What is magic number

The magic number

Rationality and irrationality in number thinking from Pythagoras to Boulez

Numbers, series of numbers, number combinations, number games, number magic: numbers have played a role in music since music began. This influence became particularly formative in the 20th century, when the sciences developed at an unprecedented pace and permeated all areas of life. Counting and arithmetic - the classic methods of measuring and ordering raw matter - have been systematically applied to the musical material: when composing with twelve-tone rows, in serial music, in stochastic, statistical and random composition methods. And of course also in computer music, where counting and arithmetic are delegated to the machine, the so-called "calculator".

The occult downside of rational counting, the speculation of numbers, has survived over the millennia to our time, as the composers transferred magical squares and number symbolism of all kinds to the musical structures. But where is the line between rationality and irrationality? Sometimes both aspects - the use of the number as a rational design principle and the belief in the supernatural power of the number - seem to merge with one another. The number as an alchemical formula for the creation of life, even if it is only symbolic as a sounding form.

Take, for example, "Study I", the first electronic composition by Karlheinz Stockhausen from 1953: Doesn't the strictly logical series of numbers that determine the inner composition of the tone mixtures look like a mysterious incantation?

1000  -  417  -  521  -  325  -  781  -  625  -  417  -  173  -  217  -  135  -  325  -  260  -  521  -  217  -  271  -  169  -  407  -  325  -  325  -  135  -  169  -  105  -  254  -  203  -  781  -  325  -  407  -  254  -  610  -  488  -  625  -  260  -  325  -  203  -  488  -  390

But be warned against errors: The number is not an "open sesame" that could unlock the meaning of music, its secret. With this series of numbers, Stockhausen only describes the generative principle that is used on the technical level to create sounds and processes; this is comparable to a composer in the 17th century who uses the figured bass to describe the chord progressions underlaid by a melody.

The close connection between musical processes and numbers is neither new nor unusual. On the contrary, it has always been a kind of seal of approval for professional craftsmanship, because number always meant order in music. The standard accusation of 20th century music: it is "too mathematical" or "too intellectual" - this accusation is ultimately narrow-minded because it does not recognize the musical necessities.

Moreover, the new music has never lacked competent criticism and self-criticism, not even in its most radical phases. That the meaning of a work of art is not grasped with tables of numbers and lists of proportions, that a seamless structure does not represent a work of art: this is the state of affairs, for example, pointed out by Theodor W. Adorno. He warned the serialists against a "abdication of the subject" and saw rationality overturning into its opposite:

"The inexpressible content is hidden in the formal a priori, the technical procedure. The general of the structure produces the particular without remainder out of itself and thereby negates it. This is how rationality gains its irrational, the catastrophically blind."

Such justified objections to an absolutization of the role of number and order do not, however, obscure the fact that composing in the twentieth century received a tremendous boost as a result of the influence of numbers. Serial and statistical methods, mathematical formulas such as the golden ratio and the Fibonacci series were used to replace the order-creating function of traditional harmony.

The series formulated by the Italian mathematician Fibonacci in the 13th century is created by always adding the previous number to a new number in the series. So it can be continued indefinitely. Starting with one it reads:

1 - 2 - 3 - 5 - 8 - 13 - 21 - 34 - 55 - 89 - 144 - 233 - 377 - 610 etc.

Stockhausen's piano piece IX, for example, is mainly based on this series and one of its derivatives. The number of repetitions of the opening chord, which noticeably shapes the physiognomy of the piece, is determined by a derivation of the Fibonacci series. At the beginning there are 142 repetitions, at the second use 87, and then with each new use there are fewer, namely 53, 32, 19, 11, 6, 3 and 1. These chord fields are separated from one another by increasingly longer insertions.

Stockhausen receives the number of repetitions by incorporating an "error" and skipping the 2: Calculated from the back, his repetition series starts with 1, then 2 + 1 =3. But then it doesn't go any further with 3+2=5, rather
and as a result:
19+13=32 Etc.
(The numbers in the compliant Fibonacci series are italic printed.)

