Johann Sebastian Bach and Die Kunst der Fuga

J.S. Bach's Prefatory Message and Implications

Introduction

In an earlier article devoted to Bach's last printed composition, the presumably authentic title was established.1 The conventional reading in all printed editions was Die Kunst der Fuge; however, the correct version would read Die Kunst der Fuga. Amongst other observations, this spelling renders the title's gematrial sum as 158, identical to the result when converting the composer's full name Johann Sebastian Bach. An earlier article indicated numerous allusions to the system "wohltemperirt" within the composition.2 In order to demonstrate here the essential notion of the unitas in baroque music theory,3 in both spellings the title's 15=7+1+7 letters are centered upon the letter T=19--the number of intervals closing the circle in tuning.4 Further to the title, the first printed edition of the composition contains a short preface comprising seven lines, called Nachricht. Thanks to a remarkable booklet of great originality and richness of ideas by Vincent Dequevauviller,5 my attention was drawn to this message. Following my study devoted to the title itself, the purpose of the present article is to scrutinize that message more profoundly and interpret the outcome. Finally, new aspects as to how many bars, ideally, the unfinished fugue would comprise, are presented.

The preface to the first edition, 1751

Die Kunst der Fuga had two early editions in rapid succession, the first one 1751 and the following one already in 1752. These editions carry different messages as preface. The text published with the first edition of Die Kunst der Fuga, is shown in Figure 1 and below.

Nachricht

Der selige Herr Verfasser dieses Werkes wurde durch seine Augenkrankheit und den kurz darauf erfolgten Tod ausserstande gesetzet, die letzte Fuge, wo er sich bey Anbringung des dritten Satzes namentlich zu erkennen giebet, zu Ende zu bringen; man hat dahero die Freunde seiner Muse durch Mittheilung des am Ende beygefuegten vierstimmig ausgearbeiteten Kirchenchorals, den der selige Mann in seiner Blindheit einem seiner Freunde aus dem Stegereif in die Feder dictiret hat, schadlos halten wollen.

Preface

The late author of this work, due to his eye disease and his death occurring shortly afterwards, was rendered incapable to terminate the last fugue wherein he identifies himself by his name upon composing the third section; therefore one wished to compensate the friends of his muse by communicating the church chorale set in four parts and adjoined at the end which the late author, in his blindness, dictated into the pen of a friend in spontaneous improvisation.

As concerns this text, Dequevauviller argues, that--contrary to appearance and expectation--it had been written and prepared in advance by J. S. Bach himself! Thus, one might wonder, what further insight the present considerations could reveal. To report still further, Dequevauviller observed that the Nachricht counts 76 words in total: for the title 1, and 75 words for the remaining body text. As is known, Bach gave a Tri-Unitary representation of the number 75 via 31+13+31 in the bar-wise structure of Duetto II (in bars, 149=37+75+37). The digits 7 and 5 of 75 may be related to the number of fifths in the unequal tuning system "wohltemperirt" of Werckmeister/

Bach.6 Let us now structure these 75 words via the unitas by writing 75=37+1+37 such that the central word upon which the text is pivoted emerges as bringen. Following this word, within this single rather long sentence, Dequevauviller mentions the partition by a semicolon.

Although the Nachricht comprising 76 words is somewhat long and continues via . . . bringen; . . . up to . . . wollen--knowing the baroque traditions, practice and procedures--it is tempting to convert that text into a number via Henk Dieben's alphabet and gematria.7 Summing up to the respective end of the words concerned, yields the result in Figure 2.

Here the gematria-sum of the last 37 words that follow after bringen, up to the final word wollen, amounts to 4466-2323=2143. It is striking that this sum 2143 is by only 5 too large, such as to yield 2138, corresponding number-letter wise to BACH. This "problem" suggests taking a closer look at the text of the Nachricht. Immediately a suspect word shows up, namely Stegereif (meaning a spontaneous improvisation). Certainly, this word, in modern German spelling would read Stegreif without the obtrusive letter E=5 that renders the sum too large by this amount. However, modern German is irrelevant in this historical baroque context and one ought to consult contemporary dictionaries to verify the spelling--or even better, texts of writers close to J. S. Bach. Provided one is familiar with those contemporary texts, one easily finds that Carl Philipp Emanuel Bach, in his treatise8 Versuch / über die wahre Art / das Clavier zu spielen, Volume 2, page 325, chapter 41, writes Stegreif, rather than Stegereif. (See Figure 3.) This succeeds in identifying the misprint of spelling within the Nachricht, first edition of Bach's Kunst der Fuga. The gematria-sum of the last 37 words can thus be corrected from 2143 to 2138, BACH--as expected.

What does this result signify? First of all, J. S. Bach's authorship of the Nachricht--in conformity with Dequevauviller--is corroborated and firmly established. Could it otherwise be imagined, that e. g., Carl Philipp Emanuel Bach--and for what reason--would have constructed that artifice? It is thus the composer's own authentic message. But in addition, one may realize that Bach has encoded here a profound personal theological statement into his last printed work. (See Figure 4.)

The composer, of faltering health, facing his death, shows himself in Christian creed aligned with and belonging to Jesus Christ. The correspondence of the initials J. CHR., 9+3+8+17=37 is "conventional," as shown frequently in several of my own papers and in the book of Harry Hahn as well.9

Dequevauviller presented the following example, Figure 5, in the context of the rupture of the unfinished fugue. In cantata BWV 106, Actus Tragicus, the choir's soprano, at the end of the second movement--final bars upon the words "Ja, komm Herr Jesu, Herr Jesu"--fades away into the last bar of--a pause! Associating with the termination of the second part of the Nachricht by 2138 = BACH with 37 meaning J. CHR., an allusion to rupture of life by death and transfiguration in Christ could be understood.

In this paragraph the firm connection linking the Nachricht to the unfinished fugue will be established. In fact, the study of Bach's message would remain incomplete, if after the count of the words and the application of Henk Dieben's gematria, the number of letters itself were not checked. Thus, the 76 words of the Nachricht, as printed, comprise 427 letters: 9 letters for the title and 418 letters for the body text. As the latter contains the misprint with the superfluous letter E, the length of the text of 75 words can be corrected to 417 letters, factorizing 3x139. On recognizes 1-3-9, the number of the circle of fifths 19, centered numerologically upon the 3 = Trinity. The number 417 depicts in juxtaposition the number of 4 well-tempered fifths, 1 tempering fifth and 7 perfect fifths of the system Werkmeister/ Bach. Converting 4,1,7 into letters yields D, A, G. My earlier article has shown that the first theme of the unfinished fugue is D,A,G,F,G,A,D. This accomplishes the proof of the connection between the unfinished fugue and the Nachricht via the number of 417 letters.

As there can be no longer any doubt about J. S. Bach's authorship of the prefatory message, this proves that according to his intentions, not only the unfinished fugue, but also the final chorale do indeed belong to the composition. Some editions omit the chorale, but future editions may take into account the present result and thus grant Bach--so to speak--the right to the architectonic structure he conceived for this composition and let the form of his last printed work be closed by the chorale.

The unfinished fugue: midpoint and length according to the unitas

The following section investigates the unfinished fugue in more detail, the first theme of which my preceding article in The Diapason (March, 2000, p. 13) associated with "wohltemperirt," as described above. Dequevauviller presents convincing arguments that Bach intentionally and expressly left the fugue unfinished! Musically, the ensuing rupture of flow depicts death dramatically and in a macabre fashion. However, Dequevauviller sees an ambiguity and remains undecided, whether there are 238 complete bars to be terminated by 38 further bars or 239 bars for which 37 bars are lacking. For details, his original paper ought to be consulted. On the contrary, it will now be shown that the rationally admissible viewpoint is that the manuscript of the fugue holds 239 bars terminating at the last bar-line Bach put there. (See Figure 6.) There are 37 bars missing.

From bar 238 to bar 239, the bass descends by a fifth A-D. In numbers, as A=1, D=4; there follows 14=BACH in juxtaposition. Why should this "signature" within these two bars of the autograph be truncated and discarded by assuming only 238 bars? As concerns this signature AD=14, see also the 8th and 9th keystroke of Fugue N° 1, C-major, Das Wohltemperirte Clavier 1. 10 Furthermore, in bar 239 itself, following the quarter note D of the bass, Bach's manuscript notates a single eighth-note of the tenor upon A, into the system of the bass. Juxtaposing again within bass-system, there now holds DA=41=J. S. BACH, representing a further and ultimate signature in bar 239 at the termination of the unfinished fugue. Its manuscript--contrary to the printed version--is written into the two systems of soprano and bass-key. Had the tenor been written, as in the edition, into its system of the tenor-key, such a signature would not have been feasible. It is incoherent to recognize on the one hand via the digits of the partition 239=1+238 the letters of 2138=BACH and on the other hand, assume the factual last bar of the fugue were 238. There is no way out: the fugue, as it exists, logically extends over 239 bars; the completed one totaled 239+37=276 bars. It may be worth noting, and must have been known to Bach, that the number 239 also corresponds to Vom Himmel hoch da komm' ich her.

How can the ideal extension of 276 bars for the fugue be made plausible, or corroborated? For this purpose, let us now evoke the principle of the baroque unitas with this even number. Thus, 276=138+138, showing that the complete ideal fugue would be pivoted upon the two central bars 138 and 139. (See Figure 7.)

Incidentally, upon separating the two syllables of NACH-RICHT (13,1,3,8-17,9,3,8,19), its first half terminates letter-numberwise as (13)138, but this observation is numerology and means nothing in itself; nor, that the letters, except the last one of the second syllable, yield 37. These midpoint-bars, at first sight, appear somewhat inconspicuous. Nevertheless, four characteristic and pertinent features will be identified therein. The literally exact midpoint (bar-wise) of the completed fugue clearly would be the bar line between 138 and 139.

* Regarding the voice of the alto, the bar-line 138/139 separates the note C from the note A: Henk Dieben's number alphabet yields C=3 - Trinity, and A=1--Unity. Hence, the completed fugue of 276 bars appears to be appropriately centered upon the Tri-Unity. This is as well the basis and principle for the tuning "wohltemperirt." At this point, a correlation with the 75 words of the Nachricht emerges. The representation 75=37+1+37 showed the midpoint, the word bringen, pivoted itself upon the letter N. (See the appendix for the details of the relation with tuning.) The letter N converts to 13--the juxtaposition unitas-trinitas. The Tri-Unity can be represented by a single letter N=13--but not its form 31. In the alto voice, flanking this bar-line, the notes C, A transform to 3, 1. Incidentally, that alto voice reminds us that J. S. Bach is told to have played himself in the orchestra the part of the viol.

*In the first central bar, 138, the two lower voices of bass and tenor attack 7 and 5 notes, respectively. These numbers correlate with the tuning system Werckmeister/Bach. It comprises 7 perfect fifths and 5 fifths "wohltemperirt." The Nachricht counts 75 words plus its heading.

*In the second central bar, 139, the tenor attacks 5 notes, and the alto 3. The system Werckmeister/Bach derives from the triad of C-major--center of tonality--wherein third and fifth beat in unison. In thoroughbass, these numbers 5 and 3 represent the intervals of fifth and third.

*The last manifestation of the central pivot point is perhaps the most esoteric, profound and comprehensive one. Going from bar 138 to the onset of bar 139, the tenor holds a suspension on E, whereas the bass, figure of a catabasis, falls into the F. This reminds about "Fa mi et mi fa est tota musica,"11 Bach set to his Canon BWV 1078. Here, at the partition point via the unitas of his "last fugue," Bach addresses, what represents for him "the totality of music"! Unfortunately, the utilization of this suspension, in particular at central points (unitas) has not yet been systematically investigated within Bach's compositions, such as cantatas as well. A different most characteristic setting, simply the sequence of the notes F-E within a descending scale, occurs at the exact center of the Four Duets,12 wherein Bach had musically and mathematically specified the tuning "wohltemperirt."

At this stage, of course, one might start searching across the unfinished fugue, to find further passages where the four aspects above occur simultaneously. Or else, define different criteria for midpoint-characteristics and check whether there are possibly other candidate-midpoints under such criteria. At about twice the bar number of such places identified, the completed fugue would terminate. However, I have not yet succeeded in finding any different midpoint more convincing and significant than the one indicated within the existing part of the unfinished fugue, bars 138/139.

For completing the fugue it is thus confirmed that 37 bars are missing, related to the number of the 37 final words of the Nachricht, that succeed the semicolon. The ideal length of the complete fugue amounts to 276 bars--in agreement with the outcome of Dequevauviller's ingenious intuition and despite his ambiguous reasoning.

The autograph manuscript terminates with: "NB Ueber dieser Fuge, wo der Nahme// BACH im Contrasubject// angebracht worden, ist// der Verfasser gestorben" (Upon this fugue, where the name BACH is applied in the contrasubject, the author passed away). Whilst keeping in mind the 37 final words of the Nachricht, together with the 37 missing bars of the fugue, this sentence converts via Henk Dieben's alphabet to 867, factorizing 17x3x17. On the way to this total, when summing the text across its word BACH, up to and including the letter C--center of tonality--the intermediary result becomes 266 = Das Wohltemperirte Clavier. Alternatively, according to the triangular alphabet, the factors of the total are 6657=3x7x317. Herein, 317 may be seen as 37=J. CHR., centered upon 1=unitas.

Conclusion

Having established the corrected sum 2138 via the number alphabet for BACH in the second part of the Nachricht now proves beyond doubt that the composer himself was its author. As a consequence, that message--as concerns the contents and extension of the composition--can be trusted and taken literally. Thus, the unfinished fugue does, of course, belong to the composition and the complete work terminates with the chorale. The parallelism between the 37 words of the message's last part (37=J. CHR.) and 2138 meaning BACH, can be interpreted as a profound theological statement within his last printed work--did Bach take the last 37 bars with him when rejoining Jesus Christ? Finally, again numerous allusions or references to the tuning system Werckmeister/Bach could be identified within Die Kunst der Fuga.

For its second edition, Marpurg replaced Bach's authentic Nachricht by a "Vorbericht." The latter, although not without praise, admiration and meritorious commercial and sales intentions, can be dismissed as gibberish if compared to the significance of the composer's own message: Bach's work and concepts cannot easily be improved upon! Fortunately enough, the first printed edition has survived.

Epilogue and outlook

Contemplating this article on Die Kunst der Fuga, I realize and admit that I am myself most and principally interested in the psychology--the obvious one and the one implied--of this personality of a composer/mathematician. It is hoped that by presenting paradigmatically these results, the psychological approach apt for studying musico-mathematical baroque mentality, not only Bach, but e. g., Werckmeister and Zelenka as well, is initiated. And thus, that the official and institutionalized European Bach-research can be relieved from its present deadlock.

