Overlooked for almost three hundred years, the three characteristics of proportional parallelism can now be seen in every collection and multi- movement work that Bach published or left in autograph fair copy, their understated simplicity unifying complex works of art.
Bar total as a multiple of 10, 100, or 1000
All of Bach’s collections, whether published or in autograph fair copy have a bar total that is a multiple of 10, 100 or even 1000. In the few collections where the bar number is not a multiple of 10, 100 or 1000, the round bar total is seen when two companion collections associated by a common title or an indisputable compositional timeline are viewed together. For example it is only when book 1 of Das Wohltemperirte Clavier (WTC), The Well-Tempered Clavier, is viewed with the companion collection that Bach prepared at the same time, the Inventions and Sinfonias, Aufrichtige Anleitung (AA), that one sees the total 3120 bars. And it is only when CÜ I and CÜ II are viewed together that the self-referential key pattern and the overall plan of 3120 bars, parallel to the earlier pair, can be seen.82
Unfortunately we do not have documentary evidence to demonstrate conclusively how Bach achieved such an exact total over such a long period of time. The occasional recorded bar totals in his music suggest he was keeping an eye on the length of a movement, either because it was a copying habit, or because he was aiming to achieve a specific numerical 81
J. F. Riederer, Catalogus derer eintausend fünfzig Paragrammatum Cabbalisticorum Trigonalium (Nürnberg, 1719). Introduction, 4.
82Table 6.8.
total, or both. Above all, though, it is the additions and extensions that Bach made as he compiled and revised a collection that provide empirical evidence of his intention to create a perfect structure imbued with the characteristics of proportional parallelism.
Numerical reference to Bach’s name
The second characteristic of proportional parallelism is a recognisable Bach signature, formed either in the total number of bars or in the overall key pattern. A common numerical allusion is the combination of the numerals 2-1-3, with numbers 14, 41 and 158.83Because the numbers 2- 1-3 are also the component parts of the first numerus perfectus 6 it is dangerous to interpret the bar totals formed from these numerals as a self- referential conceit. It is quite possible however that this numerus perfectus / parallel signature duality pleased Bach.
The signature is sometimes formed by key patterns across a work, as in both the collections for violin, the Six Solos (BWV 1001–6) and the Six Sonatas (BWV 1014–19). In each case the pattern B-A-C is clearly seen when the 1600 bars of the 2 : 1 proportion are grouped together (Tables 5.1
and5.4). In Bach’s time the musical letters B and H for B natural were
interchangeable, contrary to a common twentieth-century misconception. Parts I and II of the CÜ series have signatures in both the bar totals and the keys: 3120 bars and B-A-C-H, facilitated by Bach’s transposition of the French Overture from C minor to B minor.84
It has been suggested that Bach deliberately copied the Six Solos for violin onto forty-one sides of music as an allusion to the numerical value of one of his signature forms;‘J. S. Bach’. This is possible, although if Bach had intended forty-one sides, he could have numbered each side of music, as he did for the Missa of the B-minor Mass (P 180). In the event, he foliated the solos on consecutive recto sides from 2 to 22.
The different numerical totals of Bach’s name as a whole have been the subject of speculation and problematic interpretations since Friedrich Smend first introduced the notion to musicology in the 1940s. It would be ludicrous to claim that Bach intended every occurrence of the numbers 14, 41, 70, 100, 108 and 158 to be a self-reference. The multiple possibilities generated by each number when their divisors and multiples are‘allowed’ mean that almost any number in a Bach score could be considered
signatory. There are nonetheless some notable occurrences of these numbers in obvious structural positions which suggest they were planned. Framing the collection of Leipzig organ chorales transmitted in P 271 is a perfect 2 : 1 proportion formed by the Bach name numbers 158 : 79. The first and last chorale preludes in this collection together have 158 bars, and the two central preludes have 79. The proportion would have had to be planned before Bach began to copy the collection.
Strongly implied as one of the numerical ground plans of the posthu- mously published The Art of Fugue is a similar perfect 1 : 2 proportion on the name number 158, although multiplied by 10. Thefirst eight move- ments have exactly 790 bars, after which is the much discussed emblematic signature. The remaining movements have 1300 bars, and thefinal incom- plete fugue 239 bars, just 41 bars short of the anticipated 1580 bars. Planning to publish the collection posthumously, Bach might have intro- duced the puzzle of the unfinished final fugue so that the reader would ask both‘what’ and ‘who’ is missing; its solution, highlighted by the emblem- atic seal after bar 790, pointing to the overall 1 : 2 structure with its missing 41 bars, which in turn points to the missing author, J. S. Bach (41).
