The Sublime between Music and Mathematics

I love classical music and … I discovered that mathematics is fundamental! It is an article that is a bit difficult and also very historical, but for lovers of classical music it is worth reading it, perhaps after learning some notion of music!

Is there a relationship between mathematics and music? For the Pythagoreans no doubt yes. Leibniz argued that "music is the hidden mathematical exercise of a mind that calculates unconsciously". Lorenz Christoph Mizler, a Bach student, said that "music is the sound of mathematics". Below J. S. Bach between canons and legends.

"The Society for Music Sciences over the years had 19 members, including Telemann and Haendel. Bach entered it in June 1747: needless to say, as a fourteenth member (even in 1747, 14 appears twice). For the admission it was necessary to produce a musical composition of mathematical nature, and to present a portrait: Bach took two birds with one stone, presenting a portrait that depicts him with the score of a triple six-voice Canon in hand. At the end of each year the members of the Society had to exhibit a new composition: in 1747 Bach delivered the canonical Variations on the theme "I get off the stars", in 1748 the musical offer. In 1749 he wanted to present the art of escape, which he could not finish because of his health conditions.
Together with the Goldberg Variations, these works constitute his spiritual testament: a dematerialized music, built on the basis of abstract principles of arithmetic and geometric symmetry. As the word already says, which means "rule" or "law", the musical form that best lends itself to this type of symmetry is the canon. A series of chasing entries, repeating the first in a translated, reflexed or proportional form. The various items, although all similar, can be synchronized or staggered, higher or lower, parallel or mirror, faster or slower.
Naturally, the whole must be musically sensible and pleasant: which is all the more difficult, the more the characteristics of the various voices differ from one another.

The first major series of Bach's canons is found in the Goldberg Variations, composed in 1741 to alleviate the insomnia of Count von Keyserling, and repaid with a golden casket containing 100 louis. The story is probably a legend, due to the name of the count's harpsichordist: Goldberg in fact means "mountain of gold". The title was simply: Air with 30 variations, composed to refresh the spirit of the melomaniacs. Since the air is repeated at the end, there are a total of 32 pieces, each of 32 bars. The 9 multiple variations of 3 (from the third to the twenty seventh) are canons at increasing intervals: we start in unison, with the voices repeated at the same height and we end up at the ninth. The metric structure of the canons presents all 9 possible combinations of 2, 3 or 4 groups of notes, each with 2, 3 or 4 notes.
A veritable summa of canonical art is found in the Fourteen canons discovered in France in 1974, as an appendix to a manuscript of the Goldberg Variations.
Only two of them were known, and one was precisely the one presented by Bach for admission to the Mizler Society. Among the others, two are particularly virtuosic: the eleventh to six voices, reflected specularly two by two, and the fourteenth, in which a voice is not only slower than the first, but is also played backwards.
In 1747, a month before his entry into Society, Bach was invited to the court of Berlin by Frederick the Great.

The king proposed to him a theme on which improvise an escape, and the "old Bach" improvised two: one to three, and the other to six voices. Back home he added 10 canons and a sonata to three, always on the same theme (which in the trio is played on the flute, the instrument of the king), and titled the whole Regis Iussu Cantio Et Reliqua Canonica Art Resoluta, an acrostic for RICERCAR, which was an archaic name for the escapes. For its origins, today the work is instead called Musical Offer.
When he died, Bach was preparing the publication of the Art of Escape, composed in the last decade of his life. The drafts were ready, but they did not specify the order of the 14 escapes and the 4 canons of which the work is composed. The joints, which are all on the same theme and in the same shade (king minor), are divided naturally into four groups: 8 simple, double and triple, 3 imitation, 2 mirror and 1 large escape (the unfinished) . So again the numbers that make up the name of Bach reappear, and probably the canons had to separate the four groups. In this case they are fugal canons or canonical fugues: they combine the characteristics of the canon with those of the escape, which is a procedure based on analogous rules (such as the change of tone or mode), but freer.
The composition of canons and fugues is a mysterious and difficult problem, whose solution can be fun and stimulating. For example, a group of European musicians coordinated by
Luciano Berio is trying to rework and complete the Escape Art for the Spoleto Experimental Opera Theater, and the new version will be performed at the end of the year in various European cities. On his scores, however, Bach claimed to compose for another reason: Soli Deo Gloria, "solely for the glory of God". In short, he understood that, since in Paradise one speaks mathematics, canons and fugues must be the music that goes out of fashion up there. "(From an article by P. Odifreddi, So Bach gave the numbers HERE.)

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