Bach composed with numbers

On the morning of the 28th of July 1750, Johann Sebastian Bach woke up and saw the light .
He hadn’t seen it for a long time, because of two eyes surgeries he underwent some months earlier, which made him go blind. On the evening he died, leaving unfinished a great fugue on a theme of four notes :“B flat, A, C, B” in English notation, or “B-A-C-H” in German notation (the German notation for B is H). Bach had already mentioned in music his own name in other works of his, belonging to crucial periods of his life. Among these “The Passion according to St Matthew” or the “B Flat Minor Mass”; he considered it a portrayal of the Holy Cross because of its zigzag course on the stave.

Replacing the letters with their equivalent numbers ( 1 for A, 2 for B, etc.) “Bach” becomes “2138” and adding up the figures the result is 14. The great unfinished fugue is just the fourteenth one in his work “The Art of Fugue”, and that number frequently occurs throughout this work. For example, in the choral fantasy on the “I’m opposite to your throne” theme, last Bach had finished: the theme is mad of 14 notes, the entire melody of 41. The latter is obviously the opposite of 14, but it also coincides with “J.S. Bach”. “We could say there are no hidden meanings underneath Bachian numerology” – playfully says Malcolm Boyd in a biography (by sheer chance) consisting of 14 chapters, 41 music samples and 14 figures among which Bach’s is the last one. More earnestly, we guess a Bachian sensibility to some logical and mathematical music aspects, as it was a widespread habit at that time.

Liebniz had already stated earlier that “…music is the hidden maths exercise by an unconsciouslyclip_image002 calculator mind”. The statement was resumed by Lorenz Christophe Mizler, a Bach’s pupil who founded in Lipsia in 1738 a “Musical Science Association”, aiming to underline connections between maths and music. Mizler said the “music is the sound of maths” an in 1739 he produced an example of automatic composition entitled “A figurative bass” mathematically suggested and very clearly showed by a brand-new machine. It’s not a chance he is considered a forerunner of ”Oumupo”, Prospective Music Factory, founded in 1985 from “Oulipo”.

As time went by, Musical Science Association grew up to 19 members, among which Telemann and Handel. Bach entered it in June 1747:needless to say, as fourteenth member (in 1747 too 14 appeared two times). For the entrance examination they had to produce a mathematical shaped musical composition and a portrait: Bach killed two birds with one stone, producing a portrait representing him holding a six voices triple canon stave. Every end of the year , members had to show a new composition: in 1747 Bach produced the canonical variations on the theme “I come down from the stars”, in 1748 the “Musical Offer”. In 1749 he would have presented “The Art of Fugue” which he didn’t manage to complete , due to his health conditions.

Along with “Goldberg variations”, these works make up his spiritual testament: a dematerialized music, organized according to abstract principles of geometrical and arithmetic symmetry. As the word says itself , meaning “rule” or “law”, the musical form best fitting this kind of symmetry is the canon. A sequence of voices chasing each other , repeating the first one moved ahead, reflected or proportionally. Even though resembling each other, the diverse voices can be this way synchronized or staggered, higher or lower, parallel or specular, faster or slower.
Obviously, the ensemble should turn out to be musically reasonable: this aspect is as much difficult to find as more the voices features differs each other. The first great Bachian canon chain lies in the “Goldberg variations”, composed in 1741 in order to ease Lord von Keyserling, and rewarded with a golden box containing 100 louis d’or. This story is probably legend, due to the name of the lord’s harpsichordist: in fact, the term “Goldberg” means “a golden mount. The title was a simple: “Aria with 30 variations, composed to cheer up music lovers’ spirit”. As the aria is repeated in the end, there are 32 pieces in all, 32 measures each. The 9 variations multiple of 3 ( from the third up to the twenty-seventh) are increasing interval canons: unisonous beginning, with voices repeated at the same pitch to end over in the ninth. Canons metrical structure shows all the nine possible schemes of 2, 3 or 4 groups of notes, each with 2,3 or 4. A proper summa of the canonical art in the “Fourteen canons” discovered in France in 1974, in addition to a manuscript of the Goldberg variations.

Only two of them were well-known and one was just the one Bach presented for the admittance to Mizler Association. Among the others, two of them are particularly virtuoso: the eleventh with six voices specularly reflecting two by two, and the fourteenth, in which a part is not only slower than the former, is played backwards. In 1747, a month before he joined the Association, Bach was invited by Friedrich the Great at his Court. The king suggested him a theme to improvise a fugue on, and the “old Bach” doubled: the former three-part, the latter six-part. Back home, he added 10 canons and a three-part sonata, always on the same theme (in the trio played by a flute, king’s favourite instrument), and entitled the whole thing “Regis Iussu Cantio Et Reliqua Canonica Arte Resoluta” acrostic for RICERCAR, ancient name to indicate the fugue. Due to its origins, the work is today known as “Musical Offer”.

When he died, Bach was preparing the publication of “The Art of Fugue”, composed in the last decade of his life. Drafts were ready, but they didn’t specify the organization of the 14 fugues and 4 canons the work consists of. All on the same theme and on the same tonality (D Minor), the fugues, are divided into four groups : 8 simple, double and triple, 3 in imitation, 2 specular and 1 great fugue (the unfinished one). So the numbers composing Bach’s name appear once again, and probably canons were supposed to separate the four groups. In this case is about 4 fugal canons or canonical fugues: they merge that way canon features with fugue ones, a course based on similar rules( as tonality or mode changes), but freer.
The composition between canons and fugues is a difficult and mysterious matter, whose solution can be funny and stimulating. For example, an European band directed by Luciano Berio is trying to remake and complete “The Art of Fugue” for the Lyrical and Experimental Theatre in Spoleto (Italy) and new version will be performer at the end of the year in may European cities. But on his score, Bach declared he composed for a different reason: Soli Deo Gloria, “only for God’s Glory”. He realized that, as in Heaven they speak maths, canons and fugues should be in vogue up there.

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