Abstract
The 'Little Organ Book' (O=B) is the first major cyclical work of Johann Sebastian Bach, and was probably written between 1703 and 1720. Hitherto it has been assumed that the O=B was unfinished. Of the 164 chorale titles listed in the autograph, only 46 chorales were composed. As a preliminary
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conclusion to the O=B, Bach, while in Köthen, wrote an elaborate title page. In this apparent torso, different types of arithmetical structures can be identified, suggesting that Bach had not only regarded the O=B as being complete, but also that he had planned it in this form. The way in which Bach structured the O=B was not new. There are examples in music from the mid-14th century right up to Bach's immediate predecessors. This is demonstrated in the works of Machaut, Dufay, Frescobaldi, Schutz and Buxtehude. Already in the early works of Machaut and Dufay, golden sections and gematrically derived figures - constituting a signature of their own name - can be detected. In the d-minor Passacaglia of Buxtehude, a complete lunar year is depicted. The basis for the connection between music and mathematics was established in Pythagorean thinking, which was extant up to the time of Bach, as shown by the writings of Gioseffo Zarlino (1517-1590), Athanasius Kircher (1601-1680) and Lorenz Christoph Mizler (1711-1778). In the first place, arithmetical structures such as numeric relations and golden sections are found in organ works by Bach from different periods. Such ratios are also observed in the O=B. In addition, Magic Squares and a Magic cube could be observed for the first time, the numerical foundations of which had to be carefully planned and could not be amended after the writing of the manuscript had commenced. This makes it clear that the O=B was planned and executed in the form in which it has been handed down to us. In addition to the purely arithmetical structures, cyclic arrangements of hymns are also discernible. Within the cycles of the Cantica pro et omni tempore there is a Trinity series, a series of new beginning in Christ and a Fibonacci series. The allocation into patterns of the manuscript, and its pagination, reveal ordered structures. Within the chorale series, many numeric symbols are observed, reinforcing the arithmetical series structures. In the Christian context, the ecclesiastical patriarch Augustine Aurelius (354-430) was an important early advocate for the interpretation of numbers. He has exerted a lasting influence on the Western Church up to the present age. His meticulous interpretation of biblical numbers is shown in the original texts. In his own use of numeric symbols and gematrical procedures, Bach stands in a long tradition, which is represented by numerous examples in literature, painting, architecture and music. However, thanks to the large number of possibilities (permutations?), the interpretation of numeric symbols is irrelevant in demonstrating the planning intent and completion of the O=B. The most ambitious construct is very likely an extremely exact golden section, which relates to the entire composed piece, and coincides with the centre of Cantica pro tempore. In addition to further golden sections, a Fibonacci series is also represented. The shared networking of these arithmetical structures required a carefully considered pre-arrangement that must have taken a lot of time. Notwithstanding one´s amazement at all these different interrelated structures, it must not be overlooked that the special genius of Bach consists not so much in solving these combinatorial exercises, but rather in his capacity to create – within this tight corset of prerequisites – music of such quality.
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