In eighteenth-century Germany the universal harmony of God’s creation and the perfection of its proportions still held philosophical, moral and devotional significance. Reproducing proportions close to the unity (1 : 1) across compositions could render them beautiful, perfect and even eternal. Using the principles of her ground-breaking theory of proportional parallelism and the latest source research, Ruth Tatlow reveals how Bach used the number of bars to create numerical perfection across his published collections, and explains why he did so. Thefirst part of the book illustrates the wide-ranging application of belief in the unity, showing how planning a well-proportioned structure was a normal compositional procedure in Bach’s time. In the second part Tatlow presents practical demonstra-tions of this in Bach’s works, illustrating the layers of proportion that appear within a movement, within a work, between two works in a collection, across a collection and between collections.
British-Swedish musicologist Ruth Tatlow is an independent scholar based in Stockholm. Her research into Bach’s use of numbers led from her classic monograph Bach and the Riddle of the Number Alphabet (Cambridge, 1991) to this sequel, Bach’s Numbers, through publications on methodology, inventive techniques and the theory of proportional parallelism. In 2004 she co-founded Bach Network UK (BNUK), establishing its open-access web-journal Understanding Bach in 2006. She is currently Chair of the BNUK Council, joint editor of Understanding Bach, and a member of the Editorial Board of the American Bach Society. Her research has attracted awards and grants from numerous sources including the Swedish Research Council, the Royal Swedish Academy of Music, the Society of Authors of Great Britain, the British Council, The Hinrichsen Foundation, The Leverhulme Trust and the Royal Swedish Academy of Letters.
Compositional Proportion and Signi
ficance
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1. Bach, Johann Sebastian, 1685–1750 – Criticism and interpretation. 2. Symbolism of numbers in music. I. Title.
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Louisa, Emily
Benjamin and Anita
with love
List offigures [pageix] List of tables [x] Acknowledgements [xv] List of abbreviations [xvii]
pa r t i f o u n d at i o n s [1] 1 Bach’s numbers [3]
2 Symmetry, proportion and parallels [36]
3 Unity, proportions and universal Harmony in Bach’s world [73] 4 Bars, compositional planning and proportional
parallelism [102]
pa r t i i d e m o n s t r at i o n s [131] 5 Three collections for strings [133]
6 Four in two collections for keyboard [159] 7 Two further collections for keyboard [182] 8 Two small late collections [204]
9 Two large late collections [224] 10 Collections of concertos [255] 11 Collections of organ works [275] 12 Great passions and cantatas [294]
13 Festive cut-and-paste projects: masses and oratorios [326]
14 Lost compositional blueprints [354]
Appendix: A theology of musical proportions and Harmony in Bach’s time [370]
Bibliography [383] Index of sources [396] Index [400]
2.1 Puttenham, The Arte of English Poesie (London, 1589), 72
Proportion by situation. Courtesy of Sterling Library, Senate House Library, University of London [page40]
2.2 Puttenham, The Arte of English Poesie (London, 1589), 76 Proportion infigure. Courtesy of Sterling Library, Senate House Library, University of London [41]
2.3 Pachelbel, Hexachordum Apollinis (Nürnberg, 1699). Chronogram by Beer. Courtesy of Sibley Music Library, Eastman School of Music, University of Rochester [48]
2.4 Buttstett, Wieder das Beschützte Orchestre (Erfurt, 1718). Courtesy of Library of Congress Music Division, Washington, DC [54]
2.5 ‘Johann Sebastian Bach’ in Kircher’s musical alphabets (Kircher, Musurgia Universalis (Rome, 1650), Tome II, Book IX, 362. Alphabetum steganographicum musicum. Courtesy of Kungliga Biblioteket, Stockholm. Callmark: 147 A. a. Fol. [63]
3.1 Puttenham, The Arte of English Poesie (London, 1589), 53. Of Proportion Poeticall. Courtesy of Sterling Library, Senate House Library, University of London [76]
3.2 Mattheson, Das Beschützte Orchestre (Hamburg, 1717), 478. Courtesy of Sibley Music Library, Eastman School of Music, University of Rochester [82]
4.1 Mizler, Musicalische Bibliothek, Vol. IV, Part 1 (Leipzig, 1754), 108. Courtesy of Sibley Music Library, Eastman School of Music, University of Rochester [117]
4.2 Müller, Wohlmeynender Unterricht (Leipzig, 1743), 172–3. Courtesy of Kungliga Biblioteket, Stockholm [124]
Tables
2.1 B/H-A-C key pattern across Clavier Übung I and II. Original print [page 64]
2.2 Statistical survey of Bach’s signatures [66]
2.3 Bach’s signatures in three common number alphabets [67] 4.1 Symmetry of intervals in the hexachord [103]
4.2 Symmetry of semitones in B-A-C-H [103] 5.1 Six Solos. Autograph score, P 967 [135]
5.2 Proportioned pair: Sonata 1 and Partita 1. Autograph Score, P 967 [137]
5.3 Proportion in Partita 3 in E major. Autograph score, P 967 [138] 5.4 Six Solos. Autograph score, P 967. Six Sonatas. Copyist Altnickol,
P 229 [140]
5.5 Large-scale movements. Fuga and Ciaccona. Autograph score, P 967 [142]
5.6 Five Solos for violin. Copyist Kellner, P 804 [144] 5.7 Revisions to the Six Solos. P 804, and P 967 [145] 5.8 Six Sonatas for violin and harpsichord. Copyist Altnickol,
P 229 [147]
5.9 Six Sonatas. Early andfinal versions compared. St 162 and P 229 [148]
5.10 Changes to Sonata 6. St 162 and P 229 [148]
5.11 Parallels between two violin collections. P 967, St 162 and P 229 [149]
5.12 Six Cello Suites. Copyist Anna Magdalena Bach, P 269 [152] 5.13 D-major Cello Suite. Copyist Anna Magdalena Bach, P 269 [153] 5.14 Proportion formed between movements. Cello Suites, P 269 [154] 5.15 A second large-scale proportion. Cello Suites, P 269 [154] 5.16 Further evidence of proportioning. Cello Suites, P 269 [155] 5.17 An earlier version of the Cello Suites? P 269 and autograph score of
BWV 998 [156]
5.18 Proportioning in an earlier version of the Cello Suites [157] 6.1 WTC Book 1. Autograph score. P 415 [161]
6.2 Early preludes expanded. Clavier-Büchlein, Kayser’s copy, P 401 and P 415 [163]
6.3 Aufrichtige Anleitung. Autograph score, P 610 [165]
6.4 Earlier names and orders. Clavier-Büchlein, 1722/3 and copy by Kayser, P 219 [167]
6.5 Two additional collections. Autograph score, P 415 and copy by Kayser, P 219 [169]
6.6 WTC I. Autograph score, P 415 with Aufrichtige Anleitung, P 610 and P 219 [170]
6.7 A chronological overview. WTC I and Aufrichtige Anleitung [171] 6.8 Clavier Übung I and II. Original print [173]
6.9 Proportional integration of two sets of two keyboard collections [174]
6.10 Partitas in A and E minors. Autograph score, P 225, and print (BWV 827, 830) [175]
6.11 Partitas in A and E minor. Autograph score, P 225, 1725 [176] 6.12 Evolving plans for Clavier Übung I. Print. Autograph score,
P 225, P 226 [177]
6.13 Early ordering of Clavier Übung I and II. Print. Autograph score, P 226 [179]
6.14 Clavier Übung Parts 1-IV. Original prints [180]
7.1 Earliestfifteen movements. Clavier Übung III. Original print [185] 7.2 Twelve later movements. Clavier Übung III. Original print [187] 7.3 Clavier Übung III. Original print [187]
7.4 Twelve consecutive thematic groups. Clavier Übung III. Original print [188]
7.5 Perfect proportioning in consecutive movements. Clavier Übung III. Original print [189]
7.6 Goldberg Variations. Numerical structure. Original print [192] 7.7 Proportions in the canons. Goldberg Variations, Clavier Übung IV.