Appendix

Applying the gematria between the semicolon at the midpoint of Nachricht to its end yielded 2138. The body text of Bach's message counts 417 letters. These digits specify the three types of fifths in the system. Therefore it is tempting to apply gematria from the onset of the 75 words--excluding the title--and check the sum up to and including the central tri-unitary letter N=13 of BRINGEN. The result is 2217. Rearranging digits will make identification obvious: 1722. This is the year Bach has dated Das Wohltemperirte Clavier, showing 1 tempering-fifth (B-F#), 7 perfect fifths and two pairs of fifths wohltemperirt (C-G, G-D and D-A, A-E). As to the factorization, 2217=3x739. Obviously, 3 means the Trinity, and writing the other factor as 7-3-9, centers 79 upon the Trinity; whereas Johann Sebastian Bach corresponds to 158, its half is 79; a representation investigated already a long time ago.13 It is worthwhile to stress that BRINGEN has at is center the letter N and this was the word at half the length of the message.

Finally, the word BRINGEN itself, at the midpoint of the Nachricht, has several remarkable properties that can best ited in form of a table. (See Figure 8.)

The seven letters of BRINGEN, according to 3+1+3=7, are centered upon the Tri-Unity, N=13, the juxtaposition of unitas-trinitas, the basis of the system wohltemperirt. The sum of the first three letters, BRI, yields 28, secundus numerus perfectus. Such numbers are made up by the sum of its parts, 1+2+4+7+14=28. Or else, 6=1+2+3, primus numerus perfectus. Werckmeister, in his treatises, quotes perfect numbers up to 33550336(!). Looking now at 28 and at the midpoint 13, permits the numerological contraction and juxtaposition to 2813, a permutation of BACH = 2138.

The group of letters GEN, 7,5,13, obviously can encode the 7 perfect fifths together with the 5 well-tempered ones by a procedure14 I have called "appearance method." Otherwise, the final letters EN, appearing as 5-13, show 53 centered upon the 1 = unitas. This may be associated with 5 = fifth in thoroughbass and 3 = third. In the C-major triad of the system Werckmeister/Bach, third and fifth beat at the unison. In analogy, similar to the exercise of this appendix, the title-word NACHRICHT itself may undergo further numerological interpretation, but this is left to the reader.n

Notes

                  *               In commemoration of the 250th anniversary of J. S. Bach's death on 28 July 1750.

                  1.              Kellner, Herbert Anton, "Johann Sebastian Bach and Die Kunst der Fuga." The Diapason, March, 2000, p. 13.

                  2.              Kellner, H. A., "How Bach encoded his name into Die Kunst der Fuge together with his tuning."     The Diapason, May, 1999, 14-15.

                  3.              Dammann, Rolf, Der Musikbegriff im Deutschen Barock Laaber 31995.

                  4.              Kellner, H. A., The Tuning of my Harpsichord. Schriftenreihe 18. Verlag Das Musikinstrument, E. Bochinsky, Frankfurt/Main 1980.

                  5.              Dequevauviller, Vincent, L'art de la fugue, un "problème algébrique." ISBN 2-9513089-0-6. Association pour la connaissance de la Musique Ancienne, 75005 Paris, 10, rue Guy de la Brosse. 1998.

                  6.              Kellner, H. A., "A Mathematical Approach Reconstituting J.S. Bach's Keyboard-Temperament." BACH, The Journal of the Riemenschneider Bach Institute, Berea, Ohio. Editor Melvin Unger. Vol. 30/1, Spring-Summer 1999, 1-10

                  7.              Kellner, H. A., "Le tempérament inégal de Werckmeister/Bach et l'alphabet numérique de Henk Dieben." Revue de Musicologie Vol. 80/2, 1994, 283-298.

                  8.              Bach, Carl Philipp Emanuel, Versuch über die wahre Art, das Clavier zu spielen. First edition I/II 1753/1762; Facsimile, Ed. Lothar Hoffmann-Erbrecht 1958, 71986.

                  9.              Hahn, Harry, Symbol und Glaube im 1. Teil des Wohltemperierten Klaviers von J. S. Bach.  Wiesbaden 1973.

                  10.           Kellner, H. A., "Review of Musique et Tempérament by Pierre-Yves Asselin." Revue de Musicologie Vol. 72/2, 1986, 294-296.

                  11.           Duparcq, Jean-Jacques, personal communication, drawing my attention to this canon's text, and that the number alphabet converts "est tota musica" to 158--as well as Johann Sebastian Bach, equal to the value for Die Kunst der Fuga.

                  12.           Kellner, H. A., Barocke Akustik und Numerologie in den Vier Duetten: Bachs "Musicalische Temperatur." In "Bericht über den Internationalen Musikwissenschaftlichen Kongreß Stuttgart 1985," Ed. Dietrich Berke and Dorothea Hanemann, Kassel 1987, p. 439-449, as well as Kellner, H. A.: "How Bach quantified his well-tempered tuning within the Four Duets." English Harpsichord Magazine, Vol. 4, No. 2, 1986(87), 21-27.

                  13.           Kellner, H. A., "Das wohltemperirte Clavier" --Implications de l'accord inégal pour l'œvre et son

autographe. Revue de Musicologie Vol. 71, 1985, 143-157.

                  14.           Kellner, H. A., "One typographical enigma in Werckmeister, Musicalische Temperatur." English

Harpsichord Magazine, Vol. 3, No. 8, 1985, 146-151; in particular p. 148.

I read with very great interest and pleasure the recent contribution by Jan Overduin to The Diapason1, "Bach and Die Kunst der Fuge." Therein, the author presented about two dozen typical examples, illustrating how the composer has interwoven the musical texture of the oeuvre with the notes of his name b, a, c, h, within the various counterpoints. (In English nomenclature, the German b is designated b-flat.) By this procedure BACH has inscribed--so to speak--in many places his signature to his compositions. Beyond this most simple form, a variety of permutations of these four basic letters can also be found, or else, transpositions to other pitches, as shown by R. Kreft,2 a comprehensive special study printed in multicolor.

For the present article, three examples from Prof. Overduin's article will be extracted and discussed. Beyond the occurrence of the name BACH in these particular musical passages, possible simultaneous allusions by the composer to his mathematical system of unequal well-tempered tuning3 will be identified. This musical temperament--due to its intrinsic mathematical nature--is necessarily based on a certain set of numbers. The rationale for the present approach to study Die Kunst der Fuge is the fact that Bach has frequently structured the form of his compositions via numbers of a set belonging to the  wohltemperirt system. From this observation originated my "Vienna manifesto" of the Bach-year 1985: to analyze Bach's works with particular attention to the aspect of numbers pertaining to well-tempering.4 Utilizing this artifice, Bach attains an elaborate unity between features of the musical form and structuring in the widest sense and the harmony of tuning--initially and nominally the harpsichord. The most specific composition for this system was, of course, Das Wohltemperirte Clavier. A harpsichord can be well-tempered in not more than 19 elementary tuning-steps.5 This is the number for the closure of the circle, and the 19 intervals are 12 fifths followed by 7 octaves in the opposite direction.

In view of the essential occurrence of the name b, a, c, h for carrying out this study, the number alphabet and its gematrial correspondences needs first of all to be introduced. Thus, the letters are numbered along the Latin alphabet from A=1, B=2, C=3,  . . . I=9, J=9 [sic],  . . . K=10,  . . . U=20, V=20 [sic],  . . . X=22, Y=23, Z=24. Expressed via that numbering, B, A, C, H will appear as 2, 1, 3, 8. The adding-procedure as prescribed by the gematria, 2+1+3+8, yields the correspondence BACH=14. Likewise, J.S. BACH will be 41, the crab or inversion of the number 14.

Now the well-tempered system will be concisely laid out, putting special emphasis on the way it will be ultimately reflected here in Die Kunst der Fuge. This temperament comprises 7 perfect fifths and 5 well-tempered ones. It derives from the central key of tonality, C-major.6 In its triad C-E-G, the enlarged third beats at the same rate as the reduced fifth--an ideal mutual adaptation. To complete the description, four well-tempered fifths ascend from c and reach the second octave of the initial third e closing this chain of fifths c-g-d-a-e. From c downwards extends a chain of six perfect fifths, reaching g-flat (f-sharp). Of course, octave-transpositions must be applied in practical harpsichord tuning wherever necessary. The last tempered fifth of the system results as B-f-sharp, closing the circle. From the third e upwards ascends the seventh and last perfect fifth e-b.

The unique and distinguishing feature of wohltemperirt is its musico-theological foundation; no other tuning has anything similar to offer. Due to the beat-rates in the triad at the perfection of the unitas,7 the system is founded upon a tri-unitarian basis. The nucleus of baroque thoroughbass is the triad, itself a symbol of the Holy Trinity. Just hearing a triad, its three components merge suavely and smoothly into an agreeable, pleasant unity.8 Furthermore, the beat ratio of 1:1 of the constituent intervals can be considered as a profound symbol of the monotheistic principle--it is here where Werckmeister's ideas on the perfection of the baroque unitas are rooted.

Returning now from theological spiritualities and mathematical ratios to the music itself, by what means could Bach reflect in a composition the numbered alphabet and the gematria? A few such examples will follow now. As concerns the numbered alphabet, for the onset of its table A=1, one may refer to the well known A-major fugue of The Well-Tempered Clavier I. Its theme starts with an isolated note a, followed by three 8th-rests. Such an incipit is highly unusual, if not bizarre, and correlating with the table's A=1 appears natural and not far fetched. For the correspondence BACH=14, the C-major fugue's theme--as well as that of B-major--starts with 14 keystrokes.9 Within Die Kunst der Fuge itself, following the first four pieces, already the theme of contrapunctus 5 (and others) count 14 keystrokes. For the gematrial correspondence J. S. BACH=41, not later than the initial two keystrokes d, a, of Die Kunst der Fuge--set in d-minor--show 41 if juxtaposed.

Now it will be indicated how musical structures can convey hints or allusions to the well-tempered system. It is based upon the ratio of the unitas between the beats of the tempered third and fifth, 3 and 5 in thoroughbass. Therefore, immediately the number 135--in juxtaposition--may be used, for instance, within the bar numbered by 135. Other possibilities may be derived from the two sorts of fifths, 5 well-tempered, 7 perfect, such as in juxtaposition 57, 75 (75 could be made up via the tri-unity as 31+13+3110), or even 577. Finally, in terms of musical notation, 5 relates  to e, 7 to g, and 3 to c. As to the number 19 and concomitant abstract structuring, looking now as an example at the B-major prelude of WTC I, it counts 19 bars, starts at bar 1913 and ends at bar 1931.11

The first extract from Prof. Overduin's article is contrapunctus 4 (BWV 1080, 4), measures 135 to 138, page 15 in Davitt Moroney's edition from G. Henle.12 Starting from bar 135 (unitas-third-fifth) the tenor sounds BACH, rhythmically comparable to a sigh. The fugue terminates at 138, which incidentally corresponds to ACH, the final letters of the composer's name; in German a sighing exclamation. Perhaps the terminating pedal on d (D=4) through the last four bars may be related to the four letters of BACH. (See Example 1.)

The second example, page 46 in the Moroney/Henle edition (BWV 1080, 11), concerns contrapunctus 11, bars 90 and 91. (See Example 2.) As Prof. Overduin points out, the alto introduces by theme three the notes B, A, C, H, but he mentions that Tovey rejected this as an allusion to BACH because in fact, it is B-A-C-C-C-H sounding here. However, Tovey could at his time not be aware of Bach's tri-unitarian temperament and thus, necessarily failed to understand the significance of A,C,C,C: 1,3,3,3 in numbers. As much as within B-AC-H, 2-13-8, the number 28, secundus numerus perfectus is centered upon 13, unitas-trinitas, the present extended theme 2-1333-8 includes three times the number 3. The frame still remains B and H. An essential factorization holds, 1333=31*43: the prime numbers 31, trinitas-unitas and 43 = CREDO (3+17+5+4+14)--a tri-unitarian Credo! Starting with the second half of this bar and counting from the bass fundament upwards, presents the notes e, a, c, thus 513, nothing else than a permutation of 135. This is interpreted as fifth 5, unitas and third 3 in thoroughbass. The crucial bar in this example is 91--the crab or inversion of 19--by which number of elementary steps the circle of fifths will close. Working backward in this bar to the second quaver shows a, a, c, thus 113: a numerical triptych of unitas and trinitas. This measure 91 not only sounds BACH in the alto, but its onset reads d, g, b, converted to numbers 4,7,2.

As concerns 472, Bach was certainly intimately familiar with the notion of permutations, thinking for example, of his choral fugues or certain three-part inventions. Thus, just from a cyclical permutation of 472 results the number 247 (=13*19). According to the baroque gematria, 247=MUSICALISCHE TEMPERATUR which is the title of Werckmeister's classical treatise, 1691. Furthermore, 247=112+135 holds additively, but the implications of such observations cannot be detailed here and these results were published elsewhere already some time ago.13

The third example still deals with contrapunctus 11, bars 144 to 145, page 48 in the Moroney/Henle edition. (See Example 3.) There the alto and treble sound BA-CH and the bass and tenor in the second quaver of 144 present G, E, converting to 7 and 5, the numbers of perfect and well-tempered fifths. The bass, in fact, now sounds G,G,G,E, in numbers 7775. It may also be mentioned that contrapunctus 11 starts in a Trinitarian fashion by three bars identically structured, with eighth-note rests on the downbeat and 3 subsequent eighth notes; one has 3+3+3=9, trias trinitatis per additionem.

Finally, a typical manifestation of the unitas, a determining and crucial element in Bach's structuring of his compositions can be pointed out at this occasion. The contrapunctus 11 extends over 184 bars, an even number. The midpoint therefore falls upon the bars 92 and 93, see the preceding example. The bar 93 (=3*31, tri-unitary factorization!) sounds, from the fundament of the bass upwards, a, c, e; in numbers 1,3,5: unitas, third and fifth in thoroughbass--on  the dominant of d-minor. In the central triad of C-major of wohltemperirt, third and fifth beat at the unison! Hence, this piece is obviously pivoted symmetrically upon the very nucleus of the well-tempered musico-mathematical system.

The considerations above represent a corollary to the examples of the underlying article in The Diapason. As to the aspects described and analyzed, there is no pretension whatsoever to be exhaustive. Rather, the purpose is, hopefully, to be thought provoking, to stimulate and encourage further, more systematic and complete investigations into the direction outlined here--as much as the article published by Prof. Overduin has led to the present study.

After having reconstituted the well-tempered system Werckmeister/Bach initially in 1975,14 it was gratifying for me to see how organ builders have taken up and followed the ideas, appreciating the technological and musical qualities of this baroque temperament. These builders include Rudolf von Beckerath, John Brombaugh & Associates Inc., T. S. Buhr, Paul Fritts & Co., Gerhard Grenzing, Otto Hoffmann Organs, Claude Jaccard, Yves Koenig, Michael Korchonnoff, Dominique Lalmand, Gebr. Oberlinger, Martin Pasi, Richards, Fowkes & Co., Charles M. Ruggles, Taylor & Boody, George Westenfelder, Karl Wilhelm, Hellmuth Wolff and Munetaka Yokota.

On these organs, tuned accordingly, many distinguished musicians have performed and recorded, including Martin Balz, Luc Beauséjour, Jonathan Biggers, Gavin Black, Robert Clark, David Dahl, George Edward Damp, François Espinasse, Bernard Foccroulle, Martin Gester, André Isoir, Calvert Johnson, Donald Joyce, George Ritchie, David Rothe, Wolfgang Rübsam, Yasuko Uyama-Bouvard and others.

