The Keyboard Temperament of J. S. Bach

John Charles Francis, BSc (Hons), MSc, PhD
CH-3072 Ostermundigen, Switzerland
Francis@datacomm.ch

Introduction

Johann Sebastian Bach prepared his didactic Klavierbüchlein für Wilhelm Friedemann Bach in January 1720, following the tenth birthday of his oldest son, Wilhelm Friedemann. On its first pages, he described the clefs his son would encounter and the musical ornaments he would need, providing first an exercise with explicit fingering indications, the Applicatio (BWV 994), followed by a Preambulum (BWV 924). The intent of this latter piece, the second in Wilhelm Friedemann's book, is considered in this paper. The thesis presented is that BWV 924 is nothing less than a pedagogic aid to help Wilhelm Friedemann perform keyboard tuning. It is shown how J. S. Bach's keyboard temperament can be reconstructed from this piece.

Originally scattered throughout the old Bach-Gesamtausgabe (BGA), the Klavierbüchlein für Wilhelm Friedemann Bach was first published as a unit in 1962 under the auspices of the Neue Bach-Ausgabe (NBA) [1]. Around that time, a facsimile edition also became available [2]. Considerations regarding handwriting and the evolution of entries can be found in the NBA Critical Commentary [3]. The original manuscript is currently housed in the library of the Yale school of Music.

Johann Nikolaus Forkel, writing in 1802, noted that J. S. Bach always tuned his own keyboard and that the procedure did not take him more than fifteen minutes [4]. Andreas Sorge, commenting in 1748 on the tuning system of the renowned organ builder Gottfried Silbermann, mentioned Bach as having described four specific triads resulting from Silbermann's method as having a barbaric nature intolerable to a good ear[1] [5]. An anecdote, in similar vein, was related by Edward John Hopkins in 1895, reporting a tradition whereby J. S. Bach as auditor of Silbermann's instruments supposedly said "You tune the organ in the manner you please, and I play the organ in the key I please"; following his remark with a Fantasy in A-flat major causing Silbermann to retire to avoid his own "wolf" [2] [6]. Friedrich Wilhelm Marpurg, writing in 1776, related that J. S. Bach had confided his tuning method to a pupil, Kirnberger, who was expressly required to tune all the thirds sharp[3] [7].

In more recent times, Johann Sebastian Bach has been seen, on the one hand, as a pioneer of Equal Temperament [8] and, on the other, as a proponent of unequal Well Temperaments. In this latter regard, suggested temperaments have included those of Werckmeister [9] and Kirnberger [10]. Attempts at reconstructing Bach's temperament have been made by Kellner [11] and Barnes [12].

Figure 1: Preambulum (BWV 924) from the Klavierbüchlein für Wilhelm Friedemann Bach

Decoding BWV 924

The Preambulum (BWV 924) is written in the key of C major and consists of eighteen measures (Figure 1). The bass line of the first three bars is shown in Figure 2.

Figure 2: base line of BWV 924 (bars 1-3)

Of interest are the notes marked with trills[4], which, taken in order of occurrence preceded by the opening note, give rise to six consecutive notes on the circle of fifths C, G, D, A, E, B, with five corresponding intervals of a perfect fifth:

C - G, G - D, D - A, A - E, E - B

This observation motivates the following hypothesis: namely, that some of the trills in BWV 924 carry a meaning beyond the obvious one of musical ornamentation; their purpose being to serve as a memory aid for Wilhelm Friedemann Bach to help him recall the complex, yet fundamental, process of tuning a keyboard.

To proceed further some notational conventions are necessary: the opening C of BWV 924 will be denoted as C₁, the note one octave below it as C₀, the note one octave above as C₂; the other notes within the octave will be similarly designated. With this notation in hand, the music in Figure 2 can be represented as follows:

C₁ E₁ G₁ G₀ D₁ F₁ A₁ A₀ E₁ E₂ A₁ B₁

For reference purposes, certain notes have been underlined: namely those that fall on the circle of fifths and fall within one octave, the "tenor" octave[5]. Note, the two bolded notes are precisely those needed to construct and tune the underlined notes starting from C₁. To illustrate this point, the tuning process is now described:

1.     Tune C₁ to an external reference (where appropriate).

2.     Tune G₁ using C₁ as a reference (perfect fifth).

3.     Tune G₀ using G₁ as a reference (octave).

4.     Tune D₁ using G₀ as a reference (perfect fifth).

5.     Tune A₁ using D₁ as a reference (perfect fifth).

6.     Tune A₀ using A₁ as a reference (octave).

7.     Tune E₁ using A₀ as a reference (perfect fifth).

8.     Tune B₁ using E₁ as a reference (perfect fifth).

Table 1: tuning of five consecutive perfect fifths on the circle of fifths

The assumption will be that Wilhelm Friedemann was taught to tune his fifths justly (i.e. pure); this assumption being motivated by several factors: the ease of tuning in just fifths[6], the rapidity with which this can be done[7], and the historical precedent of employing just fifths. The resulting tuning is summarised in Table 1.