Original print [195]
7.8 Time and keyboard attributions. Goldberg Variations, Clavier Übung IV. Original print [196]
7.9 1742 plans. Clavier Übung III and Clavier Übung IV. Original prints [198]
7.10 Canons, BWV 1087. Autograph score, F-Pn Mus. Ms. 17669 [200] 7.11 Large-scale Clavier Übung series. Original prints [202]
8.1 Canonic Variations. Engraving of three original variations [206] 8:2 Canonic variations. Numerical value of engraved title page [207]
8.3 Canonic Variations. Comparison of 1747 print and autograph score, P 271 [210]
8.4 Canonic Variations, Schübler Chorales and Goldberg Canons. Original prints [213]
8.5 Collections of canons, 1739 and 1747. Original prints [214] 8.6 A collection for organ? Autograph score, P 271 [215]
8.7 Schübler Chorales. Sources and proportions. Original print [219] 8.8 Schübler Chorales. Title page. Original print [221]
8:9 Original binding of Bach’s Handexemplar of Clavier Übung III and Schübler Chorales [222]
9.1 Musical Offering. Original print [228]
9.2 Sonata and Canons in Sections C and D. Musical Offering. Original print [231]
9.3 Number values of title words. Musical Offering [232] 9.4 Parallel values. Musical Offering. Original print [233] 9.5 Title page. Musical Offering. Original print [235]
9.6 Reichardt’s handwritten title page. Musical Offering, A.Wn. S. H. J. S. Bach 102 [236]
9.7 The Art of Fugue. Original print [244]
9.8 Contrapunctus 1–10. The Art of Fugue. Original print [245] 9.9 Fourteen movements. The Art of Fugue. Original print [246] 9.10 Possible ground plans. The Art of Fugue. Original print [247] 9.11 A possible ground plan. The Art of Fugue. Original print [248] 9.12 The Art of Fugue. Autograph score, P 200 [249]
9.13 Transformation process. The Art of Fugue, P 200 and original print [249]
9.14 Transformation process. The Art of Fugue, P 200 and original print [250]
9.15 Title page. The Art of Fugue. Copyist, Altnickol, P 200, and original print [251]
9.16 Schübler Chorales; Fuga a 3 Soggetti. Original prints [253] 10.1 Early Keyboard Transcriptions. Mixed copyists, P 804 [258] 10.2 Twelve concerto transcriptions. Copyist, Johann Bernhard Bach,
P 280 [260]
10.3 Two sets of six transcriptions. Copyist, Johann Bernhard Bach, P 280 [262]
10.4 A proportionally related collection of twelve. Copyist, Johann Bernhard Bach, P 180 [262]
10.5 Brandenburg Concertos. Autograph score, Am. B. 78 [269]
10.6 Reconstruction of Bach’s numerical plans. Am. B. 78 [271] 10.7 Rastration of BWV 1050/1. Autograph score, Am. B. 78 [272] 11.1 ‘Great Eighteen’ organ chorale preludes. Autograph score, P 271,
56–99 [278]
11.2 The ‘Great Fifteen’, P 271, and Fifteen mass settings from Clavier Übung III in 1739 [283]
11.3 Additions to P 271. Organ chorales and Canonic Variations [283] 11.4 Six Trio Sonatas. Autograph score, P 271, 2–55 [286]
11.5 Two Trio Sonatas in C minor: coincidental parallelism? [289] 11.6 Large-scale parallels betweenfive collections [290]
11.7 Parallel proportions between keyboard collections: Clavier Übung III, The Art of Fugue, Schübler Chorales and P 271 [292] 12.1 St Matthew Passion. Autograph score, P 25 [296]
12.2 St Matthew Passion. Picander’s text and Bach’s autograph score, P 25 [299]
12.3 St Matthew Passion. Layout of autograph score, P 25 [305] 12.4 St John Passion. Partial autograph score, P 28 [308]
12.5 Chorales in St John Passion. Partial autograph score, P 28 [311] 12.6 Arias, Chorales and Recitatives in St John Passion. Partial
autograph score, P 28 [312]
12.7 St John Passion, layout and proportion. Partial autograph score, P 28 [314]
12.8 Cantata‘Wo soll ich fliehen hin’, BWV 5. Autograph score, BL Stefan Zweig Collection Ms. 1 [320]
12.9 Cantata‘Gott ist mein König’, BWV 71. Printed text and autograph score, P 45 [322]
12.10 Psalm 74:12 in BWV 71/1. Printed text and autograph score, P 45 [324]
13.1 Three Lutheran masses. Copyist Altnickol, P 15/1–3 [328] 13.2 Mass in A, BWV 234. Autograph score, D-DS Mus. Ms. 971 [329] 13.3 Missa in B minor. Autograph score, P 180 [331]
13.4 Symbolum Nicenum–Sanctus–Osanna in B minor. Autograph score, P 180 [332]
13.5 B-minor Mass. Autograph score, P 180 [334] 13.6 Christmas Oratorio. Autograph score, P 32 [337]
13.7 Free-texted material in Christmas Oratorio. Autograph score, P 32 [338]
13.8 Original performance schedule of Christmas Oratorio, Leipzig, December 1734–January 1735 [339]
13.9 Recitatives, Duet and Trio in Christmas Oratorio. Autograph score, P 32 [342]
13.10 Choruses and Chorales in Christmas Oratorio. Autograph score, P 32 [343]
13.11 Layout of Christmas Oratorio. Autograph score, P 32 [345] 13.12 Christmas Oratorio title page. Printed text booklet,
Leipzig 1734 [347]
13.13 Easter Oratorio. Autograph score, P 34 [349]
13.14 Ascension Oratorio. Autograph score, P 44 fascicle 5 [351] 13.15 A proportionally related pair: the Easter and Ascension
Oratorios [352]
14.1 English Suites. Copyist, Kayser, P 1072 [357]
14.2 French Suites. Copyist, Altnickol, US ML 96.B186 [357] 14.3 The Neumeister Collection, LM 4708 and Bach’s Orgelbüchlein,
P 283 [358]
14.4 Five Great Preludes and Fugues. Mixed copyists [360] 14.5 1738/9 plan for three grand organ collections [361]
14.6 Six Great Preludes and Fugues. Unknown Berlin copyist, Am. B. 60 [362]
14.7 Four solo concertos for keyboard. BWV 1052–5. Autograph score, P 234 [364]
14.8 A collection of six concertos for harpsichord. Mixed copyists [365]
Research for this book could not have been undertaken with the conven-tional tools availablefifty years ago, and I would like to begin by expressing my gratitude to the numerous unseen sponsors and librarians, who have worked diligently to put online so many seventeenth- and eighteenth-century sources, and to the scholars who painstakingly compiled and produced the Bach documents and diplomatic source research on which Bach scholars today can build with confidence.
Over the past two decades I have received generous funding from many sources, including The Leverhulme Trust, The Hinrichsen Foundation, Kungliga Musikaliska Akademien, Kungliga Vitterhetsakademien, Vetens-kaprådet and the Society of Authors, and I would like to thank the trustees of these foundations for their confidence and interest in my research questions. I am grateful too for the help of many librarians, including my Swedish colleagues at Statens Musikverk in Stockholm, the staff at the Bach-Archiv Leipzig, the Kungliga Bibliotek in Stockholm and the Senate House Library in London, and specifically David Coppen at the Special Collections of the Sibley Music Library in Rochester NY, Brigitte Geyer in Special Music Collections of the Stadtbibliothek, Leipzig, and Steffen Voss of RISM at the Bayerische Staatsbibliothek.
One of the privileges of my research journey has been the pleasure of exchanging notes with friends and colleagues, some of whom have gener-ously read and responded to my discoveries, including Cécile Bardoux-Lovén, Christine Blanken, Gregory Butler, John Butt, Raymond Erickson, Fred Fehleisen, Don O. Franklin, John Eliot Gardiner, Helen Gough, Wendy Heller, Paula Higgins, Michael Marissen, Jennie Nell, Szymon Paczkowski, Stephen Rose, Ulrich Siegele, Peter Smaill, Reinhard Strohm, Burkhard Schwalbach and Christoph Wolff. Thanks are due also to Jeffrey Sposato, Andrew Talle and Joyce Irwin for kindly trusting me with their unpublished research results, and to Barbara Reul for her invaluable help with the translations in the Appendix. And a special thank you to four colleagues whose longstanding friendship and wise words have meant so much to me during the evolution of this book: Jonathan Dunsby, Robin
Being part of a loving family has been my anchor throughout. The church family of Immanuel International, Stockholm has been an inspir-ation and has provided deep friendships. The encouragement of my parents-in-law Mary and David Tatlow, and daughter-in-law Anita is a source of great joy. I owe a profound debt of gratitude to my parents, Olive and Frank († 4.11.14) Ballard, not least because they first showed me the love of Bach’s music. Above all, though, it has been the unfailing and unconditional love, tolerance, humour and faith of Mark, Benjamin, Louisa and Emily that empowered me to bring this work to fruition.
