And yet, even though we might be deceived by a bittersweet oboe solo, it in fact conceals far more mathematics than our feelings let us believe. From a – quasi mathematical – distance, numerous cross-connections are revealed that have been known since antiquity. This is related on the one hand to the basic rules of musical harmony, and on the other to the peculiarities of human hearing.

The longing of ears

As a sense, the human ear loves order. It steadfastly searches for recognizable patterns in its environment. It responds particularly strongly to rhythms. As soon as it has located one it radically blanks out all other sensory impressions. Rhythms are extremely infectious and quickly take control of the human spirit. At worst one can only save oneself by switching the music off, turning the faucet off and removing all the clocks from a room.

Another of hearing’s peculiarities is its ability to perceive several rhythms simultaneously. In comparison the eye is rather sluggish and quickly overstrained. The ear, on the other hand, registers several rhythms with the greatest of ease. This is why in Bach’ s Art of the Fugue, we are able to follow the overlapping and diverging of several voices effortlessly, without becoming giddy in the process.

But what does the ear do exactly when it follows rhythms? In principle it is counting – even if, as a rule, subconsciously. This borders on a purely physical condition. The German mathematician and philosopher Gottfried Wilhelm Leibniz pointed out this state of affairs in the 18th century. This intuitive counting begins with sensing the beat. It is the pulse of every piece of music and for the most part we can clap along to it intuitively.

Mathematical, or to be more precise, arithmetical principles such as this run through music on many levels. Rhythms consist of precisely counted, repetitive beats. This also applies to complicated jazz rhythms. This way, timing and entrances are defined exactly. In classical music, depending on the genre, there is a fixed sequence for repeating and varying themes, examples being main movements in the rondo or the far more complex sonata form.

Though the concrete form these types of music take is a question of culture and zeitgeist, this in no way alters the fact that numbers and sequences play a major role here. This becomes particularly evident if a piece of music departs from the well-trodden path and adopts an unexpected course. Then, our listening patterns come to light. Things turn out well if we find the surprise pleasant, and badly if it annoys us.

• Part 1: The world as sounds and numbers
• Part 2: In the beginning there was the interval
• Part 3: Harmonia mundi