Nunc Dimittis

by Herbert Anton Kellner

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.

© Staatsbibliothek zu Berlin-Preußischer Kulturbesitz, Musikabteilung mit Mendelssohn-Archiv

Arthur Carkeek, professor emeritus of organ and theory at DePauw University, Greencastle, Indiana, died October 19, 2003 at the age of 80. Born April 7, 1923, in Detroit, Michigan, he was a chorister at St. Paul's Cathedral in Detroit. Following high school he attended Wayne State University and the Detroit Institute for Musical Arts simultaneously. While serving in World War II as a chaplain's assistant and waiting to be sent to Europe, Mr. Carkeek assisted in the maintenance of the organ in the Atlantic City Convention Hall, later writing his master's thesis on that unique organ. He also gave weekly radio recitals on the Convention Hall organ. Following his Army discharge, he completed his undergraduate work at DePauw University, graduating in 1948 and receiving his AAGO certificate the same year.

Arthur Carkeek graduated from Union Theological Seminary in 1950 and returned to DePauw to teach at the bidding of his former teacher, Van Denman Thompson. Upon Thompson's retirement in 1956, Carkeek became the university organist at DePauw. During his 38-year teaching career at DePauw University, Arthur Carkeek produced many outstanding students, who went on to careers as organists, university professors, clergy, organ builders, competition winners and Fulbright scholars. He was active as a performer, lecturer, panelist and writer. Receiving grants from the Great Lakes Conference and the Ford Foundation as well as sabbatical leaves from DePauw, Carkeek studied organ building with Rudolph von Beckerath and organ with Charles Letestu. He performed many concerts on historic instruments in Germany, including a recital in Altenbruch.

Carkeek produced a number of scholarly articles, most notably a series of articles on his long-time friend Rudolph von Beckerath, published in four installments in The American Organist (1996). A further article on Beckerath will be published posthumously in the Encylopedia of Keyboard Instruments, Vol. 2, The Organ Encylopedia. In 1972 Carkeek made a recording of several organs by Charles Fisk at Harvard, Old West Church (Boston) and DePauw.

In demand as an organ consultant, Arthur Carkeek constantly supported the cause of many fine instruments. He acted in that capacity at Christ Church Cathedral in Indianapolis where a Hellmuth Wolff organ was installed in the chancel and a Taylor & Boody organ was installed in the rear gallery.

Arthur Carkeek served as the director of music at Gobin United Methodist Church and St. Andrew's Episcopal Church, both in Greencastle, Indiana. In 1998 a fire at St. Andrew's destroyed the existing pipe organ that Carkeek had nurtured over the years. That instrument was replaced in September, 2002 with Op. 1 built by Joseph Zamberlan and was dedicated in honor of Arthur Carkeek.

In 2001, Arthur Carkeek was given a lifetime honorary membership in the American Guild of Organists by the Indianapolis Chapter. He was also a member of Pi Kappa Lambda and the Association of Anglican Musicians.

A Solemn Evensong and Eucharist was celebrated on October 24, 2003 at St. Andrew's. Participants included former students, DePauw faculty, and members of the choir of Christ Church Cathedral, Indianapolis. The Arthur Carkeek Memorial Concert Fund has been established at St. Andrew's Episcopal Church, Greencastle, Indiana. He is survived by his wife Maureen  (McCormick) Carkeek, a daughter, a son, and two grandchildren.

--Richard Konzen

Halbert Scranton Gillette, chairman of the board and CEO of Scranton Gillette Communications, which publishes The Diapason, died on November 22, 2003, at his home in Lake Forest, Illinois, after a long battle with cancer. He was 81.

Born in Chicago, Illinois, June 29, 1922, the son of Edward Scranton Gillette and Claribel Reed Thornton, and raised in Chicago and Winnetka, Illinois, Mr. Gillette attended The Chicago Latin School and graduated from the Philips Exeter Academy. In 1944 he graduated from Massachusetts Institute of Technology with a degree in mechanical engineering and business. He was commissioned U.S. Navy 1944-1946, and served in the U.S. mainland during World War II and in the Naval Reserves. He was chairman of the board and CEO of Scranton Gillette Communications, Inc., which was founded in 1906 by his grandfather. Mr. Gillette started as a salesman for Gillette Publishing in 1947. In 1960, two-thirds of Gillette Publishing Co. was sold to Reuben H. Donnelley, which then was merging with Dun & Bradstreet. Mr. Gillette also moved to Donnelley/Dun & Bradstreet as a publisher and a vice president. In 1970, he rejoined his father's firm, then Scranton Publishing Company, and shortly become president of the firm, which was renamed Scranton Gillette Communications.