AA J. S. Bach, Aufrichtige Anleitung (Two- and Three-Part Inventions and Sinfoniae). BD I–VII Bach Dokumente, vols. I–VII, 1963– . Benary Peter Benary, ed., Johann Gottfried Walther.
Praecepta der Musicalischen Composition, 1955.
BJ Bach-Jahrbuch, 1904– .
Buttstett, Ut, Mi, Sol Johann Heinrich Buttstett,Ut, Mi, Sol, Re, Fa, La, Tota Musica et Harmonia
Aeterna, [1716].
BWV Bach-Werke-Verzeichnis, 1950; rev. and enlarged, 1990.
BWV2a Bach-Werke-Verzeichnis, kleine Ausgabe, 1998.
CÜ I–IV J. S. Bach, Clavier Übung, parts I–IV.
D-B Staatsbibliothek zu Berlin.
Fritsch, Lexicon Thomas Fritsch,Teutsch-Englisches Lexicon, 1716.
JP J. S. Bach, The Passion according to St John. Mattheson,
Capellmeister
Johann Mattheson, Der Vollkommene Capellmeister, 1739.
Mattheson, Orchestre 1 Johann Mattheson, Das Neu-Eröffnete Orchestre, 1713.
Mattheson, Orchestre 2 Johann Mattheson, Das Beschützte Orchestre, 1717.
Mattheson, Orchestre 3 Johann Mattheson, Das Forschende Orchestre, 1721.
MP J. S. Bach, The Passion according to St Matthew.
NBA KB Neue Bach-Ausgabe: Kritischer Bericht.
NBR The New Bach Reader, rev. and
enlarged, 1998.
New Grove New Grove Dictionary of Music and Musicians, 2nd edn, 2001.
P D-B Mus. ms. Bach P[artitur]. Original score held at Staatsbibliothek, Berlin.
St D-B Mus. ms. Bach St[immen]. Original parts held at Staatsbibliothek, Berlin.
Walther, Lexicon Johann Gottfried Walther, Musicalisches Lexicon, 1732.
Walther, Praecepta Johann Gottfried Walther, Praecepta der Musicalischen Composition, 1708.
Wolff, Essays Christoph Wolff, Bach: Essays on His Life and Music, 1991.
Wolff, Learned Musician
Christoph Wolff, Johann Sebastian Bach: The Learned Musician, 2000.
WTC I–II J. S. Bach, Das Wohltemperirte Clavier, books 1–2.
Zedler, Lexicon Johann Heinrich Zedler, Grosses vollständiges Universal-Lexicon aller Wissenschafften und Künste, 1732–54.
Sonnet
Du edle Musica, die du das Hertz bewegst,
Du schönes Himmel-Kind, wer wolte doch nicht lieben; Die sind von guter Art, die dich rechtmässig üben, Die du im innern Grund Zahl, Maaß, Gewichte hegst. Und die Proportion von Erd’ und Himmel trägst: Dein Werck besteht in sechs, und deine Ruh in sieben; Du bist mit Heimligkeit und Kunst durchaus beschrieben, Die du des Himmels Bild in deine Wercke prägst.
Du must dich zwarten auch offt übel zerren lassen; Der Mißbrauch lässet dich in deinen Würden nicht; Und ob Apollo dich mit allem Ernst verficht; Sofinden sich doch die, die deine Schöne hassen. Du aber bleibest wol: Ob sie nicht achten dein; So wirstu doch das Spiel der Frommen ewig seyn.
Henr. Georg. Neuss, Past. Gvelpherbyt., 1691
The dangers of playing with numbers are many and legendary. The humiliation of the sixteenth-century mathematician and pastor Michael Stiefel, whose calculations on biblical verses enabled him to predict that Christ would return at 8 a.m. on 18 October 1533,1was matched by the shame of his cold and hungry parishioners after their disappointing vigil. The embarrassment of theologian and musicologist Friedrich Smend four hundred years later was less public, but it still had a significant influence on the inhibition of number research. His interpretation of the number‘84’ epitomises the problems:
Bach noted a symbolic number in the autograph score of the B-minor Mass. At the end of the‘Patrem Omnipotentem’ he writes the bar number of the movements ‘84’ (7 12) . . . [This] chorus is about creation (‘factorem coeli et terrae’) . . . Earth and heaven are contained in 7 [3 symbolises heaven and 4 earth]. . . We hear
the word‘Credo’ forty-nine times, and its continuation ‘in unum Deum’ eighty-four times. Yet again numbers with symbolic content appear.2
Eighty-four quickly became a buzzword for the folly of symbolic numbers in music when it was discovered that the annotation had been written into the score by one of C. P. E. Bach’s copyists3decades after J. S. Bach’s death. In
spite of the inherent risks, this book is nonetheless devoted to compositional numbers, tackling their form, purpose and meaning in Bach’s music.
I
How numbers became associated with Bach
Bach left no description of his methods of composition, or of whether or not he used numbers when he composed. One of the earliest allusions to the mathematical bases of his music dates to the early 1740s. It can be read in a published collection of musicians’ autobiographies,4in which Lorenz Mizler (1711–78) wrote that he had been influenced by ‘reading good books, listening to good music, perusing many scores by good masters and also in his association with Capellmeister Bach’.5This entry enraged
the editor of the compilation, Johann Mattheson (1681–1764), causing him to add that‘Bach no more taught Mizler the mathematical bases of music than I did myself’,6 referring to an earlier discussion of the value of
mathematics for music in which he had named Mizler’s training in math-ematics, philosophy and music.7Mattheson’s comments should have killed any later rumours that Bach was interested in the use of mathematics in music, but they did not. A century later the great music historian Philipp Spitta unwittingly revived the topic when he drew attention to the Mizler– Mattheson dialogue. Although intended to demonstrate that Bach had no interest in the mathematical basis of composition,8it had the opposite effect. In the 1920s Arnold Schering (1877–1941) further raised the profile of numbers in music when he unearthed traditions of permutation in compositional
2
F. Smend, J. S. Bach Kirchen-Kantaten (Berlin, 1947; rev. edn 1966), vol. IV, 14 and 19. 3 Score, P 180, on page 105 in Bach’s pagination. The same scribe wrote the figure 84 in the
corresponding place in the soprano solo part, St 118/2 in C. P. E. Bach’s 1786 copies of the parts of the Credo, St 118, thus ruling out the possibility that J. S. Bach wrote thefigure.
4
J. Mattheson, Grundlage einer Ehrenpforte (Hamburg, 1740). 5 BD II, Doc. 470, 380. 6 Mattheson, Ehrenpforte, 231.
7
Ibid., 230, where Mattheson issued a lengthy diatribe against mathematics in music. 8 P. Spitta, Johann Sebastian Bach: His Works and Influence on the Music of Germany
(1685–1750), trans. A. C. Bell and J. A. Fuller-Maitland. 3 vols. (London, 1884; reprint edn New York, 1951), vol. III, 24.
invention.9He steered the number discussion towards Bach studies when he demonstrated the presence of four types of symbolism in Bach’s cantata ‘Du sollt Gott, deinen Herren, lieben’ (BWV 77), spicing the commentary with provoca-tive phrases such as‘the esotericism of Bach’s vocal canons’ and ‘holy and mysterious numbers’.10Many musicologists were eager to run with Schering’s
ideas before the ground had been fully prepared. Friedrich Smend (1893–1980) was at the forefront, introducing the term ‘cabbalism’ and using number alphabets to interpret patterns he had found in Bach’s music.11A frisson rippled
through the musical world. Amateurs and less discerning scholars lovingly nurtured the ideas, while professional musicologists openly voiced their disdain. The paragram,12which used one of more than thirty different alphabets to substitute numbers for letters, was Smend’s major stumbling block. As a widely read theologian with a particular interest in church history, he had met similar numero-alphabetical techniques in Jewish mysticism, which duped him into making an association between Bach’s numbers and religious symbolism. It was a fabulous premise with which to work, as it promised to reveal the unseen depths of Bach’s spiritual motivation, although, like the allure of the sirens’ call, it proved treacherous. Nonethe-less, numerical readings of Bach’s compositions continued to be published. Many contained fanciful and fallacious interpretations, some were down-right illogical,13 and the majority fell short of their promise not only because of weak methodology, but because of their lack of solid historical or documentary evidence.14 The shaky historical foundations on which number and interpretation structures were built made their collapse inevit-able. What could have become a valuable scientific discipline of numbers within musicology became known as‘numerology’, in all its notoriety.
