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What is a Harmonograph?

A Harmonograph is a mechanical apparatus that employs pendulums to create a geometric image. The drawings created typically are *Lissajous curves, or related drawings of greater complexity. The devices, which began to appear in the mid-19th century and peaked in popularity in the 1890s, cannot be conclusively attributed to a single person, though Hugh Blackburn, a professor of mathematics at the University of Glasgow, is commonly believed to be the official inventor.


A simple, so-called 'lateral' harmonograph uses two pendulums to control the movement of a pen relative to a drawing surface. One pendulum moves the pen back and forth along one axis and the other pendulum moves the drawing surface back and forth along a perpendicular axis.

By varying the frequency of the pendulums relative to one another (and phase) different patterns are created. Even a simple harmonograph as described can create ellipses, spirals, figure eights and other Lissajous figures. Inputing sound into the device can also vary the frequency and produce interesting results. More complex harmonographs, such as ‘Harmony’, incorporate three or more pendulums , or involve rotary motion in which one or more pendulums is mounted on gimbals to allow movement in any direction.

*Lissajous curve
In mathematics, a Lissajous curve is a graph which describes a complex harmonic motion such as a heartbeat depicted on an Oscilloscope.

Fig. I
Lateral, two-pendulum, harmonograph
note the perpendicular direction of the pendulums


Fig. II
Rotary, three-pendulum, harmonograph
third pendulum has a rotary ‘gimbal’ allowing movement in any direction


Creating Harmony

In the mid to late nineteenth century there was definitely a vogue for these instruments. Victorian gentleman and ladies would attend ‘soirées’ or ‘conversaziones’, crowding round the strange automatic drawing devices and exclaiming in disbelief as the mysterious drawings appeared before their eyes.

Back in 2005 I was researching Victorian stage shows and strange contraptions when I came across the harmonograph. With my penchant for science, making things and the bizarre, it immediately captured my imagination. With it’s relationship to musical vibrations and it’s ability to translate vibration into pictures, I was hooked on the idea.

Finally last year, the gypsy in me envisaged a travelling side show to take to festivals and gatherings. A unique, curious handcart of automatic drawing devices that could visually explore the relationship between mathematics and music. The idea of ‘Harmony’ was conceived.

I managed to track down an original Victorian barrow that was sat on the edge of a field in the middle of Lincolnshire after being retired from selling the farm produce. Intrigued as to my plans for the cart, a kind woman agreed to part with it. Over a few dry weekends, the idea and understanding of how to build such an arcane device slowly emerged and we set to work.

creating-harmony-fig-3.jpgFig. III
The Veg Barrow
with it’s original cart wheels in tact

creating-harmony-fig-4.jpgFig. IV
Stripping the cart down
shows the beginnings of 2 pendulums

creating-harmony-fig-5.jpgFig. V
The rotary gimbal
before bearings were invented, providing a universal pivot

creating-harmony-fig-6.jpgFig. VI
A lateral pendulum
pendulums must be a specific length to the centre of the counter weight

creating-harmony-fig-7.jpgFig. VII
The naming of Harmony
sign writing and painting coach marks

creating-harmony-fig-8.jpgFig. VIII
The pitch roof
Graeme and Jason trying the pitch roof

creating-harmony-fig-9.jpgFig. IX
Lower ‘C’ concurrent
Harmony in action at Shambala 2010

creating-harmony-fig-10.jpgFig. X
Harmony’s first gathering
outside the Parade Tent

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.

Fig. XI

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

Booking Form

If you would like to hire Harmony, please download the following Harmony Booking Form.

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