A discography as at that time I have published in The Tracker.15 Further references to analyses of Bach's compositions are contained--together with a heuristic derivation of the well-tempered system--in the Blankenburg-Michaelstein symposium proceedings.16 For those interested in more musicological details, a bibliography is also contained within my lecture publication on historical temperaments, held at the symposium in the Vienna Hochschule für Musik und Darstellende Kunst.17.

But the basis for all improvisation must be preparation. If I haven’t prepared, I can’t improvise. If I’ve made careful preparations I can always improvise. 

—Ingmar Bergman, 1968

The question of fugal improvisation in the Baroque era has been raised in the pages of musicology literature more than once.² It still remains topical today; yet in the practice of Baroque improvisation, the improvisation of fugue has rarely become an object of independent study. Besides William Renwick’s book, The Langloz Manuscript: Fugal Improvisation through Figured Bass (2001), it is difficult to name any widely known work that is specifically dedicated to the art of fugal improvisation in the Baroque era.³ Much valuable and interesting information about this performance practice of baroque musicians is scattered throughout various books and articles, whose subject matter is not even directly related to improvisation.

The present article therefore aims: 

1) to summarize the existing research on partimento practice;

2) to describe all the stages of fugal improvisation, beginning with the mastery of separate elements and finishing with an organization of the whole, as recorded in German sources of the first half of the 18th century.

Introduction 

Today the ability of an academically trained musician to create “on-the-fly” is thought of as exceptional—for the gifted only. Yet it is well known that in the Baroque era every professional musician was expected to possess this “gift.” Within the rich diversity of improvisational genres and forms that made up the standard set for which a Baroque improviser was to be prepared, fugue held the greatest place of honor.

At that time it was not just the great musicians who were skilled at improvisation; every church organist had to be able to improvise a fugue on a given theme. . . . The ability to improvise fugue was considered a requirement for every serious musician to such a degree then that the lack of that skill could serve as reason for ridicule. . . . And, although the testing of organists did not always include fugue improvisation, both Mattheson and Adlung think that no one should be taken as an organist who has not proved his right to such a post through the improvisation of fugue.⁴

In the 18th century if you couldn’t improvise you couldn’t call yourself a keyboard player. Worse than that, you couldn’t get a job, since all organist auditions required extemporaneous performance of a fugue on a given subject.⁵

Truly, the ability to improvise fugue was a necessary skill for organists, because a fugal statement of musical material was stipulated by the very program of the liturgical service. Beginning in the second half of the 17th century, the role of the organist, on whose shoulders rested the burden of the musical life of the church, grew remarkably.⁶ The organ, which had at one time humbly accompanied church ritual, became a most important attribute of the church service—almost its main participant. This was especially true in the northern regions of Germany, where the organ gained such acoustic strength and richness of register that it became like “a second minister,” and the musical compositions that it “delivered” were self-contained “texts” addressed to the congregants. Mattheson emphasized that fugal presentation of the chorale subject on the organ helped “to arouse reverence within the listeners.”⁷

For musicians in the secular sphere, fugal improvisation as a skill was not as necessary as it was for church organists, but the ability, nevertheless, was always appreciated. In the circle of experts and enlightened amateurs, fugal improvisation on a subject proposed by someone among those present could become one of the most intriguing and entertaining elements of a musical program. Success in such improvisation provided the performer with the established reputation of master of the highest order (a reputation that could help in a further promotion).

Although fugal improvisation was a widespread practice among Baroque musicians, we are forced to gather information on its technique literally in bits and pieces. As early as 1702, Andreas Werckmeister, in his treatise Harmonologia musica, points out the reason: “many musicians are secretive and reticent with their knowledge.”⁸ Possibly, musicians divulged their knowledge about improvisation very unwillingly because they considered it a unique commodity, providing a constant supply of students. Perhaps they did not wish to destroy the myth of the divine origin of the gift of improvisation. In any case, even in treatises that are dedicated specifically to improvisation and fantasieren, there are no concrete instructions that would allow us today to understand how fugue was improvised.⁹

Nonetheless, some secrets of Baroque fugal improvisation have already been revealed by scholars. David Ledbetter writes about one of them:

By the early eighteenth century, instruction in fugue in Bach’s tradition grew out of the figured bass, rather than contrapuntal treatises, and so was approached as an improvised genre. The technique of this was practised by using fugato movements expressed as figured basses, called in Italian partimento fugues.¹⁰

To the uninitiated musician such a statement may seem paradoxical, since according to our notion fugue and figured bass represent distinct types of musical thinking and observe a different tradition of notation. However, the discovery during the last decade of a large number of examples of so-called partimento fugue or thoroughbass fugue shows that improvisation of fugue during the Baroque epoch—just like the improvisation of homophonic forms—actually had its foundation in the practice of figured bass.¹¹ The detailed study and comparison of these examples, strengthened by the testimony of contemporary treatises, allow us to take another step forward on the path to understanding the Baroque technique of fugue ex tempore.

That the overwhelming majority of improvised fugues during the Baroque epoch were thoroughbass fugues can be explained from the point of view of psychology. The texture of a “contrapuntal fugue” (i.e., polyphonic texture) is formed by combining individualized melodic lines, each vying for our attention. In contrast, the texture of thoroughbass fugue is predominantly two-dimensional—that is, it can be clearly divided into the leading voice and a complex of accompanying voices. Consequently, improvisation of a multi-part “contrapuntal fugue” necessitates the division of attention into three or more channels, whereas performance of a multi-part thoroughbass fugue demands division into just two. Experience shows that the attention of even a well-prepared musician is capable of maintaining control over only two (a maximum of three) simultaneously proceeding streams of information.¹² As such, for objective (psycho-physiological) reasons, improvisation of thoroughbass fugue is attainable for a broad mass of musicians, whereas improvisation of a multi-part “contrapuntal fugue” is negotiable to a rare few.¹³ 

Having touched on the issue of the limits of human attention, which is so relevant to musical improvisation, it would be remiss to ignore the opportunity to quote Sergey Prokofiev, in an interview published by the New York Times in 1930:

Three melodies remain about the limit that the average ear can grasp and follow at one time. This can be done when the melodies are clearly sounded and contrasted in pitch and tone color. For a short time the ear may perceive and assimilate the effect of four different parts, but this will not be long continued, if the four parts, or melodies, are of equal importance. Listening to a four or five or even six-part fugue, the ear is conscious, possibly, of the presence of all the voices, but it only perceives and follows precisely the most important of the melodies being sounded. The other parts fill in, enrich the musical background and harmony, but they become as blurred lines of the picture. They are not clearly recorded in the listener’s consciousness as separate melodic strands in the tonal fabric. This being true, it behooves the composer to realize that in the polyphonic as well as in the structural sense he must keep within certain bounds.¹⁴

Such is the point of view of a professional musician who possessed extraordinary musical faculties. As for specialists in the fields of psychology and physiology, they have yet to come to a single opinion concerning the volume and capabilities of human attention.

Analysis

The modern theory of improvisation is based on these principles: 1) “improvisation is based on memory” and “the improviser does not create the material, but builds it from prepared blocks, from long-memorized musical segments”;¹⁵ and 2) the improviser always works from a given model.¹⁶ 

What were the building blocks that Baroque performers utilized in the process of fugue improvisation? In what sequence could they combine them? To answer these questions, let us turn to concrete musical material.¹⁷

The overwhelming majority of German samples of thoroughbass fugue follow strophic form in their composition.¹⁸ In addition, organization of the musical material inside the strophes is very often based on the typical Baroque-era structure of “head and tail,” where the role of the “head” is played by a group of statements (more rarely by a single statement) of the subject and the role of the “tail” by sequence based on standard harmonic formulae of thoroughbass. The conclusion of each strophe is marked by a cadence. Such is the method used by Kirchhoff, for example, in his C-major fugue from L’A.B.C. Musical (c. 1734), which clearly presents three strophes (Example 1):

Strophe 1 includes five statements of the subject (bars 1–9), a 2–6 sequence (bars 9–11), and a 7–6 cadence (bar 12);

Strophe 2 includes two statements of the subject in the upper part in immediate succession (bars 12–15), a statement in the bass (bars 16–17) and the 2–6 sequence already used in strophe I (bars 18–20), and a 7–6 cadence (bars 20–21);

Strophe 3 contains a statement of the subject in the bass (bars 21–22), a 2–6 sequence that shifts to 7–7 (bars 22–25), and the more explicit 5–6/4–5/3 cadence (bars 25–26).

The structural similarity among the strophes is evidence of the improvisatory nature of thoroughbass fugue, the result of work that uses a single model. It was specifically the strophe that served as the universal compositional unit, by which through duplication the improviser assembled his fugue. The number of strophes was varied, according to how long the improvisation should last. The structure of the strophe, though, did not vary. In this way the improviser’s task was to quickly and neatly fill out this preassembled structure with concrete musical material.

Obviously, the improvisation of a fugue had as its starting point the harmonization of the chosen or suggested subject. A harmony, as a rule, was kept for all multi-part statements of the subject, becoming, might we say, a retained “counter-harmony” (Gegenharmonie).¹⁹ Changes to the harmonization were made only in cases where a tonal answer was necessary. Frequently, even the counterpoint to the answer (the first countersubject) was drawn out of this same “counter-harmony.” This is easily affirmed by noting the numeral for the harmonic intervals between the answer and countersubject and then comparing the result to the author’s own figures for analogous multi-part statements (Example 2).²⁰ 

In many samples of thoroughbass fugue, all entries of the subject are concentrated at the beginning of a strophe. Following one after another without dividing episodes, the statements form a compact thematic group that serves as an entire syntactic unit larger than just a single statement. The tendency toward an increase in the hierarchical degree of unit complexity is another specific quality of improvisatory technique. The combination of smaller syntactic units into larger ones helps to expand the general volume of information accessible within short-term memory.²¹

The similarity among the strophes of thoroughbass fugue is also increased by the uniformity of the order of entries. In all strophes, a descending order of entries of the parts predominates as the most convenient and intrinsic with respect to technical considerations and notation of thoroughbass.²²

The next syntactic unit of the strophe, following the group of statements, is the episode. This section of the fugue was the most comfortable for the improviser, since here he could use patterns that he had learned. Judging from extant samples of thoroughbass fugue, episodes most often consisted of sequential repetition of one, more rarely two, harmonic formulae stereotypical to thoroughbass. This observation is supported by the theoretical works of that time. As such, to attain success in the improvisation of fugue, Philipp Christoph Hartung, in Musicus Theoretico-Practicus (1749), recommends learning entire musical progressions, which one should be able to freely and confidently play from memory, and not just read from sheet-music.²³ Many of the fragments he suggests are nothing more than textural elaborations of standard thoroughbass sequences. The thoroughbass nature of Hartung’s sequences appears especially clear if we extract their harmonic scheme and supply it by figures (Example 3).

Playing sequences had to become an automatic skill, something that was simply “in the hands” of the performer. The automation of playing skills allowed the improviser to free his attention considerably so as to be directed instead to solving upcoming tasks. In other words, while the hands played out the episode, the mind could be planning out the next set of operations. Given this, the hands had to be able to play for as long as was necessary for thinking out. For this reason, the inert nature of sequential development was not a detriment to fugue played ex tempore. The existing unspoken rule in musical practice that the number of segments in a sequence (in the case of exact repetition) should not exceed three was not observed too strictly during the fugue improvisation. Theoretically, there could be any number of segments in a sequence, as it was defined less by artistic needs than by technical ones. In practice, episodes, composed of sequences made of four to five segments, were the norm for thoroughbass fugue.

The unity of thematic material was not also a problem for thoroughbass fugue. The episode could smoothly continue the subject, but could also introduce  new musical material. In any case, the primary task of the improviser in moving from one syntactic unit to another was to transition as naturally as possible. It follows then that the greater the active memory capacity of the performer and the more formulae he could recall and have “in his hands,” then the higher the likelihood of attaining agreement of intonation between the suggested subject and episodes selected from among those prepared during the process of his musical training. The ability to competently use these preparations from “homework assignments” was very likely a basic craft known to the improviser.

The degree to which the improviser relied upon such materials prepared in advance can be judged by examining, for example, the B-flat-major fugue from Johann Caspar Simon’s collection Leichte Præludia und Fugen (1746). Of its total 37 bars, 20.5 bars (i.e., more than half) are based on material connected neither with the fugue subject, nor with its countersubject. The especially obvious “home preparations” reveal themselves in the second half of the fugue, which is made up of four autonomous sections resembling, in their function, additions in the tonic key (Example 4). At first, Simon builds a sequence on the harmonic formula 7–7, embellishing the bass line with melodic figuration. He then builds a second sequence on the harmonic formula 2–6 in strict chordal texture. Further, he inserts a toccata-like fragment pulled from the fugue’s preceding prelude, a fragment that is also in its nature a sequence. Finally, he concludes the piece with a decisive cadence in solid chordal presentation (Grave). Comparing the “specific gravity” of thematic and non-thematic material in Simon’s fugue, the conclusion suggests itself. Essentially, if the improviser were not restricted by concrete devices of thematic work, then the entire fugue, excepting statements within the exposition, could be designed from elements prepared in advance.

Judging by some samples of thoroughbass fugue, the “stock” material could penetrate straight into the group of statements, replacing separate statements or pulling them out. For example, in Fugue no. 21 (F major) from the Langlo(t)z Manuscript, the second strophe begins not with the restatement of the subject, but with non-thematic counterpoint, and only the bass part enters with the theme (Example 5).

In the D-minor Fantasy from the Mylau Tabulaturbuch, a straightforward “home preparation” in the form of a typical sequence 6/5–5/3 appears in the first strophe between the fourth and fifth statements (Example 6a). Viewed separately, this fragment appears optional—since the other statements work successfully without it (Example 6b).

The energy expended by a performer for fugue improvisation could be conserved by using the same episode for various strophes. This repetition could be identical, but it could also be modified by means of various textural clichés. For example, the second and third episodes of the anonymous G-major Prelude (which is in fugue form) from the Mylau Tabulaturbuch are based on a single harmonic formula, the 7–7 progression, though the shapes of their texture are distinct. In the first case, the lower voice is diminished; in the second, the pair of upper voices (in regular imitative counterpoint). Incidentally, this prelude demonstrates direct application of Hartung’s aforementioned recommendations: the prelude’s second episode (Example 7a) differs from his sequence shown in Example 3a only by key.

The existence of a single stockpile of thoroughbass harmonic formulae inevitably led to the appearance of universal sequences that traverse the pages of thoroughbass literature from one composition to the next, regardless of authorship. Comparison of the episode sections of numerous thoroughbass fugues makes clear that of the great variety of harmonic formulae offered in contemporary thoroughbass treatises and manuals, a precious few sequential patterns predominate: 7–7, 6/5–5, 6–6, 4/2–6.