Mr. Gillette served as past president of the Chicago Business Papers Association, as well as on the board of several insurance companies. He was the former Chairman of the Board of Occidental Life Insurance. He served as alderman in Lake Forest, Illinois, 1979-1986, and served on the Public Safety and Waterfront committees. He was co-chairman of the committee that oversaw the creation of the city's current beachfront.

He was a member of the Church of the Holy Spirit in Lake Forest, and Church of the Holy Innocents, in Lahaina, Hawaii. He was also a member of the Onwentsia Club of Lake Forest; the Les Cheneaux Club, Cedarville, Michigan; and the Lahaina Yacht Club, Hawaii. Husband of Karla Ann Spiel Gillette; father of Anne, Susan, James, Halbert and Edward; grandfather of Alexander, Madeline, Carolyn, Julia, and Isabelle.

Thyra Nichols Plass died on October 27, 2003, in Bryan, Texas, at the age of 89. She was born in Green Valley, Illinois, on April 8, 1914, and lived in Bryan since 1968. Mrs. Plass earned her bachelor's and master's degrees from the University of Chicago, and her doctor of sacred music from Union Theological Seminary in New York City. A retired organist and choirmaster, she was a member of the Brazos Valley Chapter of the American Guild of Organists, the Association of Anglican Musicians, and of St. Andrew's Episcopal Church, Bryan, Texas. In addition she was a member of The Women's Club, a founding member of the Arts Council of the Brazos Valley, co-founder of the annual children's symphony concerts, and a member of OPAS Guild. She is survived by her husband Gilbert Norman Plass, a daughter, and six grandchildren.

Ronald Edward Ballard, of Little Rock, Arkansas, died on January 23, from kidney, liver, and heart disease. Born on January 9, 1947, in North Little Rock, he was a graduate of Oklahoma Baptist University. He wrote theater and concert reviews for the Spectrum and Arkansas Times and was business administrator for Stanton Road School and First Christian Church in Little Rock. At the time of his death, Ballard was organist and choirmaster of Westover Hills Presbyterian Church in Little Rock. He previously served a number of churches in Little Rock, North Little Rock; Charlotte, North Carolina; and Brenham, Texas; and taught music, history, and mathematics in North Little Rock Public Schools and Stanton Road School. He served as dean of the Central Arkansas AGO chapter in 1976–77.

George A. Brandon died on March 30 in Davis, California, following a short illness. He was 76. Born in Stockton, California on February 4, 1924, he earned a BA in history at College of the Pacific in 1945 and a Master of Sacred Music degree in 1952 at Union Theological Seminary. He served as organist-director at several churches in the New York City area, and before returning to Union in 1955 to earn a master's in religious education he served two years as organist-director of the First Presbyterian Church in Burlington, North Carolina. While at Union, he met Dona Lee Banzett, whom he married in 1954. They taught for five years at two small midwestern colleges and then relocated to Davis, California, in 1962. During the 1960s, Mr. Brandon held positions at several Davis churches, including Incarnation Lutheran, Davis Community Church, and St. Martin's Episcopal Church. In 1994 he was commissioned by St. Paul's Episcopal Church in Sacramento, California, to write the "St. Paul's Suite," an organ work based on the last chorus of Saint-Saëns' Christmas Oratorio, for a concert that celebrated the restoration and 80th anniversary of the church's 1877 Johnson organ (opus 503). Mr. Brandon was a free-lance composer, with over 300 published compositions, including anthems, hymn tunes and texts, choir responses, secular choruses, organ and piano pieces. He researched and wrote about many aspects of church music and related fields, especially early American hymnody. He was a member of the AGO, the Hymn Society of the US and Canada, the Hymn Society of Great Britain and Ireland, and the Sonneck Society. A celebration of Mr. Brandon's life and work took place at St. Martin's Episcopal Church, Davis, on May 17. He is survived by his wife of 45 years, Dona Lee Brandon, a daughter, and a sister.