Returning now to Figure 2 and keeping in mind the hypothesis that the trills embody a meaning for tuning purposes, it is observed that the trills are of two different kinds: the first four are mordants, while the latter is a, so-called, doppelt cadence u. mordant. For the time being, the focus will be on the four mordents; these being defined in the introductory table of ornaments that J. S. Bach included at the beginning of Wilhelm Friedemann's Klavierbüchlein as indicated in Figure 3.

Figure 3: a mordant relates tonic and leading tone

From this, it follows that the four mordents occurring in the first three bars of BWV 924 (Figure 2), identify four relationships between tonic and leading tone, as shown in Table 2.

Table 2: relationships implied by mordents

From the notes already tuned (Table 1), the leading notes in Table 2 must now be tuned. Since semitone relations cannot be tuned directly by ear with any useful accuracy, there is but one practical possibility: to tune by thirds. For the time being, the assumption will be to tune these thirds justly[8]; and under this condition the tuning shown in Table 3 can be derived. Note, it is possible, and indeed easier, to tune only one leading note from the third below and then proceed by tuning in perfect fifths as shown in Table 4[9]. Using either of these equivalent procedures, the original circle of fifths in Table 1 is extended as shown in Table 5.

Table 3: tuning of the notes Bach marked by mordents as just thirds

Table 4: alternative method for tuning the notes Bach marked by mordents as just thirds

Table 5: extended circle of fifths

It can be observed that the second step of the tuning procedure dovetails nicely with the first, extending the circle of fifths by four steps. A further remark concerns the interval B - F#, which was not tuned explicitly, but rather defined implicitly as a result of other tuning operations: this interval is a "wolf", which needs to be dealt with. This is optimally achieved by spreading the wolf equally across the intervals:

A - E, E - B, B - F#.

Note, that the tuning steps shown in Table 1 form part of the method of Pythagorean tuning, and so suffer from a perceived "defect", namely, that the major thirds C₁ - E₁ and G₁ - B₁ are wide in comparison to the ideal of justly tuned thirds[10]. Accordingly, a prior, we might expect such notes to be tempered (narrowed) to remedy this problem; conveniently, distributing the wolf equally, also serves to narrow the thirds.

Returning now to the score of BWV 924 (Figure 1), and once again keeping in mind the hypothesis that the trills embody tuning instructions, it is observed that the notes B₁ and E₁ in bars 3 and 5, respectively, are marked by trills which Bach called doppelt cadence u. mordant. In view of the placement of this type of trill on precisely those notes that a priori need tempering to distribute the wolf and "improve" thirds, the doppelt cadence u. mordant on E₁ and B₁ is taken to imply a tempering operation. In this regard, note the shape of these trills: both of which point downwards, the direction in which the E₁ and B₁ must be tempered to distribute the wolf and improve the thirds. No guesswork is needed concerning the amount of this tempering, since the fundamental rationale for the operation is to distribute the wolf in Table 5 evenly, i.e., to adjust these two notes such that the error in the wolf interval B₁- F₂# is spread evenly over the three[11] intervals A₀- E₁, E₁- B₁, B₁- F₂#. This requirement precisely determines the tempering needed for E₁ and B₁, and after the relevant calculations are performed, the results indicated in Table 6 and Table 7 are obtained.

Returning again to the score of BWV 924 (Figure 1), it will be seen that the G₁# in bar 4 is also marked by a doppelt cadence u. mordant, but that it points upwards, not downwards. Earlier, the doppelt cadence u. mordant on E₁ and B₁ were taken to imply a narrowing of the thirds C₁ - E₁ and G₁ - B₁, and by analogy the inverted doppelt cadence u. mordant on G₁# can be interpreted as a widening of the interval E₁ - G₁#. There are two ways such widening can be achieved:

1. A lowering of E₁: in this case the purpose of this doppelt cadence u. mordant is purely pedagogic; a lesson for Wilhelm Friedemann that narrowing the interval C₁ - E₁ has widened the interval E₁ - G₁#.
2. A prescriptive tuning instruction that the note G₁# should be raised, implying a widening of the interval C₁# - G₁#. To determine the amount of this tempering, note in Table 7 that the tempering narrowed each fifth by 1/3 of a syntonic comma. Accordingly, given the mirror-image appearance of the symbol, the fifth C₁# - G₁# should be widened by 1/3 syntonic comma, implying that G₁# should be sharpened by this amount[12].

