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16 17 Diatonic Ratio 21 158 95 53 85 32 43 54 65 98 1615 11 The PenDulum keeping time A fundamental law of physics in one formulation states that left to itself any closed system will always change towards a state of equilibrium from which no further change is possible. A pendulum is a good example. Pulled off centre to start, it is in a state of extreme disequilibrium. Released, the momentum of its swing carries it through nearly to the same point on the other side. As it swings it loses energy in the form of heat from friction at the fulcrum and brushing against the air. Eventually the pendulum runs down, finally coming to rest in a state of equilibrium at the centre of its swing. Going back 500 years, Galileo, watching a swinging lamp in the cathedral of Pisa, realised the frequency of a pendulums beat depends on its length, and not its weight the longer the pendulum the lower the frequency. So the frequency can be varied at will by fixing a weight at different heights. Most importantly, as a pendulum runs down, its frequency stays the same. Here, therefore, is a perfect way to represent a musical tone, slowed down by a factor of about a thousand to the level of human visual perception. For a simple harmonograph two pendulums are used to represent a harmony, with the weight kept at its lowest point on one, while the weight on the other is moved to wherever it will produce the required ratio. As we shall see, the harmonograph combines these two vibrations into a single drawing, just as two musical tones sounded together produce a single complex sound. When a pendulum is pulled back and then released, the weight tries to fall towards the centre of the Earth, accelerating as it does so. As the pendulum runs down, the rate of acceleration, and so the speed of travel, is reduced., but in equal proportion to the distance of travel. The result is that the period the time taken for two beats or the number of periods in a given unit of time the frequency remains unchanged. In the picture to the left the frequencies of beats x and y are the same. For the pendulum formula, see page 53. The theoretical length of the variable pendulum that will produce each harmony can be calculated, for the frequency of a pendulum varies inversely with the square root of its length. This means that while the frequency doubles within the octave, the length of the pendulum is reduced by a factor of four. Figures are given for a pendulum 80cm long, a convenient length for a harmonograph. These theoretical markers provide useful sighting shots for most of the harmonies. Note that the pendulum length is measured from the fulcrum to the centre of the weight. Length cm 20 22.8 24.7 28.8 31.2 35.6 45.0 51.2 55.6 63.2 70.3 80 Freq. s1 66.0 62.8 59.4 55.8 53.6 50.3 44.7 41.9 40.2 37.7 35.7 33 Approx. Note C B B7 A G8 G F E E7 D C8 C Interval Name Octave Maj. 7th Min. 7th Maj. 6th Min. 6th 5th 4th Maj. 3rd Min. 3rd 2nd Halftone Unison
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