The manner of sequential motion also deserves special comment. In many samples of thoroughbass fugue, the episodes are based on diatonic sequences that descend stepwise down the scale. On one hand, descending motion step-by-step possesses a certain inertness, which under the conditions of improvisation (i.e., mental and psychological tension and temporal deficit) just plays into performer’s hands. On the other hand, diatonic motion step-by-step provides the sequence freedom in the selection of the target tonality. In reality, the great tonal mobility is hidden in diatonic sequence; a trajectory of such a sequence could be easily and organically turned at any moment into one of closely related keys. Here is a small experiment: the test of the key possibilities of a 2–6 sequence from the second strophe of the C-major fugue from Kirchhoff’s L’A.B.C. Musical (Example 8).

As these examples demonstrate, it is possible to conclude the sequence in any closely related key without applying much effort. Understandably, the target key will influence the length of the sequence. Here it is very important not to lose a sense of balance and good measure. Although the versions represented in Examples 8e and 8f are technically no different than the remaining ones, these two are much less suited to actual artistic use due to their extended monotony. Should Kirchhoff have needed, in the process of improvisation, to expand the fugue by adding another strophe, he likely would have followed version c) or d) in place of the cadence on the C-major tonic.²⁴

Once the fugue’s continuation took a concrete shape in the mind of the improviser, he could stop the potentially endless development of a sequence via the most convenient cadential formula. The playing of cadences (as well as sequences) in any key of the instrument—literally, with closed eyes—was also a necessary skill for every professional keyboardist of the Baroque era. In the opinion of many 18th-century musicians, cadential formulae are the basis, the foundation of thoroughbass; it is specifically this skill that forms the starting point for practical study of the trade. The number and types of cadential formulae varies with each source. The Precepts and Principals (1738) attributed to Johann Sebastian Bach, for example, count seventeen patterns among the most frequently used (Example 9).

Immediately following the cadence, occasionally commencing upon its final tones, the new strophe begins and all events of the described process are repeated. The similarity of the strophes imparts to the unfolding of the fugue’s form a character of repeated expositions. The formal approach to realization of the strophic scheme inevitably aroused the feeling of monotony, which, naturally, stirred up criticism from contemporaries. Mattheson, who regularly attended testing of organists, wrote:

One should restrict oneself even less to the practice of some organists, who first quite respectably, without the slightest embellishment, perform the theme four times through on the entire keyboard in nothing but consonances and pastoral thirds; then begin again just as circumspectly with the consequent from its beginning; always producing the same tune; interposing nothing imitative or syncopating; but constantly only playing the naked chord, as if it were a thoroughbass.³⁰

Here are the impressions produced on Marpurg by a certain organist who attempted to play fugue ex tempore:

Someone often has the good intention to make it better. But what does he do? He slams out the figured bass, and this is terrible to hear. There are no suspensions which make the harmony pleasant, fluent and coherent. It is a jolting harmony. One hears no stretto, no motivic development of the theme. There is no order, and the number of voices one can only surmise at the end, when as, per forma, it ought to be clear directly after the first exposition of the theme through different voices of the fugue. The theme is will never be wisely advised in the middle voices. You only ever hear it above or below—as one hand accompanies another as in an aria. One never hears the theme as comfortable, nor at the appropriate time, expressively and sensitively for the mind and the ear in a sustained and affecting way. It is but a senseless din and tumult—not to mention the discord within the harmony.³¹

The picture described by Mattheson and Marpurg was characteristic of improvisations by mediocre organists. The more talented and gifted performers avoided precise repetition of strophes and brought to each new strophe a certain degree of newness, to which extant samples of thoroughbass fugue eloquently testify. In addition to the aforementioned tonal reinvention of strophes, one can quite often find such methods of refashioning as introducing a new counterpoint to the subject, “register leap” (i.e., a skipping of two or more register pitches where the subject can enter), and the use of stretto in the final strophe.

Although the opinion does exist that “the part of the fugue related to statements of the subject was created during improvisation,”²⁵ there is reason to suggest that even during these sections the performer could sometimes refer to prepared material. Judging from extant samples of thoroughbass fugue, the study of fugal improvisation included not just the regular practice of sequential progressions and cadences, but the development of a definite set of concrete approaches to working with the most common types of subjects. Describing the demands placed on candidates for the vacancy of organist at the Hamburg cathedral, Mattheson noted: 

I don’t consider it art to concern people [organists] with unknown themes; rather, it is better to take something well-known and flowing in order to work it out even better. That is what matters, and the listener will like it better than some chromatic piddling about.²⁶

If one allows for the possibility that Mattheson was not alone in this opinion, then the chances of being tested on a subject built of familiar melodic patterns, or even on a known subject, were not so small, and thus the entire improvisation could come down to a combination of prepared materials.

Let us recall, for example, the subject that King Frederick the Great suggested to J. S. Bach for an improvised fugue in Potsdam (Example 10). It is not known with certainty whether Frederick himself composed this subject or borrowed it, but judging by its melodic profile, the monarch had chosen to demonstrate to Bach his knowledge in the “learned style” (gelehrter Stil).²⁷ It must be noted that the subject contains four thematic elements, and all of them are conventional within Baroque style: a) movement in the tonic triad, b) a jump of a seventh (saltus duriusculus), c) descending chromatic movement (passus duriusculus), and d) melodic cadence. Any Baroque musician would certainly know these melodic patterns, along with the methods of their elaboration within a fugue. The elements listed here are well represented both in didactic and artistic samples of thoroughbass fugues, and what is especially important is that their musical realization (counterpoint, harmonization) often coincides.

Depending on the conditions of improvisation, “home preparations” could have various degrees of concretization. In those cases where a fugue was improvised on the occasion of a public challenge or competitive auditions, the performer had to hold his prepared materials in his memory. In everyday practice, however, it was acceptable to use the preparations written out on paper. We find examples of such preparations in a Daniel Magnus Gronau manuscript, which is held today in the Library of Polish Academy of the Sciences (Gdansk) as MS. Akc. 4125.²⁸ This manuscript contains 517 (!) sets of preparatory sketches for fugue improvisation in all twenty-four keys. Each set holds three thematic records, written one below the next on individual staves (Example 11). On the upper staff in soprano clef, the subject with figures is written out, and the beginning of the answer with countersubject is outlined in small notes.²⁹ On the second staff in bass clef, the counterpoint to the subject with figures is recorded. On the third staff, also in bass clef, the answer with figures is fixed. In this way, every set encompasses all necessary material for planning any statement of the subject, whether alone or with multiple voices, whether in the tonic or in the dominant.

Thanks to such preparations, the process of fugue improvisation is considerably simplified, since the need to search for a harmonization of the subject, a counterpoint to it, and a suitable answer is taken care of. Essentially, the performer must only care for the episode material, and the fugue, necessary for the church service, is ready.

In summary, the improvisation of fugue during the Baroque epoch was not necessarily the spontaneous nor extemporaneous fruit of inspired fancy. Much more often it was soundly prepared and planned on all levels: from the syntactic to the compositional. Even before the start of improvisation, the performer could clearly imagine the compositional structure that he must fill out using his musical material, the bulk of which could be prepared during “home” practice. One of the most widespread compositional models was strophic form, where the structure of each strophe had identical organization and included three syntactic units: the group of statements, the sequential unfolding, and the cadence. As a result, the entire improvisation could be boiled down to finding the right harmonization for the given subject and thinking up a tonal structure for the strophes; all the rest—textural formulae, cadences, sequences—the performer took from his memory practically in ready form.

Postscript

It stands to reason that the strophic form described in this article was not the only compositional model used for fugal improvisation during the Baroque. The discovery of this model, though, in other improvisatory genres of the Baroque era gives reason to consider it as universal within the improvisation practice of that time.

There is reliable evidence that the strophic form was purposefully worked out in the process of musical training. For example, Precepts and Principles contains a set of fourteen keyboard exercises for mastering the harmonic formulae most common to thoroughbass. Surprisingly, all these exercises are precisely identical in form—all are strophic (Example 12).

The outer strophes are in the tonic, while the central ones are in the closely related keys (in dominant and parallel). It is not difficult to imagine how many distinct figuration preludes could be created on the basis of only one model, varying merely harmonic content and textural formulae.³² If one involves methods of structural transformation (extension or compression of strophe), then the number of variants is multiplied.

Examples of such preludes can be found among the sources discussed in this article. Thus, in analyzing some pieces from the Langlo(t)z Manuscript or Kirchhoff’s L’A.B.C. Musical, one gets the impression that the authors had the structure of Bach’s exercises specifically in mind while they composed, so strong are the similarities. The C-minor Prelude from the Langlo(t)z Manuscript, for example, differs from Bach’s exercises due only to one additional strophe and short melodic links between the strophes (Example 13). The F-major Prelude from Kirchhoff’s L’A.B.C. Musical also contains an additional strophe, but the development within the third and fourth strophes is dynamicized thanks to structural transformations: the sequential development is truncated in the third, and the “head” motive is withdrawn in the fourth (Example 14).

The list of works of an improvisatory character that have strophic form with variations of its solutions can be further extended, but this would be a topic for a separate article. ν

The list of German sources, containing samples of thoroughbass fugue

“39. PRAELUDIA et FUGEN del Signor Johann Sebastian Bach” (Staatsbibliothek zu Berlin—Preussischer Kulturbesitz, Musikabteilung; shelf mark: Mus. ms. Bach P 296). Published in The Langloz Manuscript: Fugal Improvisation through Figured Bass, With Introductionary Essay and Performance Notes by William Renwick. (New York: 2001), pp. 35–187.

“Des König[lichen] Hoff-Compositeurs und Capellmeisters ingleichen Directoris Musices wie auch Cantoris der Thomas-Schule Herrn Johann Sebastian Bach zu Leipzig Vorschriften und Grundsätze zum vierstimmigen spielen des General-Bass oder Accompagnement. für seine Scholaren in der Music. 1738” (Brussels: Bibliothèque du Conservatoire royal; shelf mark: mr. FRW 27.244). Published in J. S. Bach’s Precepts and Principles for Playing the Thorough-Bass or Accompanying in Four Parts, Leipzig, 1738, translation with facsimile, introduction, and explanatory notes by Pamela L. Poulin. (Oxford, 1994), pp. 41–45.

Händel, Georg Friedrich. Aufzeichnungen zur Kompositionslehre: aus den Handschriften im Fitzwilliam Museum Cambridge (Composition Lessons: from the Autograph Collection in the Fitzwilliam Museum Cambridge), Hrsg. von Alfred Mann. Leipzig: Veb Deutscher Verlag für Musik, 1978. S. 53–70 (Hallische Händel-Ausgabe: Kritische Gesamtausgabe. Suppl. Bd. 1). Republished in Continuo Playing According to Handel: His Figured Bass Exercises, With a Commentary by David Ledbetter (Oxford: Clarendon Press, 1990), pp. 44–61.

Heinichen, Johann David. Der General-Bass in der Composition. Dresden, 1728, S. 516–520.

Kellner, Johann Christoph. Grundriss des Generalbasses. Op. XVI. Erster Theil. Cassel, [1783], S. 41–45.

Kirchhoff, Gottfried. L’A.B.C. Musical (Amsterdam [c. 1734]), 34 S. Republished as Kirchhoff, Gottfried, L’A.B.C. Musical, Hrsg., kommentiert und Generalbaß realiziert von Anatoly Milka (St. Petersburg: Musikverlag “Compozitor,” 2004), XXVIII, 104 S.

Niedt, Friedrich Erhardt. Musicalische Handleitung. Erster Theil. Handelt vom General-Bass, denselben schlecht weg zu spielen (Hamburg, 1700), Cap. X. Republished as Niedt, Friedrich Erhardt, The Musical Guide, Parts 1 (1700/10), 2 (1721), and 3 (1717), translated by Pamela L. Poulin and Irmgard C. Taylor; introduction and explanatory notes by Pamela L. Poulin (Oxford: Clarendon Press, 1989), pp. 48–49.

“Pral: Kirchhoff” (Staatsbibliothek zu Berlin—Preussischer Kulturbesitz, Musikabteilung, Mus. ms. 11605), published in Kirchhoff, Gottfried, Prelude and fugue for organ from the manuscript Mus. ms. 11605: first edition, edited and with a preface and commentaries by Maxim Serebrennikov (St. Petersburg: Polytechnical University Publishing House, 2009), 16 p.

Simon, Johann Caspar. Leichte Praeludia und Fugen durch die Tone: C. D. E. F. G. A. B. dur (Augsburg [1746]), 14 S.

Simon, Johann Caspar. Leichte und wohlklingende Praeludia und Fugen durch die Tone: C. D. E. F. G. A. H. moll (Augsburg [1747]), 14 S.

Simon, Johann Caspar. Musicalisches A. B. C. in kleinen und leichten Fugetten (Augsburg, 1749), 24 S.

“TABULATUR Buch 1750” (Mylau, Archiv der Evangelisch-lutherischen Kirchgemeinde; shelf mark: MS H 3a). Transcribed in Shannon, John R., The Mylauer Tabulaturbuch: a Study of the Preludial and Fugal Forms in the Hands of Bach’s Middle-German Precursors. Ph.D., Music, University of North Carolina at Chapel Hill, 1961. Vol. 2, iii, 184 p.

Notes

1. I wish to express my deep gratitude to Prof. David Ledbetter (Royal Northern College of Music), who read the final draft of this article and kindly provided me with helpful comments and constructive suggestions.

2. The topic has been actively discussed especially in the last two decades in connection with awakened interest in the Italian improvisational practice of partimento, which spread throughout Europe in the 18th century. Currently the study of partimento is gaining incredible momentum. The most comprehensive study of this field at the moment is Giorgio Sanguinetti’s book The Art of Partimento: History, Theory, and Practice (New York: Oxford University Press, 2012).

3. Although Renwick’s book contains a special subheading, Fugal Improvisation through Figured Bass, he does not treat the actual process of improvisation. His work is not a theoretical study about fugal improvisation, but an anthology of authentic musical samples for practical mastery of this skill. In fairness, the article “On the fugue improvisation” by the Russian musicologist Sergey Maltsev also should be mentioned: Sergey Maltsev, “Ob improvizacii i improvizacionnosti fugi,” in Teoriya fugi: sbornik nauchnish trudov, otv. red. A.P. Milka (Leningrad: Izd-vo LOLGR, 1986), pp. 59–60. Unfortunately, this work containing many valuable observations about the process of fugal improvisation, because of a language barrier, did not gain wide circulation.

4. Maltsev, “Ob improvizacii i improvizacionnosti fugi,” pp. 59–60.

5. David Yearsley, “Spontaneous fugue,” in Early Music, 2001, Vol. XXIX (3), p. 452.

6. See Marina Nasonova, “Prakticheskaya deyatelnost severonemetskogo organista XVII veka,” in Starinnaya muzyka: praktika, aranzhirovka, rekonstrukciya: Materialy nauchno-prakticheskoy konferencii (Moscow: Prest. 1999), pp. 117–128.

