Music is also mathematics

By Noralv Pedersen
Photo: Rune Petter Ness

Pythagoras knew it, but Bach demonstrated it: without mathematics there is no music.

Many researchers have found numerical symbolism in the works of Johann Sebastian Bach. The picture shows the Trondheim Nidaros Cathedral Choir, the Festival Orchestra and conductor Per Fridtjov Bonsaksen.

Johann Sebastian Bach's St. Matthew Passion. Saint Matthew's story about the sufferings and death of Jesus. A majestic choral fantasia opens 'the greatest work of music in the whole Christian world'. The wide sweep of the music in the opening chorus depicts a rebellious people following after Jesus, as He drags Himself along the streets of Jerusalem. Rich harmonies and slowly flowing rhythms charge the air with ominous suspense. No stage sets or props support the words. Just the music. Just the music? This musical work has also been a rich hunting ground for music experts searching for numerological symbolism and mathematics in music.

A musical architect
Scholars have spent a lot of time studying the way in which Johann Sebastian Bach (1685-1750) made use of a number of formal mathematical patterns when he composed his majestic organ fugues. - Bach used for instance the "golden section" as well as the Fibonacci succession (1, 1, 2, 3, 5, 8, 13 etc., in which each number in the succession is the sum of the two previous ones). In many ways he worked like an architect, joining the two different parts of a musical piece into one harmonious whole before the actual process of composition started.

Bach believed that a well-proportioned fugue would provide a guaranteed basis for a successful composition. This, of course, created favourable conditions for numerologists in

their search for the ideal form, says Andreas Haug. Haug, who is professor at the Centre for Medieval Studies at NTNU, referring to the polyphonic choral work which the famous Renaissance composer Guillaume Dufay (c. 1400-74) composed for the consecration of the cathedral in Florence in 1436. - Some people have claimed that the form this motet takes mirrors the architectonic number symbolism of the cathedral. They set out to prove that Dufay was portraying the proportions of the church in his music. Some interpretations of the work go as far as analysing all notes and textual syllables in the motet so as to make the composition correspond to individual parts of the building. To put so much emphasis on numerological aspects in musical research is somewhat controversial, says Haug.

Numerology
Dufay's motets are by no means the only works of music to be thoroughly analysed by scholars in search of mathematical structures.

- The theological tradition within Bach interpretation is particularly strong. This tradition has emphasized the use of symbolic numbering in his works, explains Haug. Researchers believe that Bach would probably not have been averse to making use of numerical structures in his compositions. It is quite possible that the Baroque composer made use of such effects in different ways in the St. Matthew Passion. Each movement in the choral Wozu Dienet has 11 beats, and the word Amen is used 11 times - could this be a symbol of the 11 disciples of Jesus? The word Herr is found 11 times in the choral Herr, bin ich's - perhaps another symbol for the 11 apprentices? Or alternatively perhaps a tool used to demonstrate that the twelfth disciple, Judas, remained silent, thereby constituting a dramatic and realistic move rather than a symbolic one?

Haug's attitude towards numerological interpretations is punctuated by many question marks. He claims that it is the listener who attributes meaning to numerical structures in music.

- The mathematical proportions in a musical work are hidden from the listener from the start. He or she is thus forced to make use of interpretative techniques in order to search for them, which is problematic from a methodological point of view. Some music historians have very little time for numerology, says the Professor Haug.

He sees a composer like Bach more as a rationalist than a numerologist. - The logical structure of Bach's compositions substantiates this view. The way in which certain people emphasize the obvious numerical symbolism in works such as the St. Matthew Passion can hardly be said to be particularly sophisticated, says Haug.

An interest in numerology has produced some absurd results. Bach was apparently fascinated by one particular form of symbolism in which letters and names could be expressed in numbers (A = 1, B = 2, C = 3 etc.) The sum total of the letters in the word Bach is 14, and J. S. Bach produces a figure of 41. When Bach was asked to join Die Mizlersche Societät in Leipzig, some sources claim that he waited until he could become member number 14. Furthermore, when his official portrait was being painted for the organization, he was committed to canvas with 14 clearly visible buttons on his coat. - This is an example of how numerology can detract attention from what is important in Bach's music, says Haug.