And this is where my research enters the history. An examination of Smend’s work led to a study of number alphabets; from their origins, through the quagmires of mystical cabbalism, black and white magic and two centuries of Lutheran exegesis, to the poetical paragram. The results,
9 A. Schering,‘Geschichtliches zur ars inveniendi in der Musik’, in Jahrbuch der Musikbibliothek Peters für 1925, ed. Rudolf Schwartz (Leipzig: Peters, 1926), 25–34. Schering cites Glareanus (1547), Kircher (1650), Heinichen (1738), Mattheson (1739) and Sulzer (1778).
10Schering,‘Bach und das Symbol’, BJ (1925), 44. 11
Ruth Tatlow, Bach and the Riddle of the Number Alphabet (Cambridge University Press, 1991), 130–8.
12SeeChapter 2, §III. 13
Malcolm Boyd, Bach, The Master Musicians (London: Dent, 1983), 223. 14
Notable exceptions include the number research of Ulrich Siegele and Don O. Franklin, to whom I am enormously grateful for their generosity in sharing their expertise, and lending their support as I pursued my research paths.
givenfirst in my doctoral thesis15and later revised in the monograph Bach and the Riddle of the Number Alphabet, prove conclusively that numbers and numerical interpretations were an integral part of Bach’s heritage. Word and number conceits, including the anagram, chronogram, acrostic and paragram, were popular drawing room pastimes as well as useful tools for the more serious poet or orator in need of creative inspiration.
Research for this book began where Bach and the Riddle left off, addressing the question of whether Bach and his contemporaries actually used numbers and number alphabets when they composed. It was origin-ally designed to be a comprehensive survey of theoretical evidence showing where numbers and numerical constructions fitted into compositional theory in Bach’s time, with the anticipated conclusion that composers made little or no use of numbers in practice. The structure and contents changed radically, however, with the unexpected discovery of proportional parallelism in all the collections and multi-movement works that Bach revised for publication.16 At its most basic, the theory of proportional parallelism shows that Bach created layers of 1 : 1 and 1 : 2 proportions, using the numbers of bars in the parts and sections of compositions. The original theoretical survey is included in Part I of this book, while the demonstration of Bach’s use of numbers, including evidence of the changes Bach made as he transformed early works into perfectly proportioned collections, formsPart II.
Proportional parallelism would have seemed a self-evident practice to any composer living in Bach’s time and locality, which is not to say that all composers used it. Symmetrical organisation, parallel techniques, perfect proportions and unity were all commonplace, were found in everyday life, in every academic discipline and creative pursuit, and were also described by music theorists in books about how to compose. Numerous observations of the symmetrical organisation found in Bach’s multi-movement works have been accepted into the canon of Bach scholarship. For example, in recent years Christoph Wolff has been a long-term champion for Bach’s architec-tural designs; their symmetry, order, organisation, connection and propor-tion.17 Many of these observations can now be confirmed empirically by proportional parallelism. Furthermore, since Smend’s work, there has been a widely held assumption that numbers in Bach’s music would be symbolic. Proportional parallelism shows something subtly, but significantly, different: it is the proportions, rather than the specific numbers, that hold the meaning. 15
Tatlow,‘Lusus Musicus vel Poeticus’. King’s College, University of London, 1987. 16Tatlow,‘Collections, bars and numbers’, Understanding Bach 2 (2007), 37–58. 17
Many examples in Wolff, Essays.
Several types of evidence combine to demonstrate Bach’s use of propor-tional parallelism. There is the numerical evidence found in the scores. By comparing the numbers of bars in his early and later versions, or by tracing the changes he made as he compiled a new collection from pre-existing movements, one can see how Bach introduced the layers of perfect pro-portion. There is documentary evidence to demonstrate the specific role that numbers, unity, symmetry, proportion and Harmony18 played in compositional organisation and planning in Bach’s time. And there is documentary evidence to show how these numerical concepts would have been viewed and understood at the time. There is also a body of evidence hidden by the eighteenth-century language that has been lost in transla-tion, both literally and culturally. At all times I have aimed to incorporate results from the most up-to-date diplomatic evidence and source studies. The majority of results shown in Part II confirm the conclusions drawn by these source studies, but occasionally my demonstrations and numerical reconstructions suggest a new interpretation.
The discovery of proportional parallelism in Bach’s collections raises the fundamental and challenging question of why he spent time striving to create proportional order within his compositions. The answer lies in the philosophical and theological understanding of Harmony and harmonic proportions, which had been an essential element in philosophy, science and the arts since classical times, and was still prevalent in Lutheran Germany.19 In the late seventeenth and early eighteenth centuries thought-patterns associated with the Enlightenment began to spread across Europe. Although the ancient proportional world view survived longer in some areas, its final rejection in the early nineteenth century caused an intellectual paradigm shift that would have a profound and lasting effect on the formation of twentieth-century European culture. Bach was living, working and using proportions at this tumultuous time of philosophical change.
II
Parallels, proportions and Harmony
It was thefirst four numbers, the perfect tetrachys, expressed as the ratios 1 : 1, 1 : 2, 2 : 3 and 3 : 4 and as the proportion 6 : 8 : 9 : 12,20that the ancient 18Harmony, with a capital‘H’, will be used throughout as a translation of harmonia and
Harmonie. 19
SeeChapter 3, and the Sonnet by Neuss on thefirst page of this chapter.
20Nicomachus’ tenth proportion. See Tatlow, ‘The Use and Abuse of Fibonacci Numbers and the Golden Section in Musicology Today’, Understanding Bach 1 (2006), 77–9.
Greeks esteemed as the most perfect for music and Harmony.21The first six sounding numbers, or the senarius, would later be considered perfect because, through the ratios 1 : 1, 1 : 2, 2 : 3, 3 : 4, 4 : 5 and 5 : 6, they were the source of the musical scale, and in sequence formed the unison, the octave, thefifth, the fourth and the major and minor thirds respectively, which is one reason why the Guidonian mnemonic Ut, re, mi, fa, sol, la for the six degrees of the hexachord was also considered perfect.22Later still the seven‘harmonic’ numbers 1, 2, 3, 4, 5, 6, 8, called the septenarius,23 or numeri harmonici,24 became popular, the sequence omitting the number seven, or the‘Ruh-Zahl’.25In Bach’s time Euclid’s demonstra-tion of the number six as the first perfect number was used to endorse the universal perfection of the senarius;26those who found the septenarius more perfect also came up with numerous reasons.27 A theology of creation based on proportions and harmonia gradually evolved. Harmonic proportions in the cosmos, in the world and in the measurement of the human being were understood to be a reflection of the ‘indescribable wisdom and perfection’ of the Creator God.28The proportional perfection
of musical intervals gave rise to the terms ‘perfect unison’, ‘perfect octave’, ‘perfect fifth’ and ‘perfect fourth’. The term trias harmonica was coined to describe the consonant triad, because its perfection reflected the perfect Harmony of the Holy Trinity,29 and because its
intervals, the fifth and the major and minor thirds, expressed in the
21Pythagoras considered it the perfect number as 1þ2þ3þ4 equals 10, from which numbers the proportions 1 : 1, 1 : 2, 2 : 3 and 3 : 4 form the unison, octave,fifth and fourth.