The number 14 appears relatively frequently in the large-scale totals of collections. 1400 bars in the B-minor Mass, 1400 bars of the Ascension and Easter Oratorios, possibly 1400 bars in Six Keyboard Concertos, 2800 in the MP and possibly 2800 bars in the two keyboard collections CÜ III and IV.
Interpreting the number 28 as self-referential rather than as the second numerus perfectus raises the same problem as the interpretation of permu- tations of 1-2-3. Embracing their parallel significance is perhaps the best solution.
The annotations in his copy of Calov’s Bible commentary show that Bach was interested in eschatological numbers. For example, he wrote in the margin beside Daniel 12: 7–13 the following numbers that occur in the printed text: 1260, 1290, 1335, 1941 and 2408.85 Besides 2408,86 these figures have not occurred in the bar totals in Bach’s scores and so I suspect we should take Bach’s annotations at face value, as evidence of a lively intellect, intrigued by figures while reading his Bible carefully. Friedrich Smend was keen to systematise a Lutheran interpretation of
85
R. A. Leaver, J. S. Bach and Scripture: Glosses from the Calov Bible Commentary (St Louis: Concordia, 1985), 113–15.
862408 is the bar total of BWV 1001–6 in the modern NBA layout.
numbers and apply it to Bach’s music.87 Laudable as Smend’s aim was, it proved impossible due to the difficulty of amassing reliable data and the multiple ambiguities of parallel techniques.
Parallel layers of 1 : 1 or 1 : 2 proportions
The most significant of the three characteristics of proportional parallel- ism, however, is the layers of 1 : 1 and 1 : 2 proportions that divide the bar total, the movements and the works of a collection into rationally and frequently symmetrical structures. The prevalent view of the perfection of proportions close to the unity (1 : 1 and 1 : 2) was good reason for a composer like Bach to take the time and effort to introduce them so precisely into the structural layers of his published collections.
These perfect proportions88coexist at several layers in the structure of the publications: between sections of a movement, between movements of a work, between two works in a collection, between parts of the collection as a whole and between two related collections. In the Six Solos for violin (BWV 1001–6), for example, there are five distinct layers of proportion, and in the Christmas Oratorio (BWV 248), at least three. To save space, the smallest-scale layer of proportion, formed between sections of a move- ment and commonly found in binary-form dances, will not be included in the demonstrations inPart II.
Strangest of all the proportions in Bach’s compositions is that found between two or more collections. It seems too large and too theoretical to have been intentional, and too speculative and invisible to be of any scientific value. Nonetheless, having discovered it, it has been noted with supporting documentary evidence. In the case of the violin solos, a double 1 : 1 proportion is formed with the Six Sonatas for violin and harpsichord (BWV 1014–19), which also have exactly 2400 bars. This is particularly persuasive because their next layer of proportion is identical– a double 2 : 1 proportion with four works having exactly 1600 bars and the other two 800 bars.89The largest-scale proportion is also found between Bach’s first two keyboard collections and his first two published keyboard collec- tions. In spite of the time gap, Bach created a double 1 : 1 proportion between the four collections and their identical totals (2 : 2 collections in 87Tatlow, Bach and the Riddle, 8–36.
88
By using the word‘proportion’ rather than ‘ratio’ I am following usage of the early eighteenth century, when theorists used‘proportion’, although aware of its technical inaccuracy. Walther, Praecepta (1708), Book II, 10 (Benary, 76).
3120 : 3120 bars), adding a hitherto unseen relationship between them– a relationship that perhaps Bach alone knew.
The extent
The three characteristics can be seen in all of the collections and multi- movement works Bach published or wrote out in best top copy. The word ‘collection’ in this definition is used to mean works in which two or more compositions are united by title, e.g. CÜ I and II, or in which two or more compositions are collected under one title, e.g. the six partitas in CÜ I, or the six works of the Brandenburg Concertos. The term‘multi-movement works’ includes large compositions with many movements under one title, e.g. MP and the B-minor Mass, as well as smaller works such as the Canonische Veränderungen on the melody ‘Vom Himmel hoch’, known as the Canonic Variations.
