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.