7. Johann Mattheson, Grundlage einer Ehren-Pforte (Hamburg, 1740), S. XXXIII, § 48. Based on the study of ecclesiastical protocols, Reinhard Schäfertöns concluded that the free prelude and the organ chorale prelude and fugue were central points of organ playing at the time of worship (Reinhard Schäfertöns, “Die Organistenprobe— Ein Beitrag zur Geschichte der Orgelmusik im 17. und 18. Jahrhundert,” in Die Musikforschung, 1996, 49, Jg. Hf. 2, S. 143).

8. “Denn viel Musici sind heimlich und rahr mit ihren Wissenschaften,” Andreas Werckmeister, Harmonologia musica (Franckfurth und Leipzig, 1702), S. 95.

9. In Part I of his Musicalische Handleitung (1700), F. E. Niedt promises to give a “proper instruction on how Fugues are to be improvised” in the next parts (Cap. X). Unfortunately, his death prevented him from fulfilling his intention.

10. David Ledbetter, Bach’s Well-tempered Clavier: The 48 Preludes and Fugues (New Haven and London: Yale University Press, 2002), p. 99.

11. For more details about the difference between the terms partimento fugue and thoroughbass fugue, see Maxim Serebrennikov, “From Partimento Fugue to Thoroughbass Fugue: New Perspectives,” in BACH: Journal of the Riemenschneider Bach Institute, vol. XL, no. 2 (2009), pp. 22–44.

12. It is also important to realize that there is a notable difference between the resources demanded for perception of information as opposed to its creation (which is precisely what improvisation requires). The latter takes much more energy, and therefore, resources for attention are more quickly expended.

13. One musician alive today who possesses a phenomenal gift for improvising in any style and genre is Richard Grayson. Some of his improvisations (including fugue) on a subject proposed by an audience can be viewed on YouTube.

14. From an interview with Olin Downes, in New York Times, February 2, 1930, Arts & Leisure, p. 112.

15. Mikhail Saponov, Iskusstvo improvizatsii: Improvizatsionnye vidy tvorchestva v zapadnoevropejskoj muzyke srednikh vekov i Vozrozhdeniya (Moscow, 1982), p. 57 [in Russian]. Similar statements can be found also in Maltsev, “Ob improvizacii i improvizacionnosti fugi,” p. 6; David Schulenberg, “Composition and Improvisation in the School of J. S. Bach,” in Bach Perspectives I, 1995, p. 5; William Renwick, Analyzing Fugue: A Schenkerian Approach (Stuyvesant, NY: Pendragon Press, 1995), p. 17; Pamela Ruiter-Feenstra, “J. S. Bach and Improvisation Pedagogy: Extemporaneous Composition,” in Keyboard Perspectives II (2009), ed. by Annette Richards, p. 43; Michael Richard Callahan, Techniques of Keyboard Improvisation in the German Baroque and their Implications for Today’s Pedagogy (Ph.D. dissertation, University of Rochester, Eastman School of Music, 2010), p. 10.

16. “The improviser, let us hypothesize, always has something given to work from—certain things that are at the base of the performance, that he uses as the ground on which he builds. We may call it his model.” Bruno Nettl, “Thoughts on Improvisation: A Comparative Approach,” in The Musical Quarterly, 1974, Vol. LX, No. 1, p. 11.

17. A list of German sources, containing samples of thoroughbass fugue, appears at the end of the article.

18. The strophic form of the thoroughbass fugue has roots in the verset fugues tradition and to the sectional structure of motets and ricercar. What we say about strophes of thoroughbass fugue is closely related to Joel Lester’s “parallel sections” and David Ledbetter’s “series of expositions.” See Joel Lester, “Heightening levels of activity in J. S. Bach’s parallel-section constructions,” in Journal of the American Musicological Society, Vol. 54, No. 1 (Spring 2001), p. 49–96; and Ledbetter, Bach’s Well-tempered Clavier, p. 100.

19. The term “Gegenharmonie” first appeared in Abhandlung von der Fuge by Friedrich Wilhelm Marpurg, where it is given the following definition: “Counterharmony. Thus is named the material in the remaining parts which is set against the subject.” (“Die Gegenharmonie. So heißt diejenige Komposition, die dem Fugensatze in den übrigen Stimmen entgegengesetzt wird.”) Friedrich Wilhelm Marpurg, Abhandlung von der Fuge (Berlin, 1753), S. 18.

20. Since all standard harmonic structures in thoroughbass are noted in shorthand, we have added to the original figuring (where necessary) those signatures within brackets, which were implied by default.

21. By way of numerous experiments, it has been shown that the capacity of short-term (active) memory, without which the process of improvisation is simply impossible, is limited to 7 ± 2 units of information (the so-called Miller’s number). This can be increased only by uniting disparate elements into groups. We refer to a very illustrative example from Maltsev’s article in order to demonstrate the activity of this mechanism: “For example, short-term memory can retain around seven different letters (perhaps, X, J, D, B, G, U, S), but the number of letters drastically increases if we try to remember seven words, and will increase even more drastically if we try to remember seven sentences.” (Maltsev, “Ob improvizacii,” p. 69.) As Michael Callahan emphasizes: “Experts recognize relevant patterns, and therefore perceive stimuli in larger and more meaningful units than novices do; expert improvisers notice patterns in music and conceive of musical units in large spans (e.g., entire voice-leading structures and phrases, rather than individual notes).” (Callahan, Techniques of Keyboard Improvisation, p. 22.)

22. We remind the reader that the harmonic vertical in thoroughbass is constructed upwards from a given note, therefore the part entering with the subject must always be the lowest one.

23. “Alle in bissherigen Numern muessen nicht nur vom Papier, sondern auch auswendig auf das fertigste und deutlichste gelernt werden,” in Philipp Christoph Hartung, Musicus Theoretico-Practicus, Zweyter Theil (Nuremberg, 1749), S. 12, § 42).

24. Sometimes the tasks that were given to organists for the purpose of testing were limited by a time-frame. For example, the testing of organists for the post at the Hamburg Cathedral (24 October 1725) included the presentation of an entire fugue “created for four minutes,” a prelude of “about two minutes,” a chaconne of “about six minutes,” etc. See Johann Mattheson, Grosse General-Baß-Schule (Hamburg, 1731), S. 33. It is very difficult to improvise a piece with continuous development and at the same time fit everything within a given time-frame. It is much easier to fill the established time limits with standard-size strophes, adding a necessary number.

25. Anatoliy Milka, Muzikalnoye prinosheniye I. S. Basha: k rekonstrukzii I interpretazii (Moscow, 1999), p. 151 [in Russian].

26. “Denn mit fremden Sätzen die Leute zu scheeren, halte ich für keine Kunst; lieber was bekanntes und fliessendes genommen, damit es desto besser bearbeitet werden möge. Darauf kommt es an, und es gefällt dem Zuhörer besser, als ein chromatisches Gezerre” in Mattheson, Grosse General-Baß-Schule, S. 34–35.

27. For more details on the authorship of Thema Regium see Milka, Muzikalnoye prinosheniye I. S. Basha, pp. 153–167.

28. For more details about the manuscript MS. Akc. 4125 see Andrzej Szadejko, “Daniel Magnus Gronau (1700–1747)—didaktische Aspekte in Orgelwerken am Beispiel der Signatur MS. Akc. 4125 aus der Danziger Bibliothek der Polnischen Akademie der Wissenschaften,” in Musica Baltica (Gdansk, 2010), S. 351–361. It is interesting that Szadejko views the given source solely from a didactic perspective: as exercises in counterpoint. In my opinion, considering its intended purpose, MS. Akc. 4125 has more in common with such collections as the Langlo(t)z Manuscript and the Mylau Tabulaturbuch; it is also an anthology containing musical material necessary for the church organist’s everyday activity.

29. Indeed, the written-out figures concern themselves not with the single-part statement at the beginning of a fugue, but to the latter (multi-part) statements.

30. “Vielweniger darff man sich an den Gebrauch einiger Organisten binden, die das Thema erst, ohne die geringste Verblümung, fein ehrbar und viermahl durchs gantze Clavier in lauter Consonantzien und Lämmer-Tertzien hören lassen; hernach wieder mit dem Gefährten eben so bescheidentlich von oben anfangen; immer einerley Leier treiben; nichts nachahmendes oder rückendes dazwischen bringen; sondern nur stets den blossen Accord, als ob es ein General-Baß wäre, dazu greiffen” in Johann Mattheson, Der Vollkommene Capellmeister (Hamburg, 1739), S. 388, § 97.

31. “Ein anderer hat öfters den guten Willen, es besser zu machen. Aber was thun er? Er dreschet den Generalbaß, und dieses ist sehr erbaulich anzuhören. Da sind keine Bindungen, die die Harmonie angenehm, fliessend und zusammenhängend machen. Es ist eine holperichte Harmonie. Da höret man keine enge Nachahmung, keine Zergliederung des Satzes. Da ist keine Ordnung, und die Anzahl der Stimmen erfähret man zur Noth am Ende, da man solche gleich nach der ersten Durchführung des Satzes durch die verschiedenen Stimmen hätte empfinden sollen. Dieser Satz wird niemahls in den Mittelstimmen klüglich angebracht. Man höret ihn nur immer oben oder unten wozu beständig die eine Hand die andere, so wie eine Arie, accompagnirt. Man hört das Thema niemahls bequem und zur rechten Zeit auf eine den Verstand und das Ohr nachdrücklich rührende Art eintreten. Es ist ein hanbüchenes Gelärme und Gepolter; der unharmonischen Gänge nicht zu gedenken” in Marpurg, Abhandlung von der Fuge, Theil II (Berlin, 1754), S. XXIII–XXIV).

32. About the use of ars combinatoria techniques in the 18th-century, see Leonard G. Ratner, “Ars Combinatoria: Chance and Choice in Eighteenth-Century music,” in Studies in Eighteenth-Century Music: A Tribute to Karl Geiringer on his Seventieth Birthday, ed. by H. C. Robbins Landon and Roger E. Chapman (New York: Da Capo Press), pp. 343–363.

Jan Overduin is Professor of Music at Wilfrid Laurier University in Waterloo, Ontario, where he teaches organ and church music. He began studies in The Netherlands, where he was born, and continued in Canada at the University of Western Ontario, where he received the Masters degree in performance. The list of his teachers includes Marie-Claire Alain, Peter Hurford, and Jean Langlais. He has directed many choirs including the Wilfrid Laurier University Choir and Chapel Choir, the Niagara Chamber Choir (which he founded), the Menno Singers, the Mennonite Mass Choir. He has been actively involved in church music for over 40 years, most recently as director of music at St. Matthew's Lutheran Church in Kitchener, Ontario from 1985 to 1997. As organ soloist, he has recorded numerous broadcasts for radio and has played concerts in Europe, North America, and the Far East. His discography includes nine CDs with trumpeter Eric Schultz (on the German labels 'ebs' and 'Arte Nova Classics'), a solo album recorded at Ottobeuren, Germany (on 'ebs'), and a recent CD with recorder virtuoso Matthew Jones. Forthcoming is a book on improvisation for organists, published by Oxford University Press, and a new organ edition of the Art of Fugue. Jan Overduin may be contacted at [email protected] and welcomes visitors at his website http://info.wlu.ca/ ~wwwmusic/overduin/index.htm

The Kunst der Fuge, BWV 1080, a work consisting of 14 fugues and 4 canons all on one theme, is Bach's farewell, his testament. It is a very solemn and personal work, and ends with Bach's only fugue on his name, the notes B-flat, A, C, and B-natural (B, A, C, and H in German). Never before did he use this chromatic theme on such a grand scale or with such clarity as here at the end of Contrapunctus XIV. It is as if he puts his signature not only to the KdF, but also to his life's work. In fact it is uncanny, this very clear reference to his own name. The aural effect is almost dizzying, as is the visual appearance of the last page, with C.P.E. Bach's handwritten note about his father's death: "In this fugue, where the name BACH appears as a countersubject, the composer died." Like Shakespeare in the character of Prospero in The Tempest, Bach himself appears on stage, but it is to say "good-bye."

It is fitting that Bach reserved the 14th fugue for the use of the plain theme in clearest form, because of the relationship between the number 14 and his name. By allowing each letter of the alphabet a number (a=1, b=2, c=3, d=4 and so on), Bach's name adds up to 14 (2 + 1 + 3 + 8). Moreover, the name J.S. Bach adds up to its retrograde 41 (9 + 18 + 14). Though Bach's familiarity with numbers is not documented, the cabalistic numerical ideas were common knowledge.1 The work may have been intended as his third and final offering to the Mizler Society, which he had joined in June 1747, waiting until he was the 14th member to join. He also had his portrait painted for this society with 14 buttons on his jacket. Perhaps his aim was to finish the KdF by June 1749, as his third and final offering, since a condition of membership obliged him to submit a published "scientific" work every year until the age of 65.2

The more I play this work, the more aware I become of how saturated it is with personal references or "signatures." The B-A-C-H theme in the obvious four-note form or more subtly through the use of themes that contain 14 or 41 notes permeates the entire KdF. A casual listener or player is not likely to be conscious of some of these allusions, but the fact that they are there in such abundance imbues the work with a personal intensity and warmth that can easily be felt. While some or even many of the "B-A-C-H's" may occur spontaneously as a result of Bach's use of chromatic language, there are reasons to suspect that their incorporation is part of the overall design of the work and intention of the composer. Bach is not merely scribbling his name all over the score or playing numerological games. The chromatic language itself, the use of the key of D minor, the shape of theme and its inversion with its hymn-tunes analogies, the dramatic use of silence, various other motifs--it is all these and more, together with the "signatures," that give the work its deeply personal flavor.

The following examples include only appearances of the B-A-C-H theme that use the four actual notes B-flat, A, C, and B-natural. Excluded are all transpositions of the motif, e.g. E-flat, D, F, E etc., of which there are numerous examples. All examples have the four notes in the same octave.  Again, by relaxing this restriction, the list could be greatly expanded. Included however are those statements of the motif that are decorated with unessential notes, especially between the second and third notes; the unessential notes may serve to hide the visual but usually do not obscure the aural impact of the motif. These observations do not pretend to be profound, but are merely the result of a growing familiarity with and fondness for this stupendous work. If they have any validity, it is in underlining the deeply personal nature of the KdF.

