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Bach's canon
28 July 2018

Bach's canon

It is 268 years ago today that Johann Sebastian Bach died. On 28 July 1750 the German composer breathed his last breath in Leipzig. Maurits Cornelis Escher was a big fan. The similarities between them were considerable: the mathematical order, the strictness of the rules, the symmetry, the systematic approach.

M.C. Escher, Regular division drawing no. 59, India ink and pencil on paper, November 1942

M.C. Escher, Regular division drawing no. 59, India ink and pencil on paper, November 1942

Escher was particularly fascinated by Bach’s canon. In a letter to his friend Hein ’s-Gravezande from 1940, he wrote*:

'Now, I should like to say something else to you about the connection with music, primarily that of Bach, i.e. the Fugue or, put more simply, the canon. I loved Bach and I love him too without “understanding” his technique, but since I understand a (little) bit of it, I love it all the more.

What is a canon? I do not feel qualified to offer an adequate definition, but in essence it can be distilled to a short motif—a finished product, as it were—which is repeated. This repetition manifests itself in all kinds of ways: identical, or in a different key, or in reverse, or upside down (like a mirror image), or at half tempo, and these inverted motifs are played simultaneously, creating all manner of mathematical figures. It has a great deal in common with my own motifs, which I have rotate around various axes too. Nowadays I have such a strong sense of relationship, of affinity, that when I am listening to Bach I frequently get inspired and feel an overwhelming urge to listen to his insistent rhythm, a cadence seeking something of the infinite. In the Fugue everything is based on a single motif, often consisting of just a few notes. In my work, too, everything revolves around a single, closed contour.'

Escher saw the connections between Bach’s canon and his own regular plane division. The composer had created sonic ‘manipulations’ in a manner akin to the way in which Escher created visual manipulations.** While reading the quote, it immediately evokes Escher’s tessellations, with the single motifs that are rotated and mirrored and endlessly repeated. As an ode to Bach, we are sharing such a tessellation today.

Regular division drawing no. 59, repeated

Regular division drawing no. 59, repeated

Read the article about the St Matthew Passion programme as well, with samples from his calendar mentioning Bach concerts he visited.

Source

[*] and [**] Wim Hazeu, M.C. Escher, Een biografie, Meulenhoff, 1998, page 272-273

Erik Kersten

Erik Kersten

Editor

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On 17 July 1950, Maurits and his youngest son Jan (11 years old) left for Paris, the beginning of a French trip just like the one he had made as a child.
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Eldest son Arthur had moved to Lausanne to study geology on the advice of uncle Beer. George had recently enlisted for military service. On this, Escher wrote:
'George has been doing his military service for two weeks now, much to our chagrin, after years of delaying his studies. This is lousy, because you never know whether such a boy will later find the energy to continue studying. [This fear proved to be unfounded.] So we are here with Jantje who, still in primary school, will not be leaving us any time soon. '
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In July 1960 Escher completed the last of his four ‘circle limits’. He had struggled with it for a while, but it was a publication by the Canadian professor H.S.M. Coxeter that set him on the right path. He had met this professor at the University of Toronto in 1954, during the International Congress of Mathematicians. In the article, Coxeter described how a tessellation from the centre to the edge of a circle is increasingly reduced and the motifs come to lie infinitely close together. In 1957 Coxeter gave a lecture for the Royal Society of Canada and he asked Escher by letter if he could use a few of the graphic artist’s works in the lecture. Afterwards, Coxeter sent Escher a copy of his lecture (which had been published under the name Crystal Symmetry and its Generalizations), in which he also included the figure about which Escher would become so enthusiastic. Coxeter in turn based this figure on the work of the French mathematician Jules Henri Poincaré, who visualised this form of hyperbolic geometry in his Poincaré disc.