The ten works printed during Bach’s lifetime are ‘Gott ist mein König’ (BWV 71); CÜ I (BWV 825–30), for which five partitas were published separately before publication of the collection of six partitas; CÜ II (BWV 831 and 971); CÜ III; the Aria with Thirty Variations, the ‘Goldberg Variations’ (BWV 988); Musicalisches Opfer (BWV 1079); the Canonic Variations (BWV 769); Sechs Choräle von verschiedener Art known as the ‘Schübler Chorales’ (BWV 645–50); Die Kunst der Fuge, The Art of Fugue, largely overseen by Bach although published posthumously (BWV 1080); and two canons, the Canon Triplex à 6 voci (BWV 1076) and an untitled canon (BWV 1074). Of these ten, all but the two canons are collections according to the definition above, and all display the three characteristics of proportional parallelism. Bach was also responsible for the sixty-nine engraved musical settings in the hymn book Musicalisches Gesang-Buch published by G. C. Schemelli, for which he composed some new melodies and revised some harmonisations. Although the hymn book is a collection, Bach’s contribution to its musical content does not fall within the defin- ition of a Bachian collection.
The autograph fair copies are manuscripts that Bach intended to stand as a neat and definitive version, normally written in meticulous calligraphic script and containing very few errors.90 In many cases early versions of these works survive, making it possible to trace the revisions Bach made as he created the perfect parallel proportions. Particularly interesting in this 90Yo Tomita,‘Sources. 1. Manuscript Sources’, in J. S. Bach, ed. Malcolm Boyd. Oxford Composer
Companions (Oxford University Press, 1999).
respect are the Brandenburg Concertos (BWV 1046–51); the Six Solos for violin (BWV 1001–1006), WTC I (BWV 846–69); AA (BWV 772–801); the Six Trio Sonatas (BWV 525–30); the so-called Leipzig Chorale Preludes, or the Great Eighteen (BWV 651–68); MP (BWV 244); the Missa in A major (BWV 234); the Christmas Oratorio; the Easter Oratorio (BWV 249) and the Ascension Oratorio (BWV 11).
Many of Bach’s collections are in mixed calligraphy, either mixed in the sense that the manuscript score includes a copyist’s hand, or that Bach himself mixed calligraphic styles, with corrections and sometimes with newly composed material, demonstrating that he was revising or adding to the composition even as he was making the fair copy. The three characteristics shed light on the status of such manuscripts, which include: St John Passion (BWV 245) and the B-minor Mass (BWV 232). The status of collections transmitted in the hand of a copyist for which no autograph has survived can also be tested against the three characteristics. Collections in this category include the Six Sonatas for violin and harpsichord (BWV 1014–19a), six Cello Suites (BWV 1007–12), and the Early Key- board Transcriptions of concertos by various composers (BWV 972–87). A number of well-known collections in Bach’s hand have come down to us in separate parts, or as an incomplete collection. These include WTC II (BWV 870–93); the French Suites (BWV 812–17); the English Suites (BWV 806–11) and the Orgelbüchlein (BWV 588–644). There are also several collections known today as Bach collections that are nothing more than compilations of works of similar genre and instruments put together by librarians and editors in the nineteenth and early twentieth centuries. As these were not designed or revised by Bach, they will be disregarded in this study.
Bach’s annual cantata cycles fall outside any of these categories. They are in the widest possible variety of handwriting, and the current state of documentary evidence makes it impossible to test whether Bach designed them as collections. Nonetheless examples from several cantatas have been included inChapter 12because they indicate that Bach aimed for numer- ical proportion even when he was composing quickly on such a small scale.