Immediately in Contrapunctus I, in the most obvious voice, i.e. the soprano, in measures 10-12 Bach features the four-note name theme. Bach "hides" the eighth-note E by having it dip below the alto note G, so that even though the soprano part by itself really spells B-E-A-C-H (not a word in German), the ear perceives it as B-A-C-H. (Example 1)

The B-A-C-H motif is more hidden in Contrapunctus II, though increasing chromaticism causes it to occur more frequently. It appears twice in measures 35 to 37, both times in the dotted note motif that dominates this fugue. Though the first two notes are separated from the third and fourth by a complete measure, they occur in adjacent statements of the dotted note motif, and therefore appear related and connected. (Example 2)

Measures 22-23 of Contrapunctus III contain a very clear statement of B-A-C-H, shared between the upper two voices (B-A in the soprano, C-H in the alto).  While this sharing serves on the one hand to hide the motif, it also underlines it, since the effect is that of an ornamented version: the B-A-C-H motif beautified in a flowery way. (Example 3)

One of the most poignant of all references to the name of Bach occurs in Contrapunctus IV. The shape of the regular inverted theme is such that there is a noticeable high point on the notes B-flat and its "resolution" to the semitone below. There is also a marked similarity to the hymn-tune "Aus tiefer Noth schrei ich zu dir," a hymn paraphrase of Psalm 130 (De profundis). The resemblance in fact is too obvious to ignore.3 In Contrapunctus IV, and only here, Bach transposes the second half of the theme up a whole tone, so that the high point of the theme now is C-H (C and B-natural) instead of B-A (B-flat and A). This causes a sudden modulation to another key, the dominant of the dominant, a rather wrenching and quite dramatic shift of key. It happens first in bar 61, and thereafter four more times (in other words, not every time the theme is heard). The change from the expected high point B-A to C-H may not be exactly an obvious reference to Bach's name, but certainly for the player, the alteration of the climax of the theme is all the more dramatic and personal, especially when the personal nature of the hymn "Aus tiefer Noth schrei ich zu dir" is taken into consideration as well. Not only the personal pronoun is emphasized by this veiled reference to B-A-C-H, but also the idea of "calling" (schreien).   (Examples 4a and 4b) 

"Calling" is also happening constantly throughout this fugue through the use of the pervading "call-motif" of the descending minor third. Towards the end of Contrapunctus IV occurs another and much more traditional example of the use of the B-A-C-H motif: in bars 135-136 it is slipped in once into the tenor voice, like a hardly noticeable signature. (Example 4c)

In Contrapunctus V, the KdF theme appears consistently in 14-note form, as it will continue to do in much of the rest of the KdF, with the two descending thirds smoothed out with passing notes. Thus the theme itself is being identified with the name of Bach. Moreover, it is especially interesting that the B-A-C-H motif is heard quite plainly and in the most obvious voice (soprano) exactly in bar 41. (Examples 5a and 5b)

Contrapunctus VI states the B-A-C-H motif near the beginning, in measures 4 and 5 in the soprano. Again the first two notes are separated by a measure from the third and fourth, but they are perceived to be related to each other through their rhythmic emphasis. (Example 6)

Contrapunctus VII features the B-A-C-H motif in much the same way, for example in the tenor part of measures 17-19. Within the context of a statement of the KdF theme in 14-note form (and in diminution), the notes B-A are again separated from C-H by a measure, but each pair of notes comes at a similar point, i.e. the end of two parallel phrases. (Example 7)

With the introduction of a new theme that is rather chromatic, numerous instances of B-A-C-H occur in Contrapunctus VIII. In measure 11 a very clear statement of B-A-C-H is shared between the two lower voices (Example 8a). In measures 85-86, the motif is featured in the soprano and in measure 112 in the bass (Examples 8b and 8c). The main KdF theme (inverted), which occurs as theme III in this fugue (beginning in measure 95, in the alto), consists always of exactly 14 notes. More noticeably, each measure begins with a quarter rest (Example 8d). The use of silence on the downbeat is a technique often used by Bach to symbolize eternity and/or death.4 Thus the form of the theme in this fugue forms associations not only with the name "Bach" (14 notes), but also with "death" (silence on the downbeats). A convincing example of this technique to express longing for death is often encountered in Bach's chorales, such as at the end of Cantata #56 (Ich will den Kreuzstab gerne tragen). (Example 8e)

In Contrapunctus IX, in bars 84-85, the B-A-C-H motif is shared between the soprano (B-flat, A) and alto (C, B-natural), but because the voices cross, the motif appears to remain in the same voice, the alto. (Example 9)

In Contrapunctus X, again exactly in bar 40-41, the B-A-C-H motif makes an appearance in the two upper voices. (Example 10)

The 14-note version of the main KdF theme, with rests on every downbeat, now  "rectus," boldly opens Contrapunctus XI (Example 11a). The inversion of the second theme of Contrapunctus VIII, which in this fugue appears as theme III, very clearly spells B-A-C-H. Donald Tovey rejects this allusion to the name of Bach, since strictly speaking the theme misspells his name as B-A-C-C-C-H, yet to a listener (as opposed to a mere score-reader) this is almost as obvious an allusion to the name of Bach as in the final fugue.  (Example 11b) The B-A-C-H motif occurs frequently, not only in connection with the third theme, but elsewhere as well. An example is found in measure 144, with the motif shared between the alto (B-flat, A) and the soprano (C, B-natural).  (Example 11c)     

Contrapunctus XII and XIII, the two completely invertible "mirror" fugues, leave the composer with very little room to maneuver.  The listener has no idea of the strict rules behind these wonderful pieces, especially the playful Contrapunctus XIII. Even here the B-A-C-H motif pervades everything, though not as overtly as elsewhere in the KdF. The descending semitones B-A and C-H permeate the texture, but the four notes never occur together, and seldom within the same octave. One reason that Bach chose D minor as the key for this work may well have been that it allowed him to "season" fugues like Contrapunctus XII and XIII with these notes.  For example, C-H (the more unusual of the two pairs of notes) is used six times in measures 25-26 of Contrapunctus XIIb, just after several highly exposed B-A's. Appearances of the motif within one voice and within the compass of a minor third also occur (though somewhat more separated than usual) in measures 14-16 of Contrapunctus XIIa (bass) and measures 46-47 of Contrapunctus XIIb (bass). Similar concentrations of B-A and C-H occur in Contrapunctus XIII, imbuing the whole with the flavor of the BACH motif (e.g. in Contrapunctus XIIIa: eight times B-A in measures 32-35, followed by eight times C-H in measures 37-41).

Theme II of Contrapunctus XIV consists of exactly 41 notes, as if in direct preparation for the next theme, that of B-A-C-H itself (Example 12). There are also numerous examples of the B-A-C-H motif in the earlier part(s) of this fugue, again as if to prepare us for the plain statement of Theme III in measure 183. To list just three examples: measures 16-17 (tenor), 59-60 (alto/soprano), and 133-134 (alto).  (Examples 13, 14, and 15)

The evolution of the B-A-C-H motif is but one of many marvels of the KdF.  A constant companion in the background, like a quietly-flowing underground stream,5 in Contrapunctus XIV it finally appears quite alone and "naked," like a new-born babe. It is a paradoxical moment of loneliness and pity, sadness and comfort, weakness and strength. Almost immediately it is used in stretto and inversion, and "with the boldest and most mysterious harmonies"6 that are wrenching in their effect on us. It is at this point that this great composer, for whom nothing seemed impossible, especially in this work, leaves us forever. But the unfinished ending in which the composer is "called by name" also contains the promise of what "eye has not seen, nor ear heard." (I Cor. 2:9)

Notes

                        1.                  William Wright, The Organ--The Instrument and Its Literature (University of Toronto: private publ., 1994) 96.

                        2.                  J.S. Bach, Die Kunst der Fuge, ed. Davitt Moroney (Muenchen: G. Henle, 1989) vii.

                        3.                  The main theme in "rectus" form vaguely hints at "Vater unser" (Lord's Prayer). The descending thirds in Contrapunctus IV are also striking characteristics in some chorales, e.g., "Wie schön leuchtet der Morgenstern" (How lovely shines the morning star) and "Wer weiss, wie nahe mir mein Ende" (Who knows how near is my life's end). The most ornamented of all versions of the theme, as found in the Canon per Augmentationem in contrario Motu shows a striking resemblance to the "Agnus Dei" from the Mass in B minor.

                        4.                  Many of the more ornate chorale settings such as those in Schemelli's Gesangbuch illustrate this, e.g., "Lasset uns mit Jesu ziehen," "Es ist vollbracht," and "Liebster Gott, wann werd ich sterben?"  Examples in Das Orgelbüchlein include "Alle Menschen müssen sterben," and "Wir danken dir, Herr Jesu Christ, dass du für uns gestorben bist" (BWV 623 and 643).

                        5.                  In other words, like a Bach (German: brook).

                        6.                  Donald Francis Tovey, Essays in Musical Analysis: Chamber Music (first published in 1944; London: O.U.P., 1972) 88.

Alexander Fiseisky, born in Moscow, graduated with distinction from the Moscow Conservatoire as pianist and organist. He is an organ soloist of the Moscow State Philharmonic Society, head of the organ class at the Russian Gnessins’ Academy of Music in Moscow, and president of the Vladimir Odoyevsky Organ Center. He organized and served as artistic director for organ festivals in Moscow, St. Petersburg, Kiev, and Tallinn, among others. In 1997 he was honored by President Yeltsin with the title ‘Honored Artist of the Russian Federation’. Fiseisky has given concerts in more than 30 countries. In the Bach anniversary year of 2000 he played J. S. Bach’s entire organ works, twice in the context of EXPO 2000 in Hannover, and once in a single day in Düsseldorf as a Bach marathon. Sought after as a juror in international competitions, he has directed seminars and masterclasses in Europe and the USA. He is the dedicatee of numerous compositions, including works by Mikhail Kollontai, Vladimir Ryabov, Milena Aroutyunova, and Walther Erbacher. A musicologist, he has edited anthologies of organ music of Russia and of the Baltics (Bärenreiter-Verlag). He has many recordings to his credit, including the complete organ works of J. S. Bach.

It goes without saying that the primary
task of every performer who wishes to convey the meaning of any given musical work must first be to understand the original intention of the composer. And when the works in question are those of Johann Sebastian Bach, where the invisible thread that should link us to the era in which he lived seems to be irretrievably broken, the task takes on Herculean proportions. The aim of this analysis is to attempt a correct reading of the Clavierübung III—one of the most enigmatic works in the whole literature of the organ.
This work, which was composed at the high point of the composer’s creativity (1739), impresses us by its dimensions alone. It is part of a cycle of works, comprising the Six Partitas (Part 1, composed in 1731, BWV 825–830), the French Ouverture and the Italian Concerto (Part 2, composed in 1735, BWV 831, BWV 971), as well as the Goldberg Variations (Part 4, composed in 1742, BWV 988). And the Clavierübung III itself is also a cyclical work—it consists of 21 chorale preludes and four duets framed by a prelude and a fugue in E-flat major.
Bach certainly accorded the Clavierübung III particular importance. It is no coincidence that this was the first work for organ that he had published in Leipzig. What was Bach’s purpose in writing this work, and what means did he choose to fulfil it?

The history of the composition. The intentions and aims of the composer
The Clavierübung III was written to coincide with the 200th anniversary of Luther’s visit to Leipzig and the festal Whitsun service in St. Thomas Church on the 25th of May 1539, which effectively marked the official recognition of the Reformation in Leipzig. The Clavierübung III consists essentially of arrangements of chorales from the Protestant church service, and in its structure it is reminiscent of Luther’s Catechism, which consists of two parts: the Greater Catechism deals with the principles of faith, while the Lesser Catechism is directed more towards children and the less-educated part of the population. Correspondingly, each chorale melody—with the exception of Allein Gott in der Höh’ sei Ehr’ [Glory be to God alone on high]—is presented in two versions: a greater version which uses all the resources of the organ including the pedals, and a shorter manualiter version.
And indeed, because of its special structure, the Clavierübung III has often in the past been referred to as an “Organ Catechism,” and correspondingly it is usually referred to today as the “Organ Mass.” It is clear that neither of these two names do full justice to the structure of Bach’s composition. Nor do they explain the inclusion of the four duets.
The title of the work is as follows:

Dritter Theil / der / Clavier Übung / bestehend / in / verschiedenen Vorspielen / über die / Catechismus- und andere Gesaenge, / vor die Orgel: / Denen Liebhabern, und besonders denen Kennern / von dergleichen Arbeit, zur Gemüths Ergezung / verfertiget von / Johann Sebastian Bach, / Koenigl[ich] Pohlnischen, und Churfürstl[ich] Saechs[eschen] / Hoff-Compositeur, Capellmeister, und / Directore Chori Musici in Leipzig. / In Verlegung des Authoris.

[Third Part of the Clavierübung consisting of various preludes on the Catechism and other Hymns for the organ: for amateurs, and especially for connoisseurs of such work, for the refreshment of their souls, executed by Johann Sebastian Bach, Royal Polish and Electoral Saxon Court Composer, Capellmeister, and Directore Chori Musici in Leipzig. Published by the author.]

Bach here follows the example of his predecessor at St. Thomas Church, Johann Kuhnau (1660–1722), and modestly calls his work Clavierübung [Keyboard Exercise].2 He thereby encourages us, through diligent practice (Übung in German), to understanding his purpose in writing this work.
Let us accept this invitation.
The first question, even after a cursory look at Bach’s work, is probably “What does it represent in this compositional form? Are we to understand it as a unified dramatic whole or as a collection of diverse pieces for the keyboard?”
Characteristically, the usual concert practice suggests that the Clavierübung III is not seen as an integral work: virtually nobody plays the whole composition in its published form.3 But the question nevertheless remains: Is there really no suggestion of an overall dramatic structure within the work?
An analysis would help us to answer this question. But before we tackle it, we should—even very generally—look at some characteristics of the musical aesthetics and Bach’s particular compositional style during the period when he was working on the Clavierübung III.

