The Mathematics of Harmony

Walking past a blacksmiths one day, Pythagoras heard a familiar harmony in the ringing tones of the hammers at work inside. He discovered that the weights of the hammers were responsible for their relative notes. A hammer weighing half as much as another sounded a note twice as high, an octave (2:1). A pair weighing 3:2 sounded beautiful - a fifth apart. Simple ratios made pleasing sounds. In his ensuing experiments he found that musical instruments worked in the same way whether struck, plucked or blown.

Pythagoras, some 2,500 years ago discovered that the joyful experience of musical harmony comes when the ratio of frequencies consist of simple numbers. The same numbers kept coming up 4, 6, 8, 9, 12, 16. Pair any of these numbers up and they are all pleasing to the ear, and...pleasing to the eye.

Pythagoras’ hammers hid a set of relationships that made octaves (2:1), fifths (3:2) and fourths (4:3). A Fifth and Fourth combine to make an Octave: (3:2 x 4:3 = (12:6) 2:1). The difference between them is called a Wholetone: (3:2 ÷ 4:3 = 9:8).


A natural pattern emerges, producing seven nodes (or notes) from the starting tone (or tonic), separated by two halftones and five wholetones, like the sun, moon and five planets of the ancient world.

A fundamental law of physics states that left to itself any closed system will always change towards a state of equilibrium from which no further change is possible. Pulled off  centre, a pendulum is in a state of extreme disequilibrium. Released, the momentum of the of it’s swing carries it through to almost the same point on the other side. As it swings it loses energy to friction in the form of heat and air drag. Eventually the pendulum runs down, finally coming to rest in a state of equilibrium at the centre of it’s swing. Galileo realised when watching a swinging lamp in the cathedral of Pisa that a pendulum’s frequency (or beat) was dependant on the length, not the weight: the longer the pendulum,
the lower the frequency.

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Fig. XI

Pythagorical Experiments taken from Gafurio’s Theorica Musice, 1492
(note the same numbers appearing in each experiment)

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