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. Not just musical notation, but also the relationship between music and time has a mathematical foundation.

Polyphonic music was recorded in notations showing the length of each note in a uniform and measurable way for the first time in Paris in around 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.


One particular mathematician and philosopher played an important part scientifically as well as in the understanding of music, namely Boëthius (480-526). Boëthius(1) 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(2), the Greek mathematician and philosopher.

Pythagoras in the Blacksmith's

One day Pythagoras(2) 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.

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'. 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".
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