The theological and philosophical basis of the work of J. S. Bach
Bach’s personal philosophy was heavily influenced by the philosophical ideas and the personality of Martin Luther (1483–1546). Books written by Luther accounted for a quarter of all the books in Bach’s private library. According to the personal inventory that was made after his death, Bach owned two complete editions of the works of Martin Luther in Latin and German, as well as works of his successors: Abraham Calov, Martin Chemnitz, Johannes Olearius, and others.4 The title page of an earlier version of the Clavier-Büchlein für Anna Magdalena Bach5 bears a note giving the title of the work as Anti-Calvinismus by August Pfeiffer, written in Bach’s own hand.
It is well known that Luther was a well-educated musician.6 In contrast to the majority of the reformers in the 16th century, Luther considered music to be a form of divine revelation. In the foreword to Georg Rhau’s anthology Symphoniae iucundae7 he wrote: “In summa: Die edle Musika ist nach Gottes Wort der höchste Schatz auf Erden.“8 [Summing up: Noble music is the greatest treasure on earth next to the Word of God.] He is quoted in the Encomion musices as giving a similar definition: “Musika ist eine schöne, liebliche Gabe Gottes, sie hat mich oft also erweckt und bewegt, daß ich Lust zu predigen gewonnen habe...”9 [One of the finest and noblest gifts of God is music. It has often aroused and moved me so that I have gained a desire to preach . . . ] And in a letter to Ludwig Senfl of 4 October 1530 we find the following lines in his handwriting:
Et plane judico, nec pudet asserere, post theologiam esse nullam artem, quae musicae possit aequari, cum ipsa sola post theologiam id praestet, quod alioqui sola theologia praestat, scilicet quietem et animum laetum…10
[I plainly judge, and do not hesitate to affirm, that except for theology there is no art that could be put on the same level with music, since except for theology, (music) alone produces what otherwise only theology can do, namely, a calm and joyful disposition.11]
Luther’s views were akin to those of Bach. Like the great reformer, Bach saw the world of music and the world of theology as very closely connected.12 A short handwritten treatise concerning figured bass, which Bach wrote while working on the Clavierübung III, is introduced with the following words:
Der Generalbaß ist das vollkommenste Fundament der Music welcher [auf einem Clavier] mit beyden Händen gespielt wird dergestalt das die lincke Hand die vorgeschriebenen Noten spielet die rechte aber Con- und Dissonantien darzu greift damit dieses eine wohlklingende Harmonie gebe zur Ehre Gottes und zulässiger Ergötzung des Gemüths und soll wie aller Music, also auch des General Basses Finis und End Uhrsache anders nicht, als nur zu Gottes Ehre und Recreation des Gemüths seyn. Wo dieses ists keine eigentliche Music sondern ein Teuflisches Geplerr und Geleyr.13
[The thorough-bass is the most perfect foundation of music. It is played with both hands on a keyboard instrument in such a way that the left hand plays the written notes, while the right hand strikes consonances and dissonances, so that this results in full-sounding Harmonie to the Honor of God and the permissible delight of the soul. The ultimate end or final goal of all music, including the thorough-bass, shall be nothing but for the Honor of God and the renewal of the soul. Where these factors are not taken in consideration, there is no true music, rather, devilish bawling and droning.14]

When Bach at the age of 23 left Mühl-hausen, he declared that the Endzweck [ultimate aim] of his creative work would be the regulirte kirchen music zu Gottes Ehren [regulated church music to the glory of God].15
One can further assess the musical and aesthetic views of the composer with the help of his annotations in the margins of a Bible that was published by Abraham Calov (1681–1682) in Wittenberg.16 These marginalia are quite valuable—they allow us to catch a glimpse of the personal views of their writer and open up his world for us.
Already in Exodus, Chapter 15, where the prophetess Miriam sings of the wonderful deeds of God, we can read in Bach’s own hand: “N.B. Erstes Vorspiel auf 2 Chören zur Ehre Gottes zu musiciren.” [N.B.: First prelude for two choirs to be sung to the glory of God.] As a comment on First Chronicles 29, v. 2117 we find the following statement by the composer:

Ein herrlicher Beweiß, daß neben andern Anstalten des Gottesdienstes, besonders auch die Musica von Gottes Geist durch David mit angeordnet worden.
[Splendid proof that, besides other arrangements for worship, music too was instituted through David by the Spirit of God.]18
First Chronicles 26 describes the choosing of musicians for the temple. Bach’s comment: “Dieses Capitel ist das wahre Fundament aller Gott gefälligen Kirchen Music.” [This chapter is the true foundation of all church music pleasing to God.]
And one final quote: Second Chronicles, chapter 5 contains the passage:

. . . it was the duty of the trumpeters and singers to make themselves heard in unison in praise and thanksgiving to the LORD, and when the song was raised, with trumpets and cymbals and other musical instruments, in praise to the LORD “For he is good; for his steadfast love endures for ever,” the house, the house of the LORD, was filled with a cloud, so that the priests could not stand to minister because of the cloud; for the glory of the LORD filled the house of God. (2 Chronicles 5:13–14)19

Bach annotates this text with a remarkable comment that has programmatic significance and shows not only his relationship to the composing, performing, and hearing of music, but also to the activities of a church musician in general: “Bey einer andächtigen Musique ist allezeit Gott mit seiner Gnaden Gegenwart.“ [Where there is devotional music, God with His grace is always present.]
These examples suffice to clarify where we must start if we wish to analyze the works of Johann Sebastian Bach. Albert Schweitzer wrote in his masterful fashion: “Music is an act of worship with Bach… For him, art was religion...”20 The orthodox Lutheran Bach, who was born and raised in Eisenach, Luther’s own town, where the façade of the main church of St. George was decorated with the Protestant motto “A mighty fortress is our God,” transcended in his music the boundaries of confession and creed. “In the last resort, however, Bach’s real religion was not orthodox Lutheranism, but mysticism. In his innermost essence he belongs to the history of German mysticism.”21
This mystical sensitivity to the presence of God and the desire to give witness to Him through music, coupled with his dazzling talent, enabled Bach in his later works to develop an astonishing artistic fusion, the likes of which had not been seen in the world’s cultural history.
In 1747 Bach was admitted to the Societät der musikalischen Wissenschaften [Society of the Musical Sciences], which his one-time pupil, the philosopher and music author Lorenz Christoph Mizler von Koloff (1711–1778), had founded.22 Mizler, a friend of Bach’s, was strongly influenced by Pythagorism and the rational philosophy of both G. W. Leibnitz (1646–1716) and Christian Wolff (1679–1754). He saw music as a mathematical science.23
The very fact that Bach accepted Mizler’s invitation to join the Societät der musikalischen Wissenschaften is in itself significant. The composer obviously sympathized with Pythagoras’s ideas concerning the universe and its perfect harmony: a harmony that, according to the teachings of the ancient philosopher and mathematician, was expressed in numbers,24 and shared the convictions of his progenies.
J. S. Bach became the fourteenth member of the Society after G. F. Telemann (6) and G. F. Handel (11), together with other well-known scholars and philosophers. Following the established tradition, upon joining the Society he contributed a mite of his own. In addition to the Canonic variations on “Vom Himmel hoch, da komm’ ich her” (BWV 769), the composer also donated a portrait of himself to the Society, which had been painted in 1746 by Elias Gottlob Hausmann. A microanalysis of the music manuscript that appears in this painting has been made by Friedrich Smend. The results have thrown light on significant aspects of Bach’s compositional methods, which until the middle of the twentieth century had not attracted much attention by scholars.25
Smend’s publication gave new impetus to investigating numerology in the works of the Cantor of St. Thomas Church in Leipzig.26 It is not without interest that the researchers first found support in the writings of Christian theologians, but later more and more in the works of the ancient philosophers.27

Features of J. S. Bach’s compositional method
Albert Schweitzer defined Bach as a phenomenon in the history of music: “Bach is . . . a terminal point . . . everything merely leads up to him.”28 Indeed the works of the Cantor of St. Thomas make use not only of the fruits of earlier achievements in composition, but they are also the consummation of the most characteristic tendencies in the music of his own time. He makes use of a plethora of past and present expressive techniques and puts them at the disposal of one single goal: the creation of “devotional music.”
So what exactly were the artistic methods used by J. S. Bach as a composer?
Victor Hugo once described Gothic cathedrals as “symphonies in stone.” If we apply this quotation to the works of Bach, we could say that his larger compositions are “Gothic cathedrals” in music. And when one looks more closely at how Bach approached a new composition we can actually find quite close parallels to architecture. One could contrast, for example, Bach’s methods with the processes current in Viennese Classicism. Whereas in the latter period composition proceeded in a “linear” fashion, beginning from the melody in one of the voices, the methods of Bach’s time started from quite a different point. First of all, the composer laid down a concept of the entire work, or—to use the architectural analogy—he created a “ground-plan.” Then he proceeded to fill in the details. An example of this method is provided by the Orgelbüchlein [Little Organ Book] (BWV 599–644).
This working method gave free rein to the composer’s imagination. The proportions of the composition and its “saturation” with both obvious and more hidden details—factors that played an important role in determining the overall sense of the work—could easily be incorporated in the composition from its very beginning. Great importance was attached to Affektenlehre [Doctrine of the Affections], musical-rhetorical figures, and numerology.
Bach was without a doubt a brilliant “musical architect.” There is no room in his works for anything non-essential. He worked in a similar fashion to the architects of the Middle Ages: every detail has its origin in the concept governing the whole. And as with the medieval builders, much of this work remains, even today, shrouded in mystery. There are always new avenues opening up in these seemingly well-known works for new generations of interpreters to explore.
One can of course only penetrate more deeply into this musical architecture of most of Bach’s works if the connection to the words of the chorales used by the composer is taken into account. Johann Gotthilf Ziegler (1688–1747), a pupil of Bach, wrote in 1746: “Herr Capellmeister Bach, who is still living, instructed me when playing hymns, not to treat the melody as if it alone were important, but to play them taking into account the affect of the words.”29
Johann Mattheson (1681–1764) described music as sounding speech. Naturally this form of speech required its own lexicon in the shape of the definite progressions of musical notes bearing the semantic meaning—the motives, or musical-rhetorical figures, as they are called. These were quoted by Bach’s cousin, Johann Gottfried Walther (1684–1748), in his Musicalisches Lexicon [Music Encyclopaedia] (1732) and in the Praecepta der Musicalischen Composition [Principles of Musical Composition] (1708). Another important compositional aspect was the use of rhetorical laws in the construction of the musical structure, so that the composition began to resemble a religious sermon. As already mentioned, the Affektenlehre [Doctrine of the Affections], which depended upon the use of unequal temperament and the resulting different emotional character of the various keys, played an important role in composition,30 as did, surrounded as it was by an air of mystery, numerology with its different levels of meaning.
One of these levels is to be found in allegorical symbolism. Andreas Werckmeister (1645–1706) gave the following meanings to the first eight numbers in Musikalische Paradoxal-Discourse:31 1 – God, unity; 2 – The Word, God the Son; 3 – The Holy Spirit; 4 – The world of angels; 5 – Symbol of Mankind (“sensual Mankind” [Numerus sensualis]); 6 – Third Person of the Godhead (3×2);32 7 – Symbol of purity and peace; 8 – Symbol of wholeness and perfection.
Another level is that of semantic symbolism. For example, the number 7 symbolises the Seven Last Words on the Cross.
A third level is that of cabbalistic symbolism. Each letter of the alphabet stands for a particular number: a = 1, b = 2, c = 3 and so forth. The letters i and j share the number 9, while u and v are both attributed to the number 20. This means that particular combinations of letters each have a corresponding number. For example, the number 14 is the sum of the numerical values of the letters BACH. Thus the number 14 (or similar numbers, such as 140 or 1.4) would be associated with the composer Bach, whose name was assembled from these individual letters.
Numbers were also used as a constructive element, whereby the harmonic proportions of the ratios of simple numbers, which had been known since Pythagoras’s time, were incorporated into the composition. In addition, the proportio divina, the “Golden mean,” was also used. Naturally Bach was a consummate master of all these creative methods and he used them constantly in his compositions. The most obvious example is the Clavierübung III, which occupies a key position among all Bach’s works for the organ.
Let us examine the structure of this composition more closely.

The chorale preludes
The central part of the work under consideration, as Bach’s title-page suggests, is the collection of chorale preludes. This collection covers not only the essential elements of the Protestant liturgy but also of Luther’s Catechism.
Kyrie, Gott Vater in Ewigkeit – Christe, aller Welt Trost – Kyrie, Gott heiliger Geist [Kyrie, God the Father, eternal – Christ, consolation of all the world – Kyrie, God the Holy Spirit] (BWV 669–674)
The triad of the first chorales creates a sense of unity. The models for these autonomous works were certain verses of the Gregorian chorale Kyrie fons bonitatis (10th century),33 which display the characteristic of a refrain. (Example 1) Such a compositional method is seldom found among Bach’s organ works. In the context of Kyrie – Christe – Kyrie it allowed the composer to establish by means of music the essence of the “one and indivisible” Holy Trinity.34
The first motif of the cantus firmus is characterized by a stepwise progression. In the final statement of the cantus firmus (which is the same in all three compositions), note the upwards leap over a fifth. It is perhaps of interest to note that both the stepwise movement on the one hand and the prominent role of the fifth on the other (elements that determine the mood of the first chorales of the Clavierübung III) play an important part in the dramatic construction of the whole work.
The unity of the initial Kyrie – Christe – Kyrie is underlined by the fact that they are written in a single compositional style—the stile antico. Hermann Keller described them as “Orgelmotetten kunst-vollster Art” [The most highly artistic motets for organ].35 The music suggests greatness and quiet strength. The movement of the accompanying voices working out the motifs of the cantus firmus is linear. The cantus firmus, which is kept in longer note values, appears successively in the soprano (Kyrie, Gott Vater in Ewigkeit), in the tenor (Christe, aller Welt Trost), and in the bass (Kyrie, Gott heiliger Geist), and thus symbolizes in similar fashion the three Persons of the Trinity: God the Father, who is above all, who holds all in being; Jesus Christ, the mediator between God and humankind; and the life-giving Holy Spirit.
The epic element appears organically tied to the inner dynamics of the Kyrie – Christe – Kyrie. The contemplative character of the first chorale gives way to a feeling of emotional turbulence in the second chorale. The third chorale is energy-laden, an effect achieved by the introduction of a fifth voice, the acceleration of the musical structure, and the use of chromatics.
The end of the chorale Kyrie, Gott heiliger Geist is quite remarkable: against the backdrop of the final statement of the cantus firmus in the pedals, a tie overflowing with chromatic dissonances appears in the upper voices. These six-and-a-half bars differ quite markedly from all that has gone before. The sound as it were illustrates the text, which at this point contains a plea for mercy. The word eleison is accompanied by an ostinato, which climbs in seconds and by a chromatic figura parrhesia. The music suggests a certain personal involvement. It is significant that one finds the motif BACH in crab motion here (although it appears in other notes), and finally encounters the signature of the composer: CH-BA in the alto of the penultimate bar. (Example 2)
There are altogether 60 bars in the chorale prelude Kyrie, Gott heiliger Geist, which matches Werckmeister’s concept well.36 And there is of course the additional association with the creation of the world (the six days of God’s creative work).37 It is worth mentioning that in the first prelude of the Clavierübung III the numerical symbol for the name Bach already occurs more than once. The subsequent statement of the theme in the chorale Kyrie, Gott Vater in Ewigkeit is not only emphasized by the use of parallel thirds, but also by its extension to 14 notes (the numerical value of the letters BACH).38 And the cantus firmus in the chorale prelude Kyrie, Gott heiliger Geist has a total of 41 notes (JSBACH).
The three manualiter Kyries, each in the form of a small fughetta, all elaborate the opening motif of the appropriate verse of the chorale. Each following chorale begins in the soprano with the last note of the preceding chorale, which serves to underline the inner unity of the three manualiter pieces Kyrie – Christe – Kyrie.
An interesting aspect, which is seldom found within Bach’s organ works, is how the keys of the six pieces we have looked at are related. Each of them has at least two tonal centers. We should not let the key signature with three flats of the greater chorale preludes Kyrie – Christe – Kyrie confuse us: the rules of musical notation would certainly have allowed these preludes to have been written with only two flats. It would appear that the composer intentionally adopted three flats in order to strengthen the association with the Holy Trinity.

Allein Gott in der Höh’ sei Ehr’
[Glory be to God alone on high] (BWV 675–677)
A special feature of the following section of the Clavierübung III is the fact that it has three different preludes on the chorale Allein Gott in der Höh’ sei Ehr’—the Protestant version of the Gloria in excelsis from the Gregorian Mass for Easter Sunday. An explanation for this phenomenon must be sought in the text of the chorale itself,39 as it sings the praises of the Holy Trinity. Correspondingly, Bach includes three preludes here, each of which is a very individually elaborated piece in three-part texture.
In the first prelude, elegant and rhythmical canon-like outer voices surround the cantus firmus in the alto. The next prelude is executed as a trio sonata with pedal obligato. The cantus firmus appears from time to time in one or other of the voices of this exquisite trio and blends with the natural flow of the music.40 The last chorale prelude is a small fugato in the manner of an Italian versetto, based on the first notes of the cantus firmus.41 All in all, these three versions of the angel’s praise Allein Gott in der Höh’ sei Ehr’ create a feeling of incorporality and immateriality, convincing us by their clarity and purity, and creating an impression of harmony and perfection.
In this section of the Clavierübung III there is a small, at first glance insignificant, compositional detail that is, however, very interesting when seen from the perspective of the dramatic construction of the whole. The keys of the chorale preludes—F major, G major, and A major—form an ascending motif that is the basis for all three preludes on Allein Gott in der Höh’ sei Ehr’. The composer must assuredly have chosen this sequence of keys with the aim of thus uniting the whole cycle. Numerology reveals another interesting aspect—the numerical values of F, G, and A (6 +7 + 1) comes to 14, the same value as BACH.