Two semitones still remain to be specified, Bb and F respectively, there being no specific information from any trills in the score as to how the intervals should be tuned. Assuming tuning by just fifths, then starting at D#/Eb the intervals Eb - Bb and Bb - F can be tuned. Alternatively, starting at C, the intervals F - C and Bb - F can be tuned in just fifths. In fact, it turns out that with the tuning steps considered so far, the difference is virtually imperceptible[13].

Table 6: optimum tempering to distribute the wolf

Table 7: the fifths resulting from tempering in Table 6

Table 8: method for tuning the notes Bach marked by mordents as sharp thirds

It is now shown how the tuning method above can be modified to tune the thirds in Table 4 wide, rather than justly. The procedure is as follows:

1. Perform the tuning steps indicated in Table 1.
2. Tune the interval D - F# justly (as above).
3. Temper the E and B such the three intervals A- E, E- B, B- F# are equally good (as above).
4. Retune F#, such that the interval B - F# is a just fifth[15] and continue tuning a just circle of fifths to D₁#/Eb (Table 8)[16].
5. Reflecting the upward pointing doppelt cadence u. mordant on G₁# in BWV 924, widen the fifth C₁# - G₁# by 1/3 syntonic comma[17] (option, as above).

6.     Tune the Bb and F either i) as just fifths starting from D₁#/Eb, or ii) as just fifths starting from C[18].

The tuning systems considered so far are summarised in Table 10. The derived temperaments are referred to as Temperament I, Temperament II, Temperament III and Temperament IV, respectively. Their corresponding deviations from 12-tone Equal Temperament are given in Table 11, while a comparison between the fifths of all the derived temperaments is provided in Table 12. Thereafter, the characteristics of specific temperaments are described by tables of intervals, tables of thirds and fifths, and tables of the major and minor tetrachords.

Discussion

The structure of the sequence of fifths within the derived temperaments can be represented as shown in Table 9, where J, denotes a just fifth, N, a fifth narrowed by 1/3 syntonic comma, W, a fifth widened by 1/3 syntonic comma, E, an Equal Tempered fifth and X, the fifth that is 1/3 syntonic comma smaller than an Equal Tempered fifth.

Considering the major thirds of Temperament I (Table 14) and Temperament II (Table 18), it can be observed that the two narrowest major thirds are D - F# and A - C#, and that these are tuned justly. The widest occur at F# - A# and C# - F, and are Pythagorean thirds, while the remaining thirds in Table 14 fall between these extremes in a progressive manner according to the circle of fifths. Looking now to the major (Table 15) and minor (Table 16) tetrachords for Temperament I, it will be noticed that duplication occurs: specifically, the major tetrachords starting on Ab and Eb are the same; likewise those starting on Bb and F. With regard to the minor tetrachords, the ones starting on Bb and F are identical. Comparing now with Temperament II, where, G#/Ab has been sharpened, it can be observed that all tetrachords are unique (Table 19 and Table 20). This provides an excellent rationale for sharpening G#/Ab, namely, to add explicit key colour and variety; a further rationale is the creation of a wide fifth which adds further colour to the temperament. A similar observation can be made regarding Temperament III: namely, that the major tetrachords on Ab and Eb are identical, as are those on Bb and F (Table 23). In the case of the minor tetrachords, there are also two duplications: those rooted in Ab and Eb are the same, and likewise those rooted in Bb and F (Table 24). Comparing now with Temperament IV, where explicit sharpening of G#/Ab has occurred, it is important to note that all major (Table 27) and minor (Table 28) tetrachords have been rendered unique as a result of this tempering operation. That the tempering corresponding to the upward pointing doppelt cadence u. mordant should be exactly that needed to render all the tetrachords unique, is hardly a coincidence. Accordingly, the prescriptive interpretation of the trill on G#/Ab in Figure 1 can be assumed. This eliminates Temperaments I and III from consideration.