Otto Juergen Hofmann, organbuilder of Austin, Texas, died on May 12, at the age of 82. He was born of German immigrant parents in Kyle, Texas on December 9, 1918, the youngest of ten children. He attended the University of Texas, studied physics, music, philosophy, and sociology, and had a PhD in physics from UT-Austin. Hofmann built his first slider-chest and mechanical-action organ in 1938. One of his first contracts was to rebuild the organ at St. Mary's Cathedral, Austin. He then built the organ for St. Stephen's Episcopal School. This was soon followed by an organ for the chapel of Austin Presbyterian Theological Seminary. Hofmann built, rebuilt, and restored numerous organs throughout Texas and beyond. He was involved in the early efforts of the tracker organ revival, and collaborated with Flentrop in building an organ for Matthews Memorial Presbyterian Church in Albany, Texas, in 1955. He served as president of the International Society of Organbuilders, and in 1975 was awarded the Industrial Arts Medal by the Austin chapter of the American Institute of Architects. He retired from active organ building in 1994.

James Dale Holloway died on May 17, the random victim of a shooting on the campus of Pacific Lutheran University, Tacoma, Washington. Holloway, age 40, was appointed assistant professor of music and university organist at PLU in the fall of 2000. Born in Columbus, Georgia, on July 4, 1960, Holloway received a BMus from Shorter College in Rome, Georgia; a master's degree from the University of North Texas; and after beginning doctoral studies at the University of Alabama, completed his doctorate at the University of Washington. As a church musician, he served parishes in Georgia, Texas, Alabama, and Oregon, before moving to Tacoma in 1989 to become minister of music at Trinity Lutheran Church. For ten years he taught part-time at PLU before his full-time appointment in 2000. He was a performer and lecturer at national and regional conventions of the AGO, the OHS, and the Association of Lutheran Church Musicians. He was current president of the western regional ACLM, and was a winner in 16 organ competitions at the state, regional, and national levels. A memorial fund, the "James Holloway Music Scholarship Fund," has been established at PLU in his memory. A memorial service took place on May 21 at the university. He is survived by his wife, Judy (Willis) Carr, and five stepchildren.

Antonio Ruffatti died on May 6, in Padova, Italy, at the age of 89. He co-founded Fratelli Ruffatti, organbuilders, in Padova in 1940--following a centuries-old tradition in that geographic area of Italy--a firm which continues today under the direction of his sons Francesco and Piero. In the 1950s, he succeeded in making Fratelli Ruffatti known internationally by building a five-manual organ for the Shrine of Our Lady of Fatima in Portugal. In the following decade, he built many organs in the United States. Following the classical tradition in his own country, he was among the first in Italy to call for the return to mechanical action. Fratelli Ruffatti built several tracker instruments in the early 1960s, a practice which was highly uncommon, if not controversial, at the time in Italy, but which has grown to become an important part of the current activity of the firm. He also delved into restoration techniques, and today Fratelli Ruffatti is one of the few firms licensed by the Italian government to do historical restoration on Italy's ancient instruments.

Gordon W. Brooks died on January 20 in Warren, Ohio, at the age of 82. He was born on February 22, 1916 in Mineral Ridge, Ohio, attended Niles McKinley High School, and earned the BA in organ from YSU Dana School of Music. He was a piano and organ teacher for many years and was organist at Indianola Methodist Church in Youngstown; organist at several Lutheran churches in the Youngstown and Niles area; music director at First Presbyterian Church in Warren for 14 years; and organist at Robert H. Roberts-Clark Memorial Home for 10 years. An Army Air Corps veteran of World War II, he was a member of the National Gmanfa Ganu Welsh Society, where he was organist for 50 years and formed the Welsh quartet "Cor Bach Cymreig."