22
T. Christensen, The Cambridge History of Western Music Theory (Cambridge University Press, 2002), 253–4, 276–8.
23For example in Walther, Praecepta, 36 (Benary, 83), and Buttstett, Ut, mi, sol, 26–8, citing Andreas Werckmeister, Musicalische Temperatur (Quedlinburg, 1691), and Conrad Matthäi, Kurtzer, doch ausführlicher Bericht von den Modis Musicis (Königsberg: Matthäi, 1652), 14–15. 24Walther, Praecepta,‘Musica Poetica’, 8 (Benary, 76).
25Matthäi, Modis musicis, 15, calls seven a‘rest’ number because one cannot make any musical interval out of the seventh number on the monochord, and because God rested on the seventh day. 26
A perfect number is one whose divisors add up exactly to the number itself. The number 6 has the divisors 1, 2, 3 and 1þ2þ3 equal 6. 28 is the next perfect number because its divisors 1, 2, 4, 7 and 14 add up to 28. 496 is the third, and 8128 is the fourth perfect number. Euclid described this in his Prop. IX 36. Walther, Lexicon, s.v.‘numerus perfectus’, citing Euclid.
27J. F. Riederer, Gründliche Untersuchung der Zahl Sieben. (Franckfurt; Nürnberg, 1719). 28
Appendix, 1691-IV 8, 1691-IV 9. (Sources relevant to the doctrine of music have been assigned a short year/number reference (as here) to the Appendix, where full bibliographic references and full text in parallel English-German translation are given.)
29Walther, Lexicon, s.v.‘Lippius’ does not name Lippius as the author of the term trias harmonica.
proportions 4 : 5 : 6, were contained within the God-given senarius.30The triad thus became a powerful reminder for Lutheran Christians of the centrality of music to God and His created order.
This view of the world was still alive and current in Bach’s time. Harmony embraced both silent universal proportions and sounding music. A sonnet by Heinrich Georg Neuss (1654–1716), printed in the preface to a music treatise by Andreas Werckmeister (1645–1706) and also on the first page of this chapter, illustrates the centrality of Harmony to German-speaking Lutherans in 1691. Within a few decades, though, there would be many changes in the understanding of music and philosophy in Europe. Thomas Christensen explains:
Music theory gradually receded from its Boethian heights through the robust growth of musica practica as a discipline. By the eighteenth century, music theory had become only a shell of its former glory. (Rameau felt obliged on numerous occasions to defend the honour and dignity of music theory, while at the same time conceding such knowledge might be of little practical use to musicians.) Yet for every defender of music theory– such as Rameau or Lorenz Mizler (1711–78), the founder of the ‘Corresponding Society of Musical Science’ – there were critics such as Johann Mattheson (1681–1764), who would lambaste much theoria (or, as he preferred to call it,‘musical mathematics’) as a discredited remnant of unenlightened prejudice . . . With the weapons of empirical philosophy bequeathed by Locke, writers such as Mattheson could militantly hoist the Aristoxenianflag of sensus over that of ratio.31
Philosophical ideas from France and Britain were gaining popularity within Lutheran Germany, and gradually eroding confidence in the cen-trality of the unison and proportions. The catastrophe was not unforeseen. In 1728, in a book that Bach knew well,32 the Dresden Capellmeister Johann David Heinichen (1683–1729) made the following prediction about the effect of this new philosophy upon music:
The beginning has already been made in our times; no doubt daily progress will be made in our century to this end for those supposedly paradoxical hypotheses, and finally all the remaining weak and partly-worn pillars of the musical past will be torn completely asunder.33
30
Walther, Lexicon, s.v.‘Trias harmonica oder musica’ terms a major triad ‘Trias harmonica perfecta’, and a minor triad ‘Trias harmonica imperfecta’.
31Christensen, ed., The Cambridge History of Western Music Theory, 8. 32
NBR Doc. 140, and appendix, 529. Bach was Leipzig sales agent for the treatise, and he owned a copy. 33
J. D. Heinichen, Der Generalbass in der Composition, 2nd edn (Dresden, 1728), 5, note (a); translation in Buelow, Thorough Bass Accompaniment According to Johann David Heinichen, rev. edn (Ann Arbor: UMI, 1986), appendix B, 310–11.
Bach grew up in this climate of change, when the proportional world view was under attack and gradually falling out of favour, which makes under-standing the presence and significance of proportions in his music even more challenging.34
The complexity of this musical and philosophical transition can best be understood from the writings of Mattheson, who was determined to prove that the traditional Lutheran understanding of musical Harmony was erroneous, fighting tooth and nail to prove that true Harmony must be sounding and not silent. ‘Mr Organist’, he wrote, addressing Johann Heinrich Buttstett (1666–1727), ‘why don’t you distinguish primarily between what is properly called Harmony (Harmonia propriè sic dictam), and Harmony in Music (Harmonia in Musicis)?’35
His zeal to divide the traditional Lutheran understanding of Harmony explains many of his curious statements about musical mathematics, not least his seemingly illogical charge about Mizler and Bach.36 From a single united concept that embraced both non-sounding universal harmony and the sounding harmony of pitches and intervals, Mattheson made two distinct harmonies, i.e. Harmony proper (universal harmony) and Harmony in music (the proportions of pitch, intervals and rhythm). In contrast to contemporary theorists such as Mizler and Spiess, Mattheson saw music as a science for the ear alone and not a theory to be studied in terms of proportions and arithmetic. Nonetheless, he continued to believe in many of the philosoph-ical and theologphilosoph-ical aspects of universal Harmony, sharing some funda-mental views about proportions with theorists traditionally understood to be his opponents. Attempting to persuade Buttstett of his erroneous understanding of Harmony, Mattheson demonstrated his personal belief in proportions, writing:
There is no doubt whatsoever that the Lord God is pleased with proportions, and the universe demonstrates this. . . God is pleased with musical sounds and their proportions: I doubt that as little as I doubt Christ’s Birth, because music is His creation, indeed one of His best creations and gifts.37
This shows that even Mattheson, reputedly the great opponent of musical mathematics, still profoundly believed that God was pleased with both non-sounding and sounding proportions. It was a belief that motivated him and many other authors and musicians of the period to recommend proportional organisation in musical composition. Mattheson papered
34Implications discussed inChapter 3. 35 Appendix, 1717-I. 36Mattheson, Ehrenpforte, 231. See alsonote 7. 37 Appendix, 1717-III.
over the logical cracks of his position, qualifying his recommendation to proportion pieces of music‘for there is nothing more pleasing to the ear’. Theorists who held the traditional view of Harmony as a united entity recommended proportional ordering in compositions without qualifica-tion: to them it made no difference if the proportions were heard or not.
III
The unison and Harmony applied
A proverb cited frequently in music treatises of Bach’s time reads: ‘the closer a proportion is to the unity [or equality] the more perfect it is: the further a proportion is from the unity the more imperfect’.38This simple formulation holds the essence of much that lies behind the concept of proportional parallelism in Bach’s compositions. The unity and the unison of the 1 : 1 proportion had become the ultimate expression of both equality and perfection. Using Christensen’s phrase, it was truly a ‘generative unison’.39 Whole lifestyle applications and artistic techniques were based
on belief in the unity because of its position in universal Harmony, regardless of the practitioner’s stand on the sensus–ratio or sounding/ non-sounding debate. The generative unison fell within Mattheson’s clas-sification of non-sounding Harmonia propriè, the proportions of which even he believed pleased God. How this unison generated many structural forms, including symmetry and parallelism, is the subject of Chapter Two. The 1 : 1 kinship between symmetry and parallelism sheds light on the significance of their use in the arts in Bach’s time. The dual meaning of emblems has been well researched and documented, but I set their parallel image-meaning into the larger context of belief in the philosophical, theological and aesthetic significance of the unity and Harmony.
Eurythmia was a synonym for symmetry, which, because of its 1 : 1 nature, was considered to be the epitome of beauty and perfection. According to Johann Gottfried Walther (1684–1748) and other theorists, eurythmia in music could be demonstrated numerically by numerus musi-cus. There was symmetry in poetry, both in the rhyme scheme and in the metrical organisation, which Morhof described as numerus poeticus. Bach was surrounded by a world of symmetry and parallel forms. Appreciating
38
Appendix, 1708-VI.