By the above assumption, Temperaments II and IV remain as viable options. If the third-hand account by Friedrich Wilhelm Marpurg [7] is taken at face value: namely, that Kirnberger was expressly required by Bach to tune all the thirds sharp, then both Temperaments I and II must be rejected as invalid options. That would leave uniquely Temperament IV as the implied temperament. However, the possibility remains that Marpurg's statement was merely a form of words to express Well Temperament. It is also possible that Marpurg misconstrued, or even misrepresented, Kirnberger's remarks by implying that each and every fifth is tuned sharp. Alternatively, Bach may have taught Kirnberger a different tuning system to his own. Moreover, it is possible, that Wilhelm Friedemann may have used the simpler procedure of tuning the thirds justly, while his father adopted the expedient of tuning the thirds wide. It likewise conceivable, that J. S. Bach may have used different tuning variants at different stages of his career and in different situations. Moreover, if J. S. Bach is assumed to have derived the system embedded in BWV 924 himself, that suggests an experimental disposition. and accordingly, he may have used both of these variants at one time or another.

Table 9: structure of derived temperaments in terms of fifths

One final area of uncertainty to be addressed concerns the tuning of Bb and F, and whether they should these be tuned from Eb or C (or both). In the case, of Temperaments I and II, the choice is of little import, with a resulting difference in tuning of two cents. However, in the case of Temperaments III and IV, the difference is significant. The following options can be considered:

1)     Tune Bb justly from Eb, and then tune F justly from Bb; this is a valid option.

2)     Tune Bb justly from Eb; tune F justly from C; this option can be discounted as it gives rise to an inappropriate Pythagorean third on the interval Bb - D.

3)     Tune F justly from C and then tune Bb justly from F; this option can also be discounted as it gives rise to two inappropriately placed Pythagorean thirds on the intervals F - A and Bb - D.

4)     Tune F and Bb such that the intervals Eb - Bb, Bb - F, F - C are equal; this option compromises the just fifths unnecessarily and leads to an undulating pattern of widening thirds with two peaks. Accordingly, it can be discounted.

Temperaments I, II, III and IV have been compared with other historic temperaments and the results are shown in Figure 4, Figure 5, Figure 6, and Figure 7, respectively. Each figure indicates the calculated distance[19] of the historic temperament from the temperament derived from BWV 924, providing a measure of how far the historic temperament differs, on average, from those derived from BWV 924. The following observations apply to all the derived temperaments:

1.     Of the historic temperaments considered, Equal Temperament is by far the closest approximating temperament to Temperaments III and IV.

2.     Barnes temperament [12], derived from a statistical analysis of Bach's Well Tempered Clavier[20], is a much closer approximation to each derived temperaments than Kellner's temperament [11].

3.     Silbermann's temperament [5], as might be expected, is far away from those derived from BWV 924.

4.     The temperaments of Kirnberger [10] and Werckmeister III [9] are likewise very distant.

Conclusions

This paper has reported the discovery by the author of several tuning systems derived from an analysis of BWV 924. These systems were determined under the assumption that BWV 924 was explicitly constructed by Johann Sebastian Bach to provide a pedagogic tuning aid for his young son, Wilhelm Friedemann; the purpose being to serve as a reminder of lessons already taught. Lack of information with regard to the exact teaching of J. S. Bach led to the range of candidate solutions[21], being considered.

Two possible interpretations with regard to tempering Ab/G# were possible: one pedagogic (and arguably contrived), and the other prescriptive: the former corresponded to Temperaments I and III, the latter to Temperaments II and IV. It was found that explicitly sharpening G#/Ab renders all major and minor tetrachords unique, and also offers the benefit of one wide fifth, adding colour and variety to the keys. The author notes, that in the case of Temperament IV, for example, the key of Ab, alluded to by Edward John Hopkins [6], is most charming. In general, the variety and colour introduced by sharpening G#/Ab is very pleasing. The choice between Temperaments II and IV, depends on the credence given to the remark of Friedrich Wilhelm Marpurg [7]. Accepting Marpurg's proposition, implies accepting Temperament IV.

The implications of this analysis will be relevant for musicians concerned with the historical informed performance practice of Bach's music. The results obtained have shown that several historic temperaments, including Werckmeister and Kirnberger, are unlike those derived from BWV 924; the Kellner temperament is likewise very different. Equal Temperament is a viable performance option, but lacks the benefit of justly tuned fifths and key colour. Accordingly, musicians are invited to experiment with the temperaments presented in this paper, and Temperament 4 is recommended as the starting point.

Table 10: derived temperaments based on tempering by 1/3 syntonic comma[24]

Table 11: deviation of temperaments from Equal Temperament[25]

Table 12: comparison of fifths

Table 13: Temperament I -- interval analysis

Table 14: Temperament I -- thirds and fifths

Table 15: Temperament I -- major tetrachords

Table 16: Temperament I -- minor tetrachords

Table 17: Temperament II -- interval analysis

Table 18: Temperament II -- thirds and fifths

Table 19: Temperament II -- major tetrachords

Table 20: Temperament II -- minor tetrachords