Dies sind die heil’gen zehn Gebot’ [These are the holy Ten Commandments] (BWV 678–679)
Following the lead of Luther’s Catechism, Bach now begins an extensive section of the Clavierübung III with arrangements of the Gregorian chorale on an Old Testament theme, Dies sind die heil’gen zehn Gebot’.42 This is the last pair of chorales in a major key for the remainder of the cycle and the only time that Bach uses the same key for two consecutive compositions—Mixolydian G major, which is one of the purest keys in unequal temperament. It is significant that in both the Orgelbüchlein and in Cantata 77, the chorale melody Dies sind die heil’gen zehn Gebot’ is also written in this key.
The greater chorale prelude is developed as a composition for five voices, with the cantus firmus appearing a total of five times as a canon in the tenor. Thus it appears ten times in all, symbolizing an obedient response to the Law.43
The beginning of the prelude is wonderful: over a pedalpoint we hear, emerging out of the stillness, the motif of three descending notes, which we encountered earlier in the piece, worked out as a canon in the upper voices. The measured diatonic motion, the prepared suspensions, the surrounding motifs, and the ascending triads—these are just some of the musical means the composer has used to create a world of unspoiled purity, order, and harmony, in which the unsullied inhabitants of Paradise were at home before the Fall. (Example 3)
A change in character occurs in the fifth bar44 with the introduction of a figura suspirans45 and a motif of ‘falling seconds’, supplemented by a descending chromatic figura parrhesia motif in the alto. (Example 4)
Now the music is dominated by grief, sorrow, and misfortune.46 A change occurs once more in the sixth bar with the introduction of a figura kyklosis or figura circulatio in the alto47 (Example 5), which enriches the fabric with its new nuances. Thus with the help of symbolic motifs that are organically woven into the very fabric of the music, the composer brings us closer to the meaning of the chorale.
The First Commandment, which Luther in his Great Catechism deems to be the most important, is interpreted in the second verse of the chorale:

Ich bin allein dein Gott, der Herr,
kein Götter sollst du haben mehr,
du sollst mir ganz vertrauen dich,
von Herzens Grund lieben mich,
Kyrieleis.

[I alone am your God, your Lord,
No other Gods shall you have,
You shall put your whole trust in me,
Love me from the depth of your heart.
Kyrieleis.]

There is much evidence that precisely these lines were the starting point for Bach’s plan for the whole composition.
It is interesting to note that where the text speaks of “the love of God that comes out of the depths of the heart,” Bach interrupts the cantus firmus (bars 48–50) and increases the number of repetitions from ten to twelve. The motivation for this change can best be seen as an attempt to create a connection between the Old and New Testaments, whose interpreters in the new Christian congregations were the twelve Apostles. And Bach will follow the same intention to connect, through the symbolic comparison of the numbers ten and twelve, the Mosaic Law and the teachings of Jesus again in the Eucharist part, the conclusion of the chorale prelude section of the Clavierübung III.
It is well known that in the New Testament the Commandment of Love takes on decisive significance: “Jesus answered . . . you shall love the Lord your God with all your heart, and with all your soul, and with all your mind, and with all your strength” (Mark 12:30). The composer underlines the importance of this commandment with the help of special methods that are introduced at key points. When the word Herz [Heart] appears in the chorale text, Bach highlights it (in bars 46-47) with two groups of 16th notes, and when the words lieben mich [love me] appear in bars 51–52, he uses the heterolepsis, a musical rhetorical figure that creates the effect of two being united in one.48 Thus the composer uses musical means to portray the tangible content of the text. (Example 6)
Numerology plays an especially important role in the chorale prelude Dies sind die heil’gen zehn Gebot’.49 The chorale prelude has 60 bars (corresponding to the six days of creation). A pause first appears in the pedal after 37 notes, which can be seen as the Labarum, or Chi-Ro Christogram.50 The next pause comes after 60 further notes (another apparent reference to the creation of the cosmos). The subsequent melodic structure of the pedal line up to the pedalpoint in bar 29, which creates the illusion of a reprise, contains 47 notes. In the first bar, after the pause (bar 21), we encounter a leap of two octaves in the pedal, covering the entire range of the pedal, which is very unusual. (Example 7)
It is well known that Bach often referred to the Psalter, as did Luther in his Catechism. Psalm 47:2 states: “For the LORD, the Most High, is awesome, a great king over all the earth.” The text of the cantus firmus quoted at the point of the two octave leap is: Kein Götter sollst du haben mehr [No other Gods shall you have]. Michael Radulescu suggests that we should see the leap as an original “musical comment” by the composer, which, though hidden behind the abstract numerological symbolism, is to be understood as a distinct statement: “I am larger than life, I am your King.”51
The subsequent phrase in the pedal contains 147 notes. When Luther in his Catechism explains the meaning of the Ten Commandments, he quotes Psalm 147:11: “But the LORD takes pleasure in those who fear him, in those who hope in his steadfast love.” By introducing the number 147 into his chorale prelude Dies sind die heil’gen zehn Gebot’, Bach is underlining the actuality of the psalmist’s words quoted by Luther for the theme of the Decalogue.
The final notes of the cantus firmus in the second tenor are accompanied by a descending counterpoint in the first tenor, beginning with a chromatic figura parrhesia, which contains 12 notes (bars 57–60). The last phrase in the pedal consists of 14 notes (BACH), which is preceded by two short phrases of five notes each.
After all the above we can concur with those experts who suggest that the basic idea behind this work is love for the Creator.52 Additional confirmation for the correctness of this view is the number 315, which is the sum of all notes in the pedal. Albrecht Clement considers this number to be the numerical expression of the phrase Du sollt Gott, deinen Herren, lieben. [Literally: “You should love God, your Lord” as a direct rendering of the Luther Bible’s translation of Mark 12:30.]53
Characteristically, Bach introduces this summons in the title of Cantata 77, whose opening chorus is built upon the theme of the chorale prelude Dies sind die heil’gen zehn Gebot’, viz.

Du sollt Gott deinen Herren lieben
24 + 73 + 59 + 49 + 65 + 45 = 315

The manual fughetta on the chorale Dies sind die heil’gen zehn Gebot’, written in the form of a gigue, is also dominated by the number 10, although it also contains other interesting numerical allusions.
First of all, it is a four-voice fughetta and the theme is presented ten times (4×10 = 40). The same relationship can be seen in the exposition of the fughetta: ten bars of four dotted eighth notes (10×4 = 40). The theme runs for ten beats. Thus we see the same relationship in the exposition: 10×4 = 40. The theme in the second exposition is presented in inversion and in a shortened form (six beats). The relationship is correspondingly 6×4 = 24. And finally, the last two stretti quotations of the theme (bars 32–35) give us the relationship 8×2 = 16, as the theme here is eight beats long. It is not difficult to see that the addition of 24 and 16 results in the key number 40, which is apparently a reference to the Jewish people’s forty years of wandering in the wilderness before being given the stone tablets with the Decalogue.
The theme has a most interesting structure. It consists of two parts: the main melody of the chorale emerging from a repeated ostinato note and its leaps (six beats), and stepwise motifs over a fifth (four beats). (Example 8) Christoph Albrecht described the theme figuratively as a musical picture of a “raised warning forefinger.”54 But numerology allows us to find deeper connotation in it. The second part of the theme contains 14 notes (BACH). One could consider this as a mere coincidence, were it not that we meet the melody with this numerical symbol again at other central formative points in this little piece.
This second part of the melody occurs as a theme in its own right in the 41st beat of the fughetta (JSBACH), where it fills out the eleventh bar at the junction between the two expositions. Again, this melody is consistently developed in the 14 bars that separate the two concluding quotations of the theme from the second exposition. And we would finally add that the number 14 is underlined by the sum total of all the beats in this chorale prelude: they all add up to 140.
Without a doubt it would be the very height of negligence for a performer who is looking for an authentic interpretation to ignore the manifold recurrence in the composition of the name of its creator. The composer of the manual version of Dies sind die heil’gen zehn Gebot’ obviously had definite reasons for weaving his name again and again into the musical fabric of the work.
Let us boldly assume that in this work Bach wishes to embody the idea of the divine Commandments as the cornerstone of his own life. The tenfold repeated theme of the chorale Dies sind die heil’gen zehn Gebot’ and the numerical symbol 40 harbor the idea of the Commandments. Their importance for Bach personally is attested to by the composer’s repeated use of the symbol 14.

This article will be continued.

Carl Sloane is a pharmacist by education, a freelance translator by vocation, and an amateur harpsichordist.

In his famous treatise on continuo playing,1 Francesco Gasparini gives a table of twenty-one keys which the student could expect to encounter. The absence of certain keys proves that the contents of the table are determined by the tuning which Gasparini used, and because of his standing, it would be of particular interest if the precise nature of this tuning could be established.

There are immediate indications that Gasparini had in mind some type of regular meantone (RMT)2 with the usual range of Eb to G# and the wolf between G# and Eb. These indications, supported by substantial evidence that meantone was common in Italy well into the eight-eenth century,3 include the complete absence of Ab and the apparent expectation that the table would be universally applicable. However, the matter cannot be automatically considered settled, since the the book was written at a time when meantone was being discarded in favor of temperaments with no wolf. Tempérament ordinaire, which eliminated the wolf by spreading it over several fifths, had been in use in France for some time when it was described by d'Alembert,4 and Werckmeister's tunings,5 which usually solved the problem by reducing the number of tempered fifths, had been published at least by 1691. In addition, there are apparent inconsistencies in the table itself.

The keys are illustrated in the form of figured scales which begin on the tonic, rise to the sixth degree, descend a ninth to the dominant, then leap back to the tonic. They are divided into two groups, those of "great usefulness" (gran giovamento):

G, g, a, A, Bb, b, C, c, d, D, Eb, e, E, F, f, f#, and those used in modulations:

bb, B, c#, eb, F#.

Major keys in group 1 thus run from Eb to E around the circle of fifths, minor keys from F to F#.

Tagliavini6 argues against RMT, pointing out that the absence of Db major is difficult to understand when the more highly inflected Gb major (enharmonic F#) is present. To his rhetorical suggestion that this paradox may be due to the presence of the wolf in the dominant chord of Db, he replies that C# minor should then logically be absent as well. He does not attempt to establish the criterion used to exclude keys from group 1.

The composition of group 1 does not initially seem compatible with the use of RMT: F minor, with its poor tonic chord, and E and Eb majors, with a poor chord on V and IV respectively, would not be expected in this group. On the other hand, there is plenty of contemporary evidence showing that poorly tuned intervals were used regularly in practice (Ref. 3, 156-8, 193), and on this basis, the makeup of group 1 can be logically explained.

In RMT, the most complex major keys in group 1, Eb and E, each have a single note outside the range Eb to G#; the most complex minor keys, F and F#, each have two such notes, at least one of which is on the sixth or seventh degree (see Fig. 1). Accordingly, the keys in group 1 may have been chosen on the understanding that major keys were allowed a maximum of one unavailable note and minor keys a maximum of two, the greater freedom in the minor keys being due to the variable inflection of VI and VII.

It is worth noting that even in group 2, Gasparini's key signatures never have more than three flats or four sharps, thus staying within the same limits as the major keys in group 1. (The section on modulation--(pp. 111-114)--gives key signatures which exceed these limits, but it also illustrates keys not included in the table, so that Gasparini has here presumably sacrificed some degree of rigor.) In addition, the order within each pair of parallel keys in group 1 is obviously determined by the complexity of the key signature, suggesting a certain preoccupation with key complexity and unavailable notes.

 Owing to the manner in which unavailable notes enter as one moves around the circle of fifths, exact location in the scale was probably of secondary importance in group-1 keys. But ultimately position must have become of critical importance. From Fig. 1 it is apparent that the most elementary keys not in the table would have an unavailable note on at least one of I, II or V. Although the presence of the wolf on either I or V (an unavailable note on one member of either of the pairs I/V or II/V) may have been the underlying reason for outright rejection, I think that a more likely working criterion was the spelling of I, II and V, and that the presence of F# major does not imply the inclusion of Gb, any more than the presence of C major implies the inclusion of B#. It seems likely, especially in view of the more lenient treatment of minor keys in group 1, that the presence of the wolf on the dominant of C# minor (or the wrong spelling for II) was considered acceptable. This hypothesis is admittedly not as credible as it would be if Gasparini had not figured V with a major third, because some softening of the effects of the wolf would be expected in certain positions of the chord G#-B-D#, especially of the first inversion, by the presence at the relevant pitch of a partial from the B-natural.7

To the extent that RMT is established, Gasparini's table shows that, contrary to most modern opinion, G# was not retuned to Ab for compositions in F minor.7 In addition, the table gives a valuable clue to the tuning used by Domenico Scarlatti. There is almost nothing in the Venice and Parma codices to suggest that Scarlatti retuned for F minor, in which he wrote extensively (in the Parma codex, it occurs more often than any other minor key), but there is some rather pretty evidence    that he retuned for Ab major and several of the keys in group 2.8 One is therefore strongly tempted to conclude that Scarlatti used the same tuning as Gasparini.

Notes

                        1.                 Francesco Gasparini, L'Armonico Pratico al Cimbalo (1708; reissue, New York: Broude Bros., 1967), 83-6.

                        2.                 "Regular" means only that the eleven good fifths are the same size.

                        3.                 Patrizio Barbieri, Acustica, Accordatura e Temperamento nell'Illuminismo Veneto (Rome: Torre d'Orfeo, 1987), 152-8.

                        4.                 Jean-Le Rond d'Alembert, Elemens de Musique Theorique et Pratique (1752; reissue, New York: Broude Bros., 1966), 48-9.

                        5.                 Andreas Werckmeister, Musicalische Temperatur (1691; reissue, Utrecht: Diapason Press, 1983), 78-9.

                        6.                 Luigi Ferdinando Tagliavini, "L'Armonico Pratico al Cimbalo. Lettura Critica," in  Francesco Gasparini (1661-1727)--Atti del primo convegno internazionale (Comune di Camaiore) (Florence: Olschki, 1981), 133-55, at 149-51.

                        7.                 C. Sloane, "A Further Note on Tempered Minor Chords," Journal of Sound and Vibration 170, 2 (1994): 261-2.

                        8.                  Carl Sloane, "The Case for Meantone in Scarlatti," Continuo, 16, 6 (1992): 1516.