The Mathematics of the “Goldberg Variations”

“Piano radicalism”, world spirit and village inn can be very close together. We’ve known that since Thomas Bernhard’s 1983 novel “Der Untergeher”, published a year after the sudden death of Canadian piano virtuoso Glenn Gould: “Also Glenn Gould, our friend, only turned 51, I thought as we entered the inn” , thinks the first-person narrator, who says he has just returned from Chur from the funeral of his friend Wertheimer and remembers the time that Gould, Wertheimer and himself studied piano together in the early 1950s in Salzburg “at Horowitz”.

The interpretation of the “Goldberg Variations”, which Gould actually performed at the end of the 1950s as part of the festival, becomes a milestone that is supposed to make Gould an exceptional genius compared to all the others. “When we finished our lessons with Horowitz, it was clear that Glenn was already a better pianist than Horowitz himself,” Bernhard reads – and as much as there may have been fiddling back and forth between the village inns and world virtuosity, it follows also this novel of a Bach number mysticism.


Glenn Gould brought the “Goldberg Variations” out of the corner of the esoteric. In 1955 he needed only 38 minutes and 34 seconds for the 32 parts. Most people remember the recording from 1981, which is significantly slower and takes 51 minutes and 18 seconds.

The word “Goldberg Variations” appears in italics 32 times in Bernhard’s novel, and 32 is the bar number of the opening piece “Aria”, which research has never been able to clearly attribute to Bach. However, since no contemporary wrote Bach’s 32-bar pieces, the most well-known motif in the world was left in the master’s handwriting and research continued into the meaning of the number 32 in this piece.

Bach spoke of “32 fundamental notes” in connection with the “Goldberg Variations”, one of the few things that is known for sure.

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The “Goldberg Variations”, quasi divided by three, can be heard on Wednesday as part of the Herbstgold Festival in Eisenstadt at 7.30 p.m. with Julian Rachlin, including an introduction to the mathematics of the piece.

A piece for sleepless nights?

The first Bach biographer, Johann Nikolaus Forkel, invented the rest almost as freely as Bernhard invented Gould’s schooling with his antipode Horowitz. Bach wrote these piano exercises for Johann Gottlieb Goldberg so that he could play the pieces to the sick Count Hermann Carl von Keyserlingk on sleepless nights – with the catch that Goldberg was just 14 years old when the exercises were completed and because of the skills required for this piece should have been a child prodigy.

Page from Untergeher to the Goldberg Variations


Three men and “piano radicalism”: with Thomas Bernhard in “Untergeher”

32 or 30?

In fact, the number 32 is less relevant to the mathematics of the “Goldberg Variations” than the number three or 30. If you subtract the prelude and the da capo at the end, the actual “Goldberg Variations” are 30 in number, the again divided into three parts. Every third piece is a canon and after each canon there is an interval jump up to number 27. Up to Variation 30, where Bach makes a quodlibet against the seriousness of this piece.

Quodlibet was understood in the German-speaking area in the 18th century as a funny to rough compilation of folk songs – “perhaps”, according to the musicologist Siegberg Rampe, “an arrogant hint from the composer to his contemporaries that the essence of his work cannot be determined by consistent calculation alone grasp” is. Consistently, at least that’s how the Mozarteum student Bernhard can be interpreted, must be countered in the novel by the quodlibet of an inn and corresponding inn talks of exceptional artists.

Structure of the 30 Goldberg Variations


Structure of the 30-plus-two-part “Variations” (BWV 988). Always helpful: the “Bach Handbook” (Metzler Verlag)

When the piano piece becomes a trio

The mathematics and the building the work a chance to bring out its wisdom, beauty and passion.

“If you work with violin, viola and cello”, says Rachlin about this work, which was once composed for a two-manual harpsichord, “then the instruments take on different functions: the cello provides counterpoint for the bass, the violin the melody for the right hand “. But, according to Rachlin, the viola “as a middle register” can be used like a libero to combine throughout this piece.

“‘The Goldberg Variations’ are the pinnacle of Bach’s art, to which Beethoven in turn also refers with his ‘Diabelli Variations’ (plus letter D in with Michael Korstick). The three, according to Rachlin, follow you through the entire “Goldberg Variations” because of the canon system, but also because every ninth variation is a duo. Here you can combine three: “One time the violin plays with the viola, one time the viola with the cello, one time the cello with the violin.”

Goldberg Variations as String Trio

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Attempt of the “Goldberg Variations” as a trio at Deutsche Grammophon 2007

All the variations, Rachlin recalls, were written in G major, only three of them in G minor. If you look at the complete work, then the variations presented themselves as “a pyramid consisting of a base of 15, a middle of ten and a top of five variations”, according to Rachlin. The base of the pyramid, for example, results from Variation 15, which is the first variation in minor. The leap from major to minor and from minor to major is pretty much the greatest contrast imaginable in such playing exercises at Bach’s time.

Math isn’t everything

Nevertheless, no matter how much mathematics there is in the “Goldberg Variations”, they are a work of special expression, even passion: “The 25th variation is called the black pearl because it leads into all human abysses, while the 13th variation we call it the white pearl because it radiates so much power and expression.”

“The ‘Goldberg-Variations’ are an absolute physical challenge due to the mathematics, but also the passions and motif variations inherent in them,” sums up Rachlin. Against this background, researchers to this day have not been able to agree who the target audience for these playing exercises might actually have been. Some variations are aimed at the possibilities of the amateurs of the time. According to research, however, Bach could hardly have expected that the average piano student would have been able to play through these variations as a whole.

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