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.