Wanted proximity to art and science

Inspired by the new perspectives that exact mathematical thinking opened up to them in the 20th century, many composers believed in a departure into a future in which, at a higher level, could be realized again what had disappeared from consciousness since the Greek pre-Socratics: the Unity of artistic and scientific thinking. Edgard Varèse, for example, saw a new concept of beauty emerging through the interaction of science and art. Anton Webern said in 1932 about the potential that was hidden for him in the twelve order of tones:

"If you come to this correct conception of art, then there can no longer be any difference between science and inspired work."

And exactly fifty years later, in 1982, Iannis Xenakis drew the conclusion from the rapid development of artistic intelligence in the 20th century:

"Nothing prevents us now from anticipating a new relationship between the arts and the sciences, especially between the arts and mathematics. The arts will consciously provide the 'problem' for which mathematics must and should shape the new theories. "

The works of Iannis Xenakis can be seen as exemplary in terms of the application of higher mathematical operations to the musical material. For example the piano piece "Herma" from 1961/62, in which he treated the musical material according to the laws of mathematical quantities. Other mathematically based methods that Xenakis has used over the years are the stochastic method based on the calculus of probability, composing with Markov chains, with tree structures and with so-called seven - irregular scale grids based on the combination of different sets of parameters. This combination of mathematics and music, carried out at the highest level of reflection, is anything but unconditional. It has a history going back two and a half thousand years and goes back to the Greek pre-Socratics. Xenakis refers explicitly to these, especially to Pythagoras and Parmenides.

But how did this close relationship between mathematics and music come about? This can be explained with an instrument that can be considered the archetype of all stringed instruments and at the same time enables the rational understanding of sound by means of measurement: the monochord.

The monochord consists of a string which is stretched between two bridges and which can be divided into two parts in different ways by means of a movable third bridge, so that the two string sections always produce different pitches depending on their length. The string pitch in the ratio 1: 1, i.e. with the movable bridge exactly in the middle, results in the harmony. The string division 1: 2 results in the octave, 2: 3 the fifth, 3: 4 the fourth, 4: 5 the major third, etc.

The sequence of the simplest integer ratios corresponds to what the French acoustician Marin Mersenne discovered experimentally in 1636: the overtone spectrum.

Pythagoras and the monochord

However, the monochord is much older. In the 6th century BC, Pythagoras designed his musical cosmology by assigning the same proportions to the laws of the cosmos and the human spirit: namely, the integer proportions of the tones, as they can be generated on the monochord. The simpler the numerical relationships, the more "harmonious" the proportion was. One to one, the unison, was the perfect harmony; one to two, as "decoupling", the first step towards less consonant numerical relationships.

In Pythagoras' worldview there was still the old mystical thinking, in which intuitive and rational knowledge of the world coincided. Number symbolism played a major role for him, for example in the magic number seven: There were seven celestial spheres, seven colors of the spectrum and seven planets.

In the Pythagorean doctrine of numbers, two aspects emerged: the quantitative aspect of counting and measuring, which should lead to mathematics, and the qualitative aspect of meaning which should lead to the psychological symbolism of numbers. In the two and a half millennia up to our present, one or the other aspect has always come to the fore.

Like Plato, Augustine, and with him the entire Christian Middle Ages, emphasized the quantitative aspect. The world, which can be heard in numbers, was thus subject to an objective, spiritual principle of order. The renaissance articulated the qualitative aspect of the number in its natural philosophy and saw the harmonic proportions hidden in a speculative way in all areas of beings. The idea lived on even after Copernicus refuted the Ptolemaic view of the world. The Jesuit Athanasius Kircher stated around 1650 that an "occult harmony" prevails on all levels of being, from which he concluded:

"It is not a herb that does not consonire and dissonate with another."

Rationalism then saw music as a kind of scientific research object, and for Rameau the natural overtone series could claim general validity. The doctrine of affect, in turn, distanced itself from the Pythagorean, mathematical-harmonic approach and enthroned the subject as the ultimate musical authority. However, the qualitative aspect of Pythagoreanism always lived on subliminally. This can be seen both in Bach's number symbolism and in Mozart's occult use of Masonic number proportions, especially the number three in the "Magic Flute".