39T. Christensen, Rameau and Musical Thought in the Enlightenment (Cambridge University Press, 1993), 84–90 – although Christensen applied the phrase solely to musical properties.
the underlying philosophy that may have influenced their proliferation and popularity lends further weight to our understanding of why Bach chose to use parallels and proportions in his compositional practice.
The acrostic could be used to create a parallel meaning to a poem or a hymn text when the letters of a specific name were woven into the structure, commonly in thefirst letter of the first word of each consecutive stanza. The poetical paragram is literally a 1 : 1 parallel, with the numerical value of each word in the parallel columns reaching the same total. There were parallels used for compositional invention, one of the most common being a parallel piece of music, created when a new bass line was set to a pre-existent melody, then a new melody to the new bass line, followed by a second new bass line made for the new melody, and so on ad infinitum. Hybrid parallel forms were created from any combination of parallel techniques, such as numero-alphabetical substitutions, emblematic images or arithmetical or algebraic procedures. Bach would have recognised the term lusus ingenii as a description of parallel techniques.40In a keyboard publica-tion that Bach knew well, his predecessor as Leipzig Thomaskantor, Johann Kuhnau (1660–1722), claimed that his entire keyboard publication, Musicalische Vorstellung einiger biblischer Historien, was nothing more than such a lusus, implying that he might have embedded many more compos-itional parallels than the mere musical description of the cited biblical stories. There were also parallels with printing symbols. Indeed symbolism, including number symbolism, is nothing more than a parallel expression.
Bach’s use of parallel forms can be seen in some of his album entries, where he uses a Latin dedication with a dual meaning. But above all it is the documented use of the family name in music notes– B-A-C-H – that bears witness to his knowledge and use of lusus ingenii and parallel techniques. To what extent Bach developed this practice and toyed with the numerical value of his names can only be guessed. Evidence of what appears to be a reference to his name appears so frequently throughout his published works that I decided to include it as one of the three characteristics of proportional parallelism.41 Bach’s birthday on 21 March (213) happens to be parallel to the numerical value of thefirst three letters of his surname in the natural order and milesian number alphabets. The parallel use of B and H in music for B♮ makes it possible for B-A-C also to stand for B/H-A-C.42 Permutation was commonplace and second nature to
40
Tatlow, Bach and the Riddle, 8 and 109. 41 Tatlow,‘Collections, Bars and Numbers’, 37–58. 42The idea that‘B’ always stood for B♭ and ‘H’ for B is a common twentieth-century
misconception.
any master of fugal writing. Thus when the letters and numbers 213 appear as 132 or 312 or 123, these too may be a parallel to the 213 of the Bach surname.
Chapter 3 explains how belief in the unison and Harmony became intertwined with the Lutheran doctrine of music, and how musicians with a vested interested in Harmony generated conceptual and behavioural regulations, or moral norms from the belief. Proportions in any artistic pursuit gave an objective means of capturing perfection and beauty. Striving to achieve this perfection was an important life goal for those who believed in Harmony, because, as Werckmeister wrote:‘God Himself created nature through His omniscient counsel so that everything might strive to achieve unity and therein take pleasure.’43
It was his belief in Harmony that caused Buttstett to put in writing what he understood to be the true, correct, biblical view of music,44 to rectify Mattheson, whose views in Orchestre 1‘contravened the word of God’.45 Buttstett most probably knew Werkmeister’s book The Noble Art of Music: Its Greatness, Use and Abuse, with its notion of God-pleasing music, which implied its opposite, that music could be composed and used in a way that was definitely not pleasing to God. This belief led to behavioural guidelines for musicians, in which some activities, such as playing in pubs and accompanying the drinking rounds, were classified as sinful.46More
inter-esting than the prohibitions, however, are the positive practices that this belief encouraged. Developing piety, godliness and virtues were among the practices that Bach and his contemporaries were challenged to strive to achieve. The guideline given by Bach’s second cousin Walther, that good Harmony will result not only when it is‘composed after the artistic rules, but above all and primarily when it is used in virtuous and God-pleasing practices’,47implies that he adhered to a belief in the moral and behavioural
applications of Harmony. Bach’s choice of wording for the title pages of his Clavier Übung publications implies the same. The hours of hard work Bach must have put into revising his compositions to introduce the unity and layers of literal 1 : 1 or 1 : 2 parallel proportions also speak of a man striving to achieve something specific. He would have saved a lot of time and patience had he left what were perfectly good works unrevised and unproportioned. By making his collections harmoniously structured (i.e. by introducing unsounding parallel proportions, as in Mattheson’s 43
Appendix, 1687-I. 44Chapter 3, §II. Note the phrase‘das rechte Fundamentum Musices’. 45Buttstett, Ut, Mi, Sol, 3.‘Es ist auch solches kühnes Unterfangen wieder Gottes Wort.’ 46Appendix, 1691-IX, 1691-XI and 1723-III. 47 Appendix, 1708-II.
Harmonia propriè) Bach may have been ensuring they would survive the Rapture and last for eternity. Buttstett believed this; Mattheson did not.
Chapter Four presents many new sources to show how belief in propor-tions and the unison influenced methods of musical composition. The ideal of the 1 : 1 and 1 : 2 proportions as the most perfect persisted among German-speaking theorists until the mid eighteenth century. Mattheson wrote more extensively than any other theorist of the period about com-positional planning, and, regardless of his view that harmony must be sounding, he recommended that the composer should organise an entire composition so that‘every part demonstrates a true proportion, uniformity and unison’.48
Perhaps the most direct description he gives is thefifth in his list of rules for loveliness in a melody or in a whole composition, where he writes that the composer should ‘Observe well the proportion of all sections, parts and terms’.49His belief in well-proportioned music extends to rhythm, the numerus musicus, and to the number of bars as they display a specific progressio arithmetica. Theorists recommend that proportions are formed by the number of bars, the number of beats within a bar, the rhythmic flow within a phrase and the number of bars making up a section. Mattheson makes clear that, although he has for practical reasons used a melody to illustrate this, the same applies to the order and dispos-ition of a complete musical work.50
All this presupposes that the composer, in this case Bach, kept an eye on the number of bars in his composition. And we know he did. When transcribing a part or an entire score, the copyist used the number of bars to determine the length of thefinished manuscript, to dictate the size and number of staves to be drawn on each sheet of paper, and to check that the correct number of bars had been copied. Bach used this method, some-times recording the cumulative total of bars in his scores.
IV
Historically informed methodology
At every stage of the research presented in this book I have aimed to use tools the composer himself could have used, to formulate any discoveries in terms and concepts the composer would have recognised, and to ensure 48
Tatlow,‘Theoretical Hope’, Understanding Bach 8 (2013), 39–40; Mattheson Capellmeister, Part II, chapter 14, §30, 240.
49Mattheson, Capellmeister, part II, chapter 5, §52, 141. 50Mattheson, Capellmeister, part II, chapter 14, §4, 235.
that, as far as possible, every claim or concept is supported by appropriate historical sources.51 Twenty-first-century definitions, preconceptions and ‘common sense’ can seriously hamper the historian trying to discover how a musician living in Bach’s time and location thought about specific subjects. It was a systematic use of the principles of what I have termed historically informed theory (HIT) that enabled me to discover propor-tional parallelism while searching documents and scores for answers to questions about Bach’s possible use of compositional numbers. There are many subtle traps in the process. Recognising aspects of Bach’s physical environment and studying the philosophical and theological ideas of his time can easily encourage us to falsely assume that we have an accurate grasp of life in the early eighteenth century. Similarly, recognising concepts in scores and music treatises of Bach’s time can persuade us that we understand music as Bach and his contemporaries did. Although many of Bach’s techniques and concepts are familiar to us, the image is distorted by all manner of personal and collective modern assumptions. Transla-tions, whether modern or original, frequently mask the intended meaning or the implications of a word or phrase that would have spoken clearly to the original reader. The problems become more complex when we attempt to understand the specific mindset of a historical figure such as Johann Sebastian Bach, whose compositions have become such an integral part of modern-day cultural experience. Although his life story and position in music history are so familiar, can we really know whether what he read caused him to compose differently?
