Entropy is the key to a tuneful piano

That’s Maths: In piano tuning, entropy measures disharmony or discord: the aim is to make it as small as possible

An ingenious method of tuning pianos based on the concept of entropy was recently devised by Haye Hinrichsen of Würzburg University. Entropy was first recognised in the mid-19th century in thermodynamics and later in statistical mechanics. It measures disorder.

Around 1948, Claude Shannon developed a mathematical theory of communications and used entropy to indicate information content. In the context of piano tuning, entropy measures disharmony or discord: the aim is to make it as small as possible.

Sound is a series of variations of air pressure. All sounds travel at the same speed, but they have many different patterns of pressure. They may be analysed into wave- like components, each being a pure sine wave with fixed amplitude and frequency. The amplitude gives the loudness of the wave, whereas the frequency – the number of pressure oscillations per second – gives the pitch or tone.

Equal temperament

Western music is based predominantly on a tuning system called equal temperament, where the ratio of frequencies of neighbouring notes is constant. With 12 semitones in an octave, this ratio is the 12th root of two. Thus, by advancing a full octave, the frequency is doubled. The system allows music in all keys, but at a price: all intervals other than octaves are slightly out of tune. With the exception of purists, we have learned to tolerate the slight imperfections of equal temperament.