In Romanticism the Pythagorean idea of ​​a world harmony reappears, namely in the idea of ​​an intuitively perceptible unity of all that is. Novalis conjured up the possibility of "beautiful, mystical and musical mathematics", and Robert Schumann set the lines by Friedrich Schlegel as the motto for his Fantasy Op. 17:

"Sounds through all the notes
in the colorful earth dream
drawn a low sound
for whom he secretly listens. "

The 20th century then tries to reconstruct the harmonic world view of Pythagoras from the spirit of the 19th century, when Rudolf Steiner describes the ratio 1: 2, the octave, as the "musical expression of our ego riddle", in which lower and would meet a higher ego, or if Jacques Handschin suspects that "where we cannot recalculate the circumstances, there is a divine order".

Twelve-tone arithmetic ...

With the development of the twelve-tone technique by Josef Matthias Hauer and Arnold Schönberg, the number has a completely new, concrete function in music beyond all speculation: the function of the generative principle for the definition of micro- and macro-structure. After all: Hauer saw his explorations in the area of ​​twelve tones still embedded in some kind of "music of the spheres", which should give his efforts the character of objective commitment. In spite of this, his thinking was geared towards the concrete shape of the sound and was strictly constructive and melodic. He said:

"The main focus of my work is that I repeatedly play or sing all twelve tones of the closed circle of fifths and fourths of our tempered semitone scale. I do not think and hear in absolute tones, but in tropes (twists), in 'constellations' of the twelve tones, in movements of the intervals to one another. "

With the piano piece "Nomos", Hauer wrote the first twelve-tone work in history in 1919. A repeatedly repeated series of twelve pitches is overlaid with a five-tone rhythmic model, and when the last note of the series coincides with the last note of the rhythm model, the cycle is closed and the piece is finished.

It was Josef Matthias Hauer who wrote the first twelve-tone composition, but the development of this compositional technique is mostly associated with Schönberg to this day. This is probably due to the fact that Schoenberg was able to interpret the Twelve Ordinance culturally in a more comprehensive sense than Hauer, because for him it had far more than just a methodological meaning.

... and Kabbalah

According to an investigation by Juan Allende-Blin, cross-references to the Kabbalah cannot be directly demonstrated in Schönberg's twelve-tone concept, but they can be strongly suspected. The Kabbalah is that branch of Jewish mysticism that deals with the interpretation of the history of its origins. Schoenberg's derivation of all the manifestations of the series from a single basic figure is, according to Allende-Blin, an equivalent to the monistic thinking of the Kabbalah, which in turn is related to the Jewish concept of God. Then the cancer form corresponds to a figure of thought that is frequent in Jewish mysticism, as does the emphasis on the law. Added to this is the outstanding importance of the number in Kabbalah. This aspect of the number naturally plays a fundamental role in the series composition, albeit in a purely technical sense and detached from its speculative-symbolic content.

In the middle movement of the "Serenade" op. 24 with the Petrarch sonnet "O could I ever recover the revenge on you", one of Schönberg's first twelve-tone compositions, there is a similar procedure to that of Hauer's earlier in the composition "Nomos" Applied: The number of tones in a row is twelve, but the number of syllables in a line of poetry is only eleven; The series cycle and verse cycle are therefore shifted from one another.

Schönberg himself pointed out that the number eleven plays an important role in his serenade and that he consciously used the numerical relationships for musical construction. The theme of the set of variations has a length of 11 bars and consists of 2 times 14 tones. The whole movement is 77 bars long, i.e. 7 (= 14: 2) x 11 bars.

The number eleven, located between the round numbers 10 and 12, which are also very important for Kabbalah, is used in symbolism as a symbol for the incomplete. But it probably also has to do with the zodiac, because one of the twelve signs of the zodiac is always behind the sun and is therefore invisible. Hence the relation to the dream of Joseph in the Old Testament: He saw in the dream that the sun, moon and eleven stars were bowing in front of him. Was Schoenberg, who, as he said, quite consciously used the number eleven in his serenade, was also aware of this symbolism?