Intellectual influences
There are many resources that help decipher cultural codes in Bach studies, including dictionaries and reference works published in the Leipzig area during Bach’s lifetime. The most comprehensive is undoubtedly Zedler’s encyclopedia (Zedler, Lexicon), and for understanding how German was translated into English in Leipzig at this time, Fritsch’s German–English dictionary (Fritsch, Lexicon). To avoid falling into the very time-bound trap one is trying to escape, it is essential to remember that Fritsch’s synonyms reflect early eighteenth-century English usage.
Books that Bach knew or that he might have read are also invaluable resources. Although he made no inventory of his library,52 we know he 51Tatlow,‘Theoretical Hope’, 33–60.
would have had access to many more books than he possessed. Borrowing literature from family members,53or from wealthy pastors and supporters, commonly supplemented personal ownership at this time. Bach could also have visited public lending libraries in Eisenach, Erfurt, Lüneburg, Weimar, Cöthen and Leipzig, where the majority of readers were middle-class professionals, as the detailed records of the public library in Weißenfels show. Among their thirty-six volumes of music books and scores were Werckmeister’s Musicae mathematicae hodegus curiosus and Athanasius Kircher’s (1602–80) Musurgia Universalis, if Bach had not been able to borrow these classics from friends.54
It is important not to ignore the fact that ideas may also have been transmitted by personal contact, through word of mouth, or in private correspondence. Bach probably never met Andreas Werckmeister. Their connection was through Johann Gottfried Walther. Bach and Walther were cousins through their respective mothers, Maria Elisabetha Lämmer-hirt (1644–94) and Martha Dorothea LämmerLämmer-hirt (1655–1727).55The boys
were born within six months of each other and seem to have kept in regular contact throughout their lives. As a 19-year-old in 1704 Walther journeyed to Halberstadt and Magdeburg specifically to get to know the‘famous musician Mr Werckmeister’.56 The purpose of the trip may have been to discuss the latest musical ideas and possible content for Walther’s own music treatise that he would complete four years later.57
He acquired Werckmeister’s complete works and other ‘stumme Leh-rmeister’58
on this trip, and they may have exchanged the names and addres-ses of useful contacts.59There may also have been an exchange of musical
53
See Johann Gottfried Walther Briefe, ed. K. Beckmann and H.-J. Schulze (Leipzig: VEB, 1987), 34–54 for Walther’s 1729 catalogue of music books and scores in a letter to Bokemeyer, dated 4 April 1729.
54M. Raabe, Leser und Lektüre im 18. Jahrhundert, Vol. 4 (München; London: K. G. Saur, 1989), 4 and 534–5. The library purchased Zedler, Lexicon later in the century. H. Bokemeyer c. 1726, J. G. Schwanberger in 1766, and Bach’s biographer J. N. Forkel from 1799, were all readers at this library.
55
A related Martha Lämmerhirt married the Erfurt theorist and organist J. H. Buttstett, whose writings are discussed inChapter 3.
56Walther, Briefe, 219. Letter 37 to Johann Mattheson, 28 December 1739. Werckmeister died on 26 October 1706, two years after the visit.
57
Walther, Praecepta.
58Walther, Briefe, 68 and 219 and editorial comment on page 84, naming Robert Fludd’s Utriusque Cosmi (1617) and Kircher’s Musurgia Universalis (1650) in addition to Werckmeister’s works.
59For example, Werckmeister may have recommended to him the work of his Wolffenbüttel-based theologian friend Heinrich Georg Neuss.
manuscripts. In 1729 Walther told Heinrich Bokemeyer (1679–1751) that he owned over 200 organ pieces, the majority given to him by Werckme-ister (manuscripts in German tablature by Buxtehude),60and the remain-der given to him by Bach,61 whom he describes as his ‘Vetter’ and ‘Gevatter’.62Bach would no doubt have seen Walther’s Buxtehude
manu-scripts and probably have been interested to hear what his cousin had discussed with Werckmeister.
Given Walther’s personal association with cantor Johann Heinrich Buttstett and the cousins’ mutual association with Erfurt, there can be no doubt that Bach and Walther would have been interested readers of the -Buttstett–Mattheson battle publications,63
and we know that by 1729 Walther owned copies.64 Whatever their response to the dispute, Bach and Walther would definitely have understood Buttstett’s thought-processes and the doctrinal issues that caused Mattheson’s zealous and largely unprovoked invective.
Fifteen years later, while Walther was compiling material for his musical dictionary alongside his duties as court organist in Weimar,65Bach and his second wife found themselves based in the cosmopolitan university city of Leipzig. Located conveniently on the trade routes, Leipzig attracted inter-national visitors to the thrice-yearly fairs, and had a thriving publishing industry. Together with Halle, Leipzig was a host city to the massive lexicographical project initiated by Johann Heinrich Zedler (1706–51), who aimed to record all world knowledge of the time. It was also in Leipzig that Walther decided to publish his dictionary, which was available from 1729 for the pre-publication entries beginning with letter ‘A’, with full publication from 1732, when Bach was its Leipzig sales agent.66
60Walther, Briefe, 62–3, and BD II, Doc. 263, 193.
61Bach might have received manuscripts when Walther died in 1748, if the Neuss manuscripts were in Walther’s possession. However, there is no documentary evidence to support this suggestion.
62Fritsch, Lexicon.‘Er ist mein Gevatter’ means ‘He was godfather to a child of mine’. In 1712 Bach stood godfather to Walther’s son Johann Gottfried the younger. Vetter means ‘cousin’. Walther and Bach were not cousins, but Walther’s assignation shows that he thought of Bach as a close relative.
63It would be interesting to know how Walther reacted to the publications, in view of his bad experience of Buttstett as a teacher.
64
Walther, Briefe, 35, letter 4 April 1729 includes two catalogues of books, including‘(11) Matthesonii Opera omnia, (32) Buttstedts [sic] Ut re mi fa sol la, and (41) Matthesons Orchestre 1. 2. u. 3 Theil’.
65
J. G. Walther was organist at the church of St Peter and St Paul, Weimar from 29 July 1707, and Court Musician from 1721 until his death on 23 March 1748.
Bach owned a copy of the dictionary, and also acted as sales agent for Heinichen’s 1728 treatise Der Generalbass in der Composition, with its masterly introductory analysis of current conflicts within music.67 His promotion of the treatise suggests that he was not offended by Heinichen’s manner or approach to music. The same cannot be said of his attitude towards Mattheson, whose publications were conspicuously absent from Bach’s sales stock. Moreover, Bach seems to have ignored two invitations to contribute a short biography to Mattheson’s Grundlage einer Ehrenpforte, in spite of Mattheson’s praise of Bach in Orchestre 2 (1717),68 and their
possible meeting in St Catherine’s church, Hamburg, in 1722.69
Why Bach did not promote Mattheson’s books, when to do so would have been commercially advantageous, and why he did not foster the contact and reply to Mattheson’s invitation when others from the Thüringian region responded,70 are tantalising questions. Perhaps Bach formed a negative opinion of Mattheson’s morality and theology through the battle with Buttstett, or found his comments inaccurate or blasphemous, or even considered Mattheson himself to be an abuser of music, an author who sinfully sought his own glory rather than God’s.71 Unfortunately we
cannot know.
Views of composition
Even though Bach accepted Mizler’s invitation to join his corresponding society, whose aim was to study the science of music, Mizler understood that Bach never intended to ‘occupy himself with deep theoretical speculations on music’,72 choosing rather to contribute musical com-positions that demonstrated theoretical principles. When Bach was teaching he could not avoid covering the theoretical basis of music. Several short sets of his music theory guidelines have survived,73copied by his pupils C. F. Richter and Carl August Thieme (1721–95) in 1738.74
The guidelines are a free adaptation of sections from Niedt’s 67
NBR Doc. 140, and appendix, 529. Wolff Learned Musician, 342.
68Mattheson, Orchestre 2, 222 footnote comment. 69Wolff, Learned Musician, 213–15. 70
Mattheson, Ehrenpforte, s.v.‘Walther’, 387. 71Appendix I, 1691-V, 1716-I. 72
BD III, Doc. 666, 89; NBR, 306.
73NBA Supplement, ed. P. Wollny (2011), 37–38, rules for figured bass from P 225, 123–4; 41–44, rules for syncopation in double counterpoint; 45–64, rules for canons.
