This article first takes up the question of the authentic title of Bach's last published work. Thereafter, this paper demonstrates a relation between the theme of the unfinished fugue of this composition and the well-tempered tuning "wohltemperirt". Already an earlier study1 in The Diapason identified possible allusions within Die Kunst der Fuga to the tuning Werckmeister/Bach for which Das Wohltemperirte Clavier was composed. Bach used the set of numbers belonging to the system "wohltemperirt" to arrange for the setup of his compositions, succeeding in this way to unify the musical harmony with architecture. Such structuring can be of a very abstract nature and concern even the melody of themes, number of keystrokes, or bar-lengths of movements, etc. The present case as well will show a rather complex and abstract means of structuring, but unambiguously recognizable.
Due to the way Die Kunst der Fuga was edited and published, there have always been some doubts and queries as to the exact intentions of the composer, including the authenticity of the title. It reads--in the printed edition--Die Kunst der Fuge. However, there also exists a manuscript version of the title page written by Bach's son-in-law, Johann Christoph Altnikol (Deutsche Staatsbibliothek Berlin, Mus. ms. Autogr. Bach P 200), that once belonged to the collection of Georg Poelchau. Altnikol, close to Bach, assisted the composer suffering from failing health and eyesight in preparing the publication. On Altnikol's manuscript, Figure 1, the title page reads Die Kunst der Fuga [sic]:
Now the gematria will be applied2, as commonly used by baroque composers and writers--even preceding Bach--such as Kuhnau3 or Werckmeister4. The title of Johann Sebastian Bach's last work, in Altnikol's spelling, shown together with the composer's full name thus reads:
Thanks to Altnikol's manuscript, the likely title in its authentic spelling is established, as it appears improbable that this gematrial equivalence represents a mere co-incidence. Within Bach's holograph of Das Wohltemperirte Clavier the composer himself always spelled "Fuga" and never "Fuge". On the contrary, the spelling of Die Kunst der Fuge [sic] results in 162=2x9x9. No gematrial significance whatsoever appears.
At Bach's time the most common numbering of the alphabet was from 1 to 24, A=1, I, J=9, U,V=20 and Z=24, as shown5 in Figure 2. It is this numbering that leads to the conclusion above.
However, the triangular alphabet was in use as well and also known to Bach, and both alphabets are tabulated in Figure 3, the triangular values follow from any Z of Dieben's alphabet via Zx(Z+1)/2:
The finality to present this table is, of course, to check Die Kunst der Fuga via this alternative, triangular alphabet. The latter converts Werckmeister's Musicalische Temperatur to the value of 19756 (whereas according to Henk Dieben the outcome is 247=13x19). With the number 1975 one may first recognize 19 as twelve fifths and seven octaves that close the circle, 19 intervals, 12+7. The remaining juxtaposition 75 shows the 7 perfect and the 5 well-tempered fifths of the system. Werckmeister's title thus allows the gematrial conversion via two distinct alphabets! After this introductory preparation, the supposedly authentic title will be now converted via the triangular alphabet as well. One finds Die Kunst der Fuga = 1225=(5x7)x(5x7) broken up into prime numbers. A double conversion yields a perceptible result in this case as well: 5 well-tempered fifths and 7 perfect ones of Werckmeister's and Bach's system "wohltemperirt". As 5x7=35, the figures 3=third and 5=fifth in thorough-bass appear--in the C-major triad these intervals beat in unison. The first prelude of Das Wohltemperirte Clavier comprises 35 bars. This title, Die Kunst der Fuga, can certainly be said to be scrupulously selected/constructed--and certainly not understood up to now.
The second part of this study will investigate the structure of the first theme of the last, unfinished fugue, in relation to the tuning Werckmeis-ter/Bach upon which Das Wohltemperirte Clavier is based. In this system wohltemperirt, 5 tempered fifths and 7 perfect ones close the circle of 7 octaves. This amounts to a division of the Pythagorean comma by five. Refining to more detail, there are the 4 usual well tempered fifths c-g-d-a-e, together with 1 tempering fifth B-f#, and the remaining 7 fifths are perfect7. A corresponding set of these numbers of fifths characterizing the musical temperament looks like 4, 1, 7.
At this stage, a small, but necessary, mathematical excursion follows. From Euclid and Greek mathematics, the "perfect numbers" were considered. These numbers are made up by the sum of their divisors, such as for the first perfect number, 6=1+2+3. Likewise, for the next one 28=1+2+4+7+14, being divisible in turn by 1, 2, 4, 7 and 14. It is remarkable that Werckmeister and other musicians as well8 were apparently fascinated by these numeri perfecti and in several of his treatises he quotes 6, 28, 496, 8128, 33550336. There exists an infinity of perfect numbers, but up to now it is unknown whether there are any odd perfect numbers--all perfect numbers known to date are even. These numbers considered perfect may be the reason that baroque and earlier composers grouped their works by numbers of 6 for publication.
