# Harmonograph

37 36 furTher harmonics seven limit and higher number ratios As the numbers in the ratios increase it becomes harder to distinguish the harmonies one from another at a glance the loops have to be counted, and slight variations produce little of aesthetic value. A typical example, 75, is shown opposite top. Rotary motion produces a series of increasingly complex drawings, influenced by relative frequency, amplitude and direction. In contrary motion the total number of loops equals the sum of the two numbers of the ratio. With concurrent motion the nodes turn inwards, and their number is equal to the difference between the two numbers of the ratio. The contrary drawings below show a fourth 43, another fourth, a major sixth 53 and a major third 54. The lower pictures opposite, drawn over a hundred years ago, show unequal amplitude drawings of the perfect eleventh 83 an octave and a fourth and the ratio 73 which is found in sevenlimit jazz tuning not covered in this book. Two octaves and a major third 41 x 54 equal 51, the fourth overtone, which differs from four fