches of the twelve semitones as both ratios and decimals, along with the corresponding temperament in cents. The automated analysis, performed using the symbolic equation solving software Mathematica, yielded the remarkable result that the diagram encoded two cammerton-cornet-ton transpositions, confirming a tentative hypothesis that it represented two circles of fifths corresponding to the respective end points of the diagram. Moreover, both of the cammerton-cornet-ton solutions conformed to our knowledge of historical pitch standards.  It was further indicated that a specific marking by Bach, the ‘C’ of ‘Clavier’, showed the transpositions R2-1 (cammerton) and R12-2 (cornet-ton) as Bach’s preferred solution. R12-2 was derived as a right-to-left reading of the diagram and it will be shown that the mirror[2] form of R2-1 corresponds to a left-to-right reading. A statistical analysis was also performed according to the frequency of key signature usage in the Bach-Werke-Verzeichnis, considering both organ and other keyboard works. Strong correlations were found with the width of the major thirds for R12-2 and R2-1, and indeed the strongest correlation, not exceeded by any considered historic temperament, occurred at the mid-point of R12-2 and R2-1, i.e. the intervening key. The current paper presents the results in a more musician-friendly manner, showing how the diagram relates to the circle of fifths. It also provides comparison with earlier interpretations proposed by Sparschuh, Zapf and a later one proposed by Lehman. In addition, it presents more details in terms of comparison with historical temperaments. The rationale behind the construction of the temperament is presented and the theoretical framework corresponding to the practical equal-beating tuning method is described.

Introduction

The suggestion that the sinuous circles of J. S. Bach’s 1722 cover sheet for Das Wohltemperirte Clavier may depict a tuning scheme was made some years ago by Andreas Sparschuh, a mathematician from the Technical University, Darmstadt[3]. In an award winning presentation to the German Tuners Association, Sparschuh outlined his numerical reading of Bach’s design:

Start-1-1-1-0-0-0-2-2-2-2-2-End

The clavichord specialist Michael Zapf and the tuning theorist Herbert Anton Kellner, both independently met with Sparschuh. While Zapf was convinced that Bach’s diagram was indeed a tuning system, he was sceptical regarding Sparschuh’s initial interpretation, considering it at odds with historical tuning practice. Zapf subsequently made a proposal of his own, treating Sparschuh’s numbers as an equal-beating specification in terms of seconds-per-beat[4]. Keith Briggs from BT Labs investigated the mathematics of Zapf’s proposal and noted that, by closing the circle-of-fifths, the pitch of Bach’s keyboard might be determined[5]. A proposal by the author, interpreted Sparschuh’s numbers as beats-per-second, and included both ends of Bach’s diagram in the analysis to close the circle-of-fifths[6]

{1}-1-1-1-0-0-0-2-2-2-2-2-{2}

A subsequent proposal by Lehman[7] interpreted Sparschuh’s numbers as 1/12 and 1/6 of a Pythagorean comma, but without a consistent reading for the end points[8].

Tuning Circumstances at Bach’s Time

Lorenz Christoph Mizler’s main claim to fame today is as the founder of the society whose 14^th member was Johann Sebastian Bach. On the subject of tuning, he once noted that while Werckmeister’s temperament was the best in his day, it had nevertheless been improved upon since Neidhardt’s time[9]. However, we know that the best tuning practices of Mizler’s day did not satisfy Bach; for as his son Carl Philipp Emanuel noted, no one else could tune the harpsichord of his father to his satisfaction[10]. In this regard, Lindley proposes[11] that the elaborate theoretical tuning models expounded by the likes of Neidhardt and Sorge, were unable to capture the subtle nuances that Bach customarily achieved in tuning[12]. He, suggests, moreover, that the cause of the theoretical deficiency was that Sorge and Neidhardt would never split their basic unit of measurement for tempering, namely 1/12 of a Pythagorean comma. Lindley notes that this fraction is unable to satisfy the following three conditions simultaneously:

We will see that the tuning for Das Wohltemperirte Clavier achieves all of these goals and that, as Lindley intimates, in so doing a finer division of the Pythagorean comma is needed.