Patricia "June" Kean died last November in Springfield, Missouri. She was born on June 22, 1933, in Fort Worth, Texas, and was married to Barry Kean in 1959. She was preceded in death by her husband, and is survived by three sons, three grandchildren, and two sisters. Dr. Kean completed the BMus at Oklahoma City University in 1954, the MMus in piano at Eastman School of Music in 1957, and the DMA at North Texas State University in 1973. She served for many years as organist at University Heights Baptist Church and Minister of Music at Calvary Temple, as well as organist for King's Way United Methodist Church in Springfield. She had played recitals in Argentina, Austria, Germany, Poland, Ukraine, China, and the US. A long-time member of the AGO, Dr. Kean served as Dean of the Springfield chapter. She was also a member of Pi Kappa Lambda, the Springfield Music Club, and the Missouri Teachers National Association.

Lawrence I. Phelps died on February 22 of double pneumonia at a hospital in Boston. He was 75 years old. He was born on May 10, 1923 in Boston, and studied conducting and organ at the New England Conservatory of Music. He apprenticed with G. Donald Harrison at the Aeolian-Skinner Organ Company in 1944 and worked for the company for five years, and then spent a year as a voicer and tonal finisher with Walter Holtkamp. In 1949 he became an independent consultant and was engaged by the Christian Science Board of Directors to oversee the design, installation and tonal finishing of two organs for the First Church of Christ, Scientist, Boston. He was appointed tonal director of Casavant Frères in 1958, set up a division at Casavant for mechanical action organs in 1961, and remained with the firm until 1971. Phelps operated his own firm in Erie, Pennsylvania 1973-78. Among the organs he built are the IV/74 mechanical action organ for the Cathedral of SS. Peter and Paul in Providence, Rhode Island, and the 1973 installation at St. Luke's Church, Ft. Collins, Colorado. From 1982-1995 he was tonal director of Allen Organ Company. Prior to this he was responsible for nearly 800 pipe organs, 60 of which were mechanical action. He left Allen to become curator of organs at the Mother Church in Boston. His first marriage was to Ruth Barrett, organist of the Mother Church, Boston. He later married British organist Gillian Weir. He was a frequent lecturer at organ builder conferences and wrote many articles for a number of organ journals including The Diapason. A memorial service will be held on April 17 at 2 pm at Trinity Church, Copley Square, Boston.

--David Burton Brown

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.

Lorene S. Banta (Mrs. Cornelius Gordon S. Banta) died on November 22, 2001 in Winter Park, Florida. Born on May 18, 1914, Dr. Banta and her husband moved to Winter Park in 1978 from Massachusetts when they retired from teaching at Phillips Academy in Andover. In addition to degrees in music, Banta received the PhD in classical literature from the University of Michigan, and held several church positions in the Boston area as well as in Orlando. Dr. Banta was associate professor at Queen’s College, North Carolina, until she left to be married in 1948 in Washington Cathedral. She and her husband then moved to Phillips Andover where they taught for 30 years. A member of the AGO, Dr. Banta served for many years on the staff of The American Organist magazine. A memorial service was held on December 13 at the First Congregational Church of Winter Park.

Paul Hume, longtime music critic for the Washington Post, died on November 26 in a Baltimore nursing home at the age of 85. Hume will always be remembered as drawing the famous rebuke from President Harry S. Truman when he panned a voice recital given by Truman’s daughter Margaret on December 5, 1950. Paul Chandler Hume was born in Chicago. He studied piano for seven years, organ for four years, and voice for seven years, and graduated from The University of Chicago with a major in English. In 1946 he joined the staff of radio station WINX and in 1947 became the Post’s music critic. He retired in 1982. In addition to his career at the Post, Hume taught music history at Georgetown University from 1950 to 1977 and was visiting professor at Yale University from 1975 to 1983. His books include a 1977 biography of Verdi and a 1956 study of Catholic church music. In the early 1950s Hume was the baritone soloist at Washington National Cathedral, where he also gave organ recitals. For 25 years he was director of the Georgetown University Glee Club.

Heinz Lohmann died on March 11, 2001 in Berlin, Germany, at the age of 66. Organist at Berlin’s Heilsbronnen Church from 1971 until several years ago, when Parkinson’s disease prevented him from playing, he was also a composer, teacher, and author of articles and books. Lohmann made 38 recordings, notably of music by Max Reger. His editions of the organ works of Bach, Walther and Zachow are published by Breitkopf & Härtel.