Evidence from Bach’s revisions
In the early days of my research I assumed that Bach had a single, rather than several, well-defined concepts of a work when he made the first sketch, and that the numerical plan, if he had one, would be found in the earliest versions. Gregory Butler has persuasively demonstrated, from
erasures in the original engraved prints, that Bach made many dramatic changes to the original concept of CÜ III to the extent of altering the entire stucture of the collection after he had sent it, presumably ‘complete and finished’, to the publisher.91 Every single change has significant implica-
tions for the overall bar total and the internal proportions. Tracing the revisions through to the perfect proportions in thefinal publication of CÜ III suggests that Bach had a numerical plan for each stage. Making changes to handwritten works was not such a serious or expensive business as amending engravings. However, paper was expensive and Bach’s early versions were in no way lacking structural forethought. The clearest example of Bach discarding an excellent numerical plan in favour of something even more perfect can be found in the construction of the B-minor Mass, from the 1733 Missa, through the implied construction of parts II–IV, to the final 1749 Missa tota.92
When Bach was revising a collection he frequently reordered its con- stituent parts. Studying the original order can sometimes show what the earliest numerical plan was, although it is not always possible to trace the original beneath an amended version. In the case of AA (BWV 772–801), however, we have both Wilhelm Friedemann’s Clavier-Büchlein and Kayser’s later copy of the earlier ordering. The embryonic 500 : 500 bar structure of Friedemann’s copy is visible, although several of the move- ments are left unfinished, as if father were instructing son in the art of proportion. In his own calligraphic autograph copy of the collection, Bach reordered the thirty movements to mirror and complement the key pat- terns of WTC I and in this reordering the constructional blocks of 500 and 500 bars disappeared.93
Editors have been at a loss to explain why Bach made small and seemingly insignificant bar changes to what was already perfectly good music. Hans Eppstein noted that these changes frequently occur in the final work of a collection. Observing that in the sixth sonata (BWV 1019) Bach defied the formal logic set up in the first five sonatas of this collection for violin and harpsichord, Eppstein asked ‘What might be the cause of Bach’s action?’94 Bach’s documented revisions to create layers of parallel
proportion in all his polished collections provide an explanation.
91
G. Butler, Clavier-Übung III: The Making of a Print (Duke University Press, 1990). 92
Tatlow,‘Parallel Proportions’, 142–62. 93 Tables 6.4and6.6.
94H. Eppstein,‘Fragen der Ordnungsprinzipien’, in Leipziger Beiträge zur Bach-Forschung 5, ed. U. Leisinger (Zürich: Olms, 2002), 133.
The numerical demonstrations in Part II often show blocks of bars, or ‘building blocks’, with a rational total, sometimes even proportioned. These readily recognisable units are usually unrelated to thefinal proportion of thefinished structure, but may well have facilitated the construction of the larger-scale whole. For example, the consecutive 1300-bar unit visible in the published version of The Art of Fugue, the contiguous blocks of 500 bars and 500 bars in Kayser’s copy of the Inventions and Sinfoniae, and the 900 bars in Six Solos for violin could all have helped Bach as he created the proportioned collections.
The arrangement of movements forming perfect proportions can appear consecutively, or in a symmetrical, interlocking, dovetail pattern with two ends and a middle, as in Puttenham’s ‘proportion by situation’.95
Symmetry is the quintessential 1 : 1 proportion over a central axis. Of course Bach did not design all the symmetrical arrangements that can be found in his scores: random groups and random numbers will naturally form proportions and symmetry. Nonetheless the recurrence of some very specific patterns and symmetrical features strongly implies intentionality.
Summary
Although the theory of proportional parallelism has been formulated in the twenty-first century, the ambition has been to keep the thought-processes consistent with eighteenth-century ideas and practice. Although it was the intentional application of the principles that led to the discovery of pro- portional parallelism, there remain many dangers in pursuing a method based on numbers. The number of notes or bars in a score offers the illusory promise of a direct line to the composer’s thought-processes, a promise beyond the aspiration of the majority of music analytical tech- niques. If the composer was not aware of how many notes or bars he wrote, the patterns the analyst isolates will reveal nothing about the compositional method, and subsequent interpretations based upon these patterns will be little more than fantasy. The main problem, though, is not the patterns. Patterns are good and important, helping to explain how music works and how a composition is constructed. The problem is when the analyst assumes, without external evidence, that the patterns are proof of the composer’s intentional design. External source evidence supporting the 95George Puttenham, The Arte of Poesie (London, 1589). This work is discussed inChapter 2.
method, the data and the interpretation is essential before the analyst can with any integrity claim that the composer designed the patterns. And even then there must be room for doubt.
The age-old issue of how we listen and think also affects our interpret- ation of a musical score. Bach and many of his contemporaries heard and understood music in terms of modes, hexachords, tones and keys.96 We have experienced atonality. Bach knew that the harmonic proportions he used in his compositions were central to the created order in the universe. Today we believe otherwise.
The many philosophical and musical differences between Bach and ourselves are increasingly irreconcilable. A face-to-face conversation with Bach would be the simplest way to answer many of these questions, but as this is impossible, the musicologist has to confront the problem and decide: either to give up any ambition to hear and think as Bach did, content to discover twenty-first-century resonances in his music; or to continue to