Marpurg, while arguing the case for Equal Temperament, noted that there is only one ideal version of this temperament, whereas there are many unequal schemes[15]. He therefore suggested there would be chaos until Equal Temperament was universally achieved, as otherwise each would be inclined to use whatever they considered best. We will see that in Das Wohltemperirte Clavier, J. S. Bach offers a tuning system that is also an ideal of its kind.

J. S. Bach’s use of the generic term ‘Clavier’ (keyboard) leaves unspecified the precise instrument to be used. As noted by Richard Jones[16], this, in conjunction with the deliberately circumscribed keyboard compass, suggests that the work was intended to be universally accessible to keyboard players regardless of the particular type of instrument (harpsichord, clavichord, or organ[17]) that they might have at their disposal. An important issue arises here, however, in relation to the adoption of an unequal temperament for ‘well-tempered’ instruments; namely coexistence at cammerton and cornet-ton pitch[18]. In the case of equal tempering, this is no concern as each key is identical. However, if unequally tempered instruments at cammerton and cornet-ton pitch are tuned identically to an identical temperament, then there will be inevitable intonation problems as the tuning of one instrument is two places removed from the other on the circle-of-fifths. It follows that there cannot be just one temperament for Das Wohltemperirte Clavier, given the generic term Bach uses, but rather there must be two variants: one for cammerton and another for cornet-ton. We will see that the temperament for Das Wohltemperirte Clavier consists of two transposed cammerton, cornet-ton variants, indicated by respective left-to-right and right-to-left readings of Bach’s diagram.

By virtue of certain telling features in the cammerton and cornet-ton tuning variants, Bach’s scheme can be shown to be an equal-beating one. Its transposition by two places on the circle-of-fifths requires adaptation to one of the beat rates, which Bach explicitly provides. While the equal-beating methods presented by Jorgensen have been viewed cautiously in some circles, they are nevertheless pertinent to Bach’s method. Jorgensen, in his impressive tome, Tuning, traces the historical understanding of beats, noting that in the Seventeenth Century there was no information published that they should increase in frequency when intervals are played higher up the scale[19]. To the musicians at that time, he notes, the quality of the fifths in a scale when played harmonically, seemed identical when their beat frequencies were the same; deceiving them into believing that tempered fifths were of the same size when their beat frequencies were identical[20]. Nor, it seems, had the situation changed by the end of the Eighteenth Century, for in the context of Thomas Young’s rules for well temperament of 1799, Jorgensen notes that it would be authentic practice to apply the much easier equal-beating methods[21].

C. P. E. Bach writing in 1753 concerning the tuning of the clavichord and harpsichord[22] refers to tempering most[23] of the fifths. He observed that the beats of fifths may be more easily heard by probing fourths[24]. In minimalist terms, he characterised unequal well-tempered tuning[25]:

Both types of instrument must be tempered as follows: In tuning the fifths and fourths, testing minor and major and chords, take away from most of the fifths a barely noticeable amount of their absolute purity. All twenty-four tonalities will thus become usable. The beats of fifths can be more easily heard by probing fourths, an advantage that stems from the fact that the tones of the latter lie closer together than fifths.[26]

Clearly, this generic description does not indicate which fifths should be tempered, nor by how much. Moreover, although one is admonished to listen for the beats of fifths and fourths, there is no indication as to how fast those beats must be in order to take away from the fifth a barely noticeable amount of its absolute purity.

J. S. Bach’s Tuning Procedures

The diagram on the cover sheet of Das Wohltemperirte Clavier, read left-to-right, indicates a tuning scheme for cammerton-pitched instruments. Bearings can be set by proceeding around the circle-of-fifths towards the flats using only fourths and fifths, so as to remain within the octave starting on Middle C[27]. Bach’s diagram indicates three types of interval: i) intervals beating once per second, ii) intervals beating twice per second and iii) pure intervals without beats. In the Eighteenth Century, such tempi could be readily determined from a pocket watch or pendulum clock; today, a metronome at 60 beats per minute will achieve the same result. The protrusion at the left of Bach’s diagram depicts the beat-rate of the interval closing the circle-of-fifths, representing a check that the temperament has been set correctly. That interval arises as a side effect of the tuning scheme, and it beats exactly once per second. A realisation of the tuning procedure constrained to one octave, is illustrated in Figure 1, yielding a temperament denoted as R2-1[28]

Figure 1: Canonical form of the cammerton tuning procedure

See: Das Wohltemperirte Clavier -  Frequencies for tuning instruments to the cammerton temperament

An alternative formulation, yielding the same temperament, is presented in Figure 2[29].