In the complicated mysticism of numbers in Kabbalah there are cross-references to Pythagoreanism; In the 1st century AD, Philo of Alexandria merged Old Testament and Pythagorean ideas and thus created the basis for a number-oriented interpretation of the Bible. The Kabbalah can also base its speculations on a very special fact: the 22 letters of the Hebrew alphabet are also numbers. This means that there are no limits to the semantization of the number. From this, the Kabbalah has developed a tremendously ramified network of symbolic relationships between numbers, letters and terms.

Closely related to the number 22 is the number 10 for Kabbalah, which was also of central importance for the Pythagoreans. It refers to the ten Sefiroth, the ten powers of the divine being, which are communicated in creation. The kabbalistic "tree of life" shows these ten potencies as tables that are connected to one another by 22 bridges, the 22 "true paths". This number is then found again in the 22 Major Arcana of the Tarot.

Number symbolism in Emmanuel Nunes ...

One of today's composers whose works reflect such a number symbolism is Emmanuel Nunes. As a Portuguese, Nunes is part of the Sephardic-Iberian tradition, in which ancient Greek, Jewish and Arabic traditions of thought were mixed. Nunes' works show a highly developed compositional calculus, an expression of both a Pythagorean sense of numbers and Cabalistic speculation. His composition "Wandlungen", for example, five Passacaglias for 25 instruments, is based on the number five, the large orchestral work "Tif 'Ereth" on the number six; its line-up comprises six solo instruments and six orchestral groups.

The number four, the ideal number of the Pythagoreans, forms the basis of the composition "Hesed IV". It is written for solo string quartet and orchestra. "Hesed" is the name for the fourth of the ten Sefiroth, the ten divine potencies. It symbolizes the attitude of affection, of love. With this combination of number and word, which comes from Kabbalah and which in the case of "Hesed IV" manifests itself in the title word, Emmanuel Nunes charges his music with complex, albeit covert semantics. Here the number is the bearer of a supra-personal symbolic content.

... and with Alban Berg

On the other hand, numbers have also been used again and again for the encrypted representation of private mythologies of all kinds. Probably the most famous example of a composition in which individual biographical secrets have been encoded using numerical proportions - including note names - is Alban Berg's "Lyrical Suite". Two numbers play a dominant role in this: the twenty-three and the ten. Together with the intervals A-B and B-F as well as various musical quotations, they form a dense network of structural references that extends over all six movements. The third movement, labeled Allegro misterioso, is a three-part scherzo based on the model A-B-A '. Its three shaped parts are 69, 23 and 46 bars long, i.e. 3 x 23, 1 x 23 and 2 x 23 bars. The metronome markings are multiples of 10, namely 150 and 200.

The role that the number 23 played for Alban Berg is well known: he considered it his destiny number. On July 23, 1908, at the age of 23, he suffered his first asthma attack. Later he was a co-founder of the Viennese music magazine "23", he dated the completion of many of his scores on the 23rd of the month and used this number in several works as a structural variable.

For the number 10, on the other hand, the other important symbolic number in the "Lyrical Suite", there was no reference value for a long time. But in 1977 a sensational find was published: Notes from Berg's own hand, in which the composer himself had deciphered the secrets of the score. Since then it has been known: The ten refers to Hanna Fuchs-Robettin, a Prague industrialist's wife, with whom Berg had a secret love affair in 1925. The structurally important interval H-F was now also clear: it was the woman's initials.

Peter Maxwell Davies and the Magic Square

Magic number squares, as they have been handed down mainly from the Chinese and Arabic cultures, have always been used by artists for construction purposes. One of its characteristics is that the rows of numbers, read vertically, horizontally or diagonally, always add up to the same sum. The square of 16 fields, which Albrecht Dürer depicted on his copperplate "Melencolia", is famous and the sum of which always results in the number 34 assigned to Jupiter.

The English composer Peter Maxwell Davies, who likes to hide the high density of construction of his music behind seemingly simple large forms, has been working with magic squares since the early 1970s. In his six symphonies to date he has combined them with various mathematical methods: with the arithmetic of waveforms, with the Fibonacci series, with the mathematical proportions of the church architecture of the Renaissance architect Brunelleschi, with the calculations of the flight movements of a bird.