Music as audible numbers
- In addition to mathematics being seen as numerical symbolism, music is closely linked to absolute physical entities, such as frequency and the relation between intervals (an interval is the space between two notes). Already in Antiquity this was seen as the natural or cosmic premise on which music relied, says Haug. He illustrates the fact that not just musical notation, but also the relationship between music and time has something to do with mathematics and with one of the most significant transformations in music history.

- We find polyphonic music recorded in notations showing the length of each note in a uniform and measurable way for the first time in Paris at around the year 1200. In order to indicate how the separate voices were to be coordinated in the work, composers had to make use of notations which were also able to show the length of each note. This meant that they were further able to measure any temporal aspect by dividing the length of each note into smaller units. This type of polyphonic music was not called "polyphony" as it is today, but "musica mensurabilis", or "measurable music". Musica mensurabilis opened new possibilities within musical forms which, as we have seen, both Bach and Dufay developed further, says Haug. He also mentioned one particular mathematician and philosopher who played an important part scientifically as well as in the understanding of music, namely Boëthius (480-526). Boëthius divided science into seven disciplines: grammar, dialectics, rhetoric, arithmetic, geometry, astronomy and music.

He viewed the first three disciplines as a single unit or whole, a "trivium", and the remaining four as another whole, which he named the "quadrivium". He based this categorization on the fact that the trivium had to do with language, whereas the quadrivium - which included music - had to do with numbers. The basis for Boëthius's world of ideas was the notion of music as audible numbers. He illustrated this with a legend about Pythagoras, the Greek mathematician and philosopher.

Pythagoras in the smithy
One day Pythagoras passed a smithy as he was walking in the forest. He stopped to listen to the beautiful sound of hammers striking the anvil. After he had worked out that it was the head of the hammer which was responsible for such exceptional sounds, he examined all the hammers and found that the weight of each individual hammer was in a particular proportion to that of another: 12, 9, 8 and 6 pounds. You could hear an octave each time the largest and the smallest hammer struck the anvil (12 : 6), a pure fifth emerged in the relationship between the 12- and 8-pound hammers; he found the straight fourth in the relationship between 8 and 6 or 12 and 9, and the whole note rang out when the two middle hammers were being used (9 : 8).' This means that the reason why these sounds were in harmony was attributable to the mathematical relationship that existed between them.

Based on the story about Pythagoras, Boëthius concluded that music is a matter of numbers. The medieval conceptualization of music also led to the view that music is a matter of numerical relations translated into sounds. This conceptualization depicts a woman playing the monocord, a one-stringed instrument which was used exclusively for working out the relationship between notes by pressing the finger down in various positions on the string.

- Boëthius's and Pythagoras's approaches to music can be seen as ways of discovering already existing natural phenomena, phenomena which are all created by God. When medieval man envisaged music in terms of a divine assumption concerning the relationship between tones, and was thus in the last instance to claim that God made use of mathematics to create the world, he quoted Solomon: 'but thou (God) hast ordered all things in measure and number and weight' (Wisdom of Solomon, 11, 20). In the beginning of the 14th century, a medieval theoretician of music wrote that 'music is about tones which are related to numbers and vice versa (about numbers which are related to tones)'. This notion is also found in the works of the philosopher and mathematician Gottfried Wilhelm Leibniz (1646-1716), who was the creator of analytic geometry. He wrote: "Music is the hidden arithmetical reckoning of the unconscious spirit".

Sources: Finn Benestad: Musikk og Tanke (1993) and article in Dagbladet, 2000; 'Bach's St. Matthew Passion' by Amalie Christie.

* Contact at NTNU: Andreas Haug
Tel: + 47 73 59 74 78
E-mail: andreas haug@sfm.ntnu.no