Figure 2: Optimised cammerton tuning procedure using single beat rate of 1 beat per second[31]

Instruments at cornet-ton pitch sound one tone higher than their cammerton counterparts and, to coexist harmoniously in an ensemble, their temperament must be transposed downwards by a whole tone. Read right-to-left, Bach’s diagram yields a transposed tuning scheme for cornet-ton-pitched instruments. Bearings are set by proceeding around the circle-of-fifths towards the sharps, the protrusion at the right of Bach’s diagram giving the expected beat rate for the implicitly tuned interval closing the circle-of-fifths as two beats per second. A realisation of the tuning procedure within one octave is detailed in Figure 3, yielding a temperament denoted as R12-2[30].

Figure 3: Canonical form of the cornet-ton tuning procedure[32]

Pitch and Temperament

The pitch at which Bach’s tuning procedures were designed can be obtained by reverse-engineering his diagram using the indicated beat rates (see appendices for details). The corresponding temperaments can then be derived from this pitch information. The pitch and temperament are summarised in Table 1 and Table 2, respectively, proving that the circles in Figure 1 and Figure 3 are exact transpositions.

Table 1: Pitch of Das Wohltemperirte Clavier[33]

See: Das Wohltemperirte Clavier -  Frequencies for tuning instruments to the cammerton temperament

Table 2: Temperament of Das Wohltemperirte Clavier[34]

Temperament Design

The temperament for Das Wohltemperirte Clavier arises naturally as an ideal providing an optimally smooth progression from worst to best thirds across the circle-of-fifths, so satisfying Lindley’s first condition. The thinking that led to its creation can be readily reconstructed from the well known consideration that four consecutive intervals on the circle-of-fifths determine the width of the corresponding major third. The Pythagorean third (~408 cents) is excluded, so that a sequence of four pure fifths does not occur, so satisfying Lindley’s second condition. The widest third must then consist of three consecutive pure intervals on the circle-of-fifths, followed by a tempered interval to narrow the third. This observation yields the sequence 0-0-0-1 on the circle-of-fifths, corresponding to the worst-case third. Optimising for a gradual progression towards better thirds as Lindley suggests, leads to a progressive narrowing of the major thirds by one unit at a time. The results in Table 3 are now predicated.

Table 3: Derivation of Das Wohltemperirte Clavier tuning by progressive narrowing of thirds

Considering the resulting sequence 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, it can be seen that the sum of the digits is 14[36]. Accordingly, as the Pythagorean comma must be distributed over the circle-of-fifths, each tempering unit theoretically corresponds to 1/14^th part of a Pythagorean comma, the tempering fractions being 1/7[37] and 1/14. Adopting the historical precedent of placing the best thirds in the key of C, yields the theoretic temperament R12-14P in Figure 4. The equal-beating method for tuning this temperament (R12-2) is depicted in Bach’s diagram, and realised in Figure 3[38].

Figure 4: R12-14P - theoretic representation of R12-2 in terms of Pythagorean comma fractions

The cornet-ton temperament in Figure 4 must be transposed by a whole tone for cammerton-pitched instruments to give R2-14P as shown in Figure 5.

Figure 5: R2-14P - theoretic representation of R2-1 in terms of Pythagorean comma fractions

The equal-beating method for tuning this temperament (R2-1) is depicted in Bach’s diagram, and has been realised in Figure 1 and Figure 2.

The difference in beat rates between cammerton (R2-1) and cornet-ton (R12-2) tuning procedures that occurs at the interval closing the circle-of-fifths, is explicitly represented in Bach’s diagram, and can easily be derived as indicated in Figure 6 from the following well-understood tuning principles:

1. An interval without beats in a given octave is without beats in the octave above.
2. Inverting a fourth (e.g. C1-F1) such that its lower note sounds an octave higher (i.e. F1-C2 for the given example), yields an interval which beats at the same rate.
3.  Inverting a fifth (e.g. C1-G1) such that its lower note sounds an octave higher (i.e. G1-C2 for the given example), yields an interval that beats twice as fast.