For Davies, number squares are not a mere number pattern, but a generative principle with which he can determine the shape and tone of entire works through a variety of transformations. The first work in which he systematically worked with a number square is the chamber music work "Ave Maris Stella" from 1974. Its basic thematic material is the Gregorian Marian hymn of the same name.

But its processing is based on a nine-part number square. With its derivations it determines the strongly contrapuntal piece in all its dimensions. According to the composer, this magical square is traditionally associated with the moon. And here is a symbolic context. Because in earlier traditions Mary was also identified with the ancient Babylonian deity Ishtar, who defeated the moon god and was therefore depicted standing on a crescent moon.

The number as a serial ordering principle

Freely handled and symbol-laden number manipulations, as practiced by Peter Maxwell Davies with his magic squares, differ considerably from the use of numbers in the early days of serialism. Here the number had a strictly quantitative meaning and its task was limited to organizing the sound material systematically.

The composers wrote their own number squares, which they did not understand as "magical", but as prosaic tables of values ​​for documenting the mechanics of the series. School examples for such pieces are the "Study I" by Stockhausen introduced at the beginning or the "Structures" for two pianos by Pierre Boulez. The rows that control the various parameters in "Structures" can be arranged in symmetrical squares of 12 by 12 values ​​for both pianos. They are abstract sequences of numbers that express something of the objective spirit that pervades this early serial work.



The systematic, symmetrical number structures of early serial music were soon replaced by more complex processes. It was no longer the tones themselves, but the relationships between the tones and thus the sound processes that were defined by numbers.

Klaus Huber's criticism of number fetishism

The works of Xenakis demonstrate that a mathematical conception need not stand in the way of vital music-making. A small piece that Klaus Huber wrote for Mstislav Rostropovitch in 1976 and that he premiered together with other pieces on the 70th birthday of Paul Sacher can also be regarded as an example of the connection between precise calculation and relaxed virtuosity: "Transpositio ad infinitum". The piece is based on the six tone letters of the name "Sacher": Es, A, C, H, E, Re (= D). The generative principle of the piece is called: Rapid increase in what is already there. In his work protocol, Huber noted:

"Growth, growth! Desired amount: 1111 tones / notes. - 'As fast as possible'. - Turn six into seventy-two, turn seventy-two (through chain transposition) into three hundred and sixty. Et cetera. - Values: eight different values ​​grouped in seven different numbers . (...) - Throw random processes! - Always stay quantitative (round up or down!) "

The way in which Klaus Huber condenses the current serial processes into a kind of turbo-serialism reveals an ironic distance, as does his text on the piece. The irony aims at the added value of a thoroughly rationalized production, as we know it from business, but which also characterizes in symbolic form a method of composition that subjects the musical material to an inexorable rational strategy.

Number and chance with John Cage

Criticism with numbers of the belief in numbers of a hyperrationalist conception of music came from John Cage as early as the 1950s. Its first appearance in Europe together with David Tudor in 1954 deeply unsettled the European avant-garde, which is committed to seriality, and heralded the beginning of aleatoric music. In this, chance was given a limited right. Cage, however, practiced composing by chance with incomparable radicalism. In his piano composition "Music of Changes" from 1951, he achieved results that were as complicated as those of the Darmstadt serialists with their series operations by means of a random process based on the Chinese I-Ching.

As early as 1952, Henry Cowell, Cage's teacher, had noted in a fundamental essay on Cage's random process that Cage knew not only the I-Ching but also the number table with the 88 number squares, according to which counterdances could be put together by random processes in the 18th century.


Mozart had already played with this number table, and Cage would probably have had fun watching Mozart doing it. In the late 1960s, John Cage used Mozart's template as the basis for his randomized composition "Harpsichord" for 1-7 harpsichords and 1-52 computer-generated tapes, and Mozart would probably have had fun watching Cage too.

© Max Nyffeler 2007
This text is based on a radio broadcast for the SWR on November 29, 1999.

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