74
Ibid., 3–38. Facsimile in Pamela L Poulin (ed.), Johann Sebastian Bach: Precepts and Principles
for playing the Thorough-Bass of Accompanying in Four Parts. Leipzig, 1738 (Oxford: Clarendon Press, 1994), 59–107.
Musicalische Handleitung, whose descriptions exude a clear and strong theology of music. We do not know exactly what Bach would have considered deep theoretical speculations, or what exactly Mizler meant by Bach demonstrating theoretical principles in his music, but I suspect both men would have considered proportional parallelism a practical expression and demonstration of the theoretical principles of music.
Trying to paint a true picture of the concepts Bach might have used when he spoke or thought about the construction of music is hindered by the lack of surviving personal papers. Therefore to use concepts that his peers used in books he would have read is the next best route towards catching a glimpse of Bach’s mental imagery and intellectual framework. Several authors described the composition and construction of music in terms used by rhetoricians, dividing the creative process into three: dis-position, elaboration and decoration. Some used the imagery of the human being, describing the different parts of music and art in terms of the soul, the body and the clothing. The most popular image, however, was that of architecture, and this is the image I will use to explain the parameters of my own analyses of Bach’s collections in Part II.
In music treatises that Bach knew and read, the composer’s task is likened to that of the architect as he designs and builds a new mansion. Just as the architect draws up a plan of a building, detailing the size and position of specific rooms and features within a house, the composer should do likewise, and draft a detailed plan of the parts and sections of a composition. The numerical analyses in Part II are the musical equivalent of an architectural plan. Proportional parallelism has enabled me to recon-struct Bach’s lost constructional drawings. The reconstructed plans show the literal 1 : 1 and 1 : 2 proportions that Bach designed using the ratios of the numbers of bars between movements, within works and between works in a collection. Sometimes two collections are parallel, just as an architect might design two or more proportionally related pairs of mansions in a palace complex. Sometimes old buildings retain vestiges of older structures, which allow a historical architect to reconstruct the position of an original wall, or the shape of an original foundation before its rebuilding or renovation. In the same way, it has occasionally been possible, through vestigial numerical evidence, to detect Bach’s early plans for a collection, and to reconstruct their original shape and size. Whenever numerical evidence of this kind is available, it is discussed in the context of the latest source evidence.
Architectural ground plans are simply a parallel image or an analogy to Bach’s musical ground plans. When reading the ground plans in Part II, it
is important to remember that the specific numerical dimensions of Bach’s collections are of little importance: it was the proportions and not the numbers that he was striving to achieve. In emulating the 1 : 1 unison and the 1 : 2 octave, he was aspiring to create a musical structure that would render his compositions the most perfect, the most beautiful, the most eurythmic, and the most lasting.
Evidence in Bach’s scores
The success of a historically informed theory is dependent upon accurate data. If Bach added bars and movements of specific lengths to collections in order to make them perfectly proportioned, it is only by ascertaining how exactly he counted the bars in his works that his numerical planning will be discovered. Many number experiments of the past have fallen at this point, as they use artificial data gathered from the layout of rationalised modern editions, and therefore show nothing more than that symmetrical ordering and numerical coincidence are naturally occurring phenomena. To be able to claim that a composer deliberately incorporated numerical patterns into a work, the data must be pure, based upon the composer’s original scores.
Schmieder’s thematic catalogue (BWV) has been a fabulous resource for Bach scholars for over sixty years. The beauty of the catalogue for number research is that it includes a bar total for each movement that Bach composed. Taking over from Wolgast and Ruthardt, Wolfgang Schmieder aimed to include all known works by Bach when he began to compile his catalogue in 1937. The first edition was published in 1950 after the loss of many sources in the Second World War and in time for the celebration of the bicentenary of Bach’s death. Forty years later Schmieder produced a revised edition, assigning a BWV number to newly discovered works and demoting those of doubtful authenticity to an appendix.75
However, much as the catalogues are a useful reference for the number researcher, their authority can be misleading, because the figures can disguise Bach’s compositional intentions. All repeats are included in the bar totals of the Schmieder catalogue, whereas in the shortened version, BWV2a, they are not. Printing errors apart, the bar numbers in the
75‘Schmieder, Wolfgang’, in J. S. Bach, ed. Malcolm Boyd. Oxford Composer Companions (Oxford University Press, 1999).
catalogues reflect the consistencies of a rationalised editorial policy, rather than the inconsistencies Bach introduced when he laid out his scores. Similarly the latest scholarly editions with full critical commentaries can never be a substitute for working from Bach’s original scores, although their clean modern typeface is a welcome aid to deciphering Bach’s often complex orthography. Bach occasionally noted the bar total at the end of a movement or page, and he quite regularly wrote the number of bars’ rest into orchestral parts. This documented usage shows that he counted bars much as we do today. However, there are several ambiguous features in his scores that can complicate the acquisition of accurate data. These features include the da capo or dal segno indication, the repeat, the half bar line feature and what I call the TS feature, when the time signature is used as a bar line. I will discuss each in turn.
The da capo or dal segno
The ABA structure of a da capo aria or movement is normally notated with crystal clarity in Bach’s compositions. Bach usually, but not always, indi-cates a return to the A section at the end of the last bar of the B section with the words‘Da capo al fine’ or ‘
D
’ or with a simple sign %. Sometimes he adds thefirst bar of the A section after the last bar of the B section, in which case the performer returns to the second bar of the A section. The lengths of the sections are not affected by the notation, but Bach’s layout requires space for one more written bar. If he was working out his proportions from the groundplan of the score, rather than the repeats, it is important to observe this feature literally.Bach’s
D
indication in his fair copy of the ‘Gavotte en Rondeau’ BWV 1006/3, P 967,76 illustrates the tremendous importance of score layout for number studies. Had I relied upon the NBA edition or the Schmieder catalogues (100 and 108 bars respectively) I would have missed Bach’s design of the 92-bar layout, and the large-scale proportions within the Six Solos for violin (BWV 1001–6), and the theory of propor-tional parallelism might never have been formulated. Bach’s two different notations of theD
return in the Allegro movement used in both BWV 1049 and BWV 1057 show his flexibility in a situation that might originally have been caused by a copying oversight. In BrandenburgConcerto no. 4 (BWV 1049/1) he notated the movement as 352 (
D
75) bars, whereas, in his arrangement of the same movement as a concerto for solo harpsichord and orchestra (BWV 1057/1), he notated it more logically as 344 (D
83), with the da capo return at the end of the last bar of the B section.The repeat
Bach used the repeat sign liberally in his scores, writing at a double bar either two or four dots, centrally between the five lines of the stave. Observing the repeat indications in a movement frequently doubles its length, making the bar total of an entire collection extremely large and unwieldy. A composer creating parallel layers of proportion with the number of bars is therefore most likely to ignore repeats in his bar totals.
Where the repeated bars are integral to the meaning of a text, as is the case in texted chorales, Bach frequently notated twelve bars with a repeat, even though the text demands sixteen bars. I was first confronted with this problem when exploring the numerical structure of the St Matthew Passion (MP). According to the shorter Schmieder thematic catalogue, the Passion has exactly 2800 bars, afigure that includes all the da capo indications and the repeats in the chorales. However, despite the seduc-tive beauty of a 2800 bar total, it was necessary to explore the logical possibility that Bach was also aware of the literal layout of the score with the number of written bars, including the non-repeated chorales and without the da capo repeats: this showed a parallel structure with 2400 bars.
A repeat can also be indicated in words. In the full score of the St Matthew Passion, for example, there is no music for Chorale 17‘Ich will hier bey dir stehen’. Instead, Bach inserted a written indication that Chorale 15 should be repeated with a new text.77 Similarly the ‘Osanna’ of the B-minor Mass (BWV 232 IV/1) is not written out a second time; instead Bach writes the words‘Osanna repetatur’ after the ‘Benedictus’. In the Goldberg Variations Bach uses words to indicate a repeat of the opening Aria after the Quodlibet variation 30, as he does in the Christmas Oratorio, to indicate a repeat of the opening chorus BWV 248/24 after Chorale 248/35.
77Table 12.1.