Now the characteristic set of numbers 4, 1, 7 of "wohltemperirt", (4 well-tempered fifths, 1 tempering fifth B-f# and 7 perfect fifths), will be assembled with its crabwise permutation 7, 1, 4 and centered upon 6, primus numerus perfectus. The perfectly symmetrical constellation 4-1-7-6-7-1-4 results. Gematria-wise, letters are associated to these numbers. The outcome of this procedure is D-A-G-F-G-A-D. Here one recognizes the unfinished fugue's theme of the composition, Fig. 4.1 and 4.2.
In the final rhythmic layout of these notes, by the fourth note, a sudden ritardando halving of the tempo might be felt, giving to the theme its tragic austerity, Fig. 4.3.
As regards the significance of the digits within the set 1, 4, 7, it must be remembered that the prelude of the tempering tonality B-major in WTC I counts 417 keystrokes9. Within the Well-tempered Clavier I and II comprising an even number of 5750 bars--counting through the entire composition--the number of the two central bars are 2875 and 287610. Thus, the midpoint will be bar 741 of Part II (2876-2135=741), again with the digits 1, 4, 7: the numbers of fifths.
In conclusion, these considerations provide a clue to the "authentic" title of Bach's last composition printed. Also, the characteristic numbers specifying "wohltemperirt" were identified as the theme of the last, unfinished piece of Die Kunst der Fuga. At this occasion, two recent publications investigating the composition might be quoted, by Hans-Jörg Rechtsteiner11 and above all, Vincent Dequevauviller12, the latter distinguished by a remarkable originality of ideas, sadly missing in the Bachjahrbuch--due to the unfortunate publishing policy of the present editors.
This paper was written in commemoration of Bach's passing away 250 years ago on 28. 7. 1750. By that date, at the middle of the 18th century--written according to European usage--will look as 28.7.'50, the central bar of Das Wohltemperirte Clavier reappears. Likewise, Bach's birthday on 21.3.1685, late in the 17th century, appears as 21.3.'85. Here, reconverting via the number alphabet of Henk Dieben, 2138 appears: BACH.
Notes
1. Kellner, Herbert Anton, "How Bach encoded his name into Die Kunst der Fuge together with his tuning." The Diapason, May 1999, 14-15.
2. Kellner, H. A., "Le tempérament inégal de Werckmeister/Bach et l'alphabet numérique de Henk Dieben." Revue de Musicologie 80/2, 1994, 283-298.
3. Kellner, H. A., "Welches Zahlenalphabet benutzte der Thomaskantor Kuhnau?" Musikforschung 33/1, 1980, 124-125.
4. Hermann, Ursula, "Andreas Werckmeister--Lebensweg und geistiges Umfeld." In: Bericht über das Werckmeister-Kolloquium aus Anlaß des 340. Geburtstages von Andreas Werckmeister am 30. November 1985, Michaelstein/Blankenburg 1986, Studien zur Aufführungspraxis und Interpretation von Musik des 18. Jahrhunderts, Heft 30, p. 5.
5. Harsdoerffer, Georg Philipp, Poetischer Trichter, Nürnberg, Wolffgang Endter MDCL. Faksimile Wissenschaftliche Buchgesellschaft Darmstadt, 1975.
6. Duparcq, Jean-Jacques, private communication.
7. Kellner, H. A., The Tuning of my Harpsichord. Schriftenreihe Heft 18. Verlag Das Musikinstrument, E. Bochinsky, Frankfurt/Main 1980.
8. Werckmeister, A., Musicalische Temperatur, p. 14, 19; see also Walther, Johann Gottfried, Musicalisches Lexicon, oder, Musicalische Bibliothek. Leipzig, 1732, Ed. R. Schaal, Kassel 1953.
9. Kellner, H. A., "Das wohltemperirte Clavier--Tuning and Musical Structure." English Harpsichord Magazine 2/6, April 1980, 137-140, in particular p. 139.
10. Kellner, H. A., "Das wohltemperirte Clavier--Implications de l'accord inégal pour l'oeuvre et son autographe." Revue de Musicologie 71, 1985, 143-157.
11. Rechtsteiner, Hans-Jörg, Alles geordnet mit Maß, Zahl und Gewicht. Der Idealplan von Johann Sebastian Bachs Kunst der Fuge. Peter Lang, Europäische Hochschulschriften, Reihe 36, Musikwissenschaft, Vol. 140, 1995. P. 44, Completing the unfinished fugue by 133 bars is absurd; Davitt Moroney's completion by 30 bars in his Henle-edition is much more reasonable.
12. Vincent Dequevauviller, L'art de la fugue, un "problème algébrique," Etude sur les caractéristiques numériques et les raisons de l'inachèvement de la dernière oeuvre de Jean-Sébastien Bach. Association pour la Connaissance de la Musique Ancienne, 1998, ISBN 2-9513089-0-6. The proposed completion of the unfinished fugue by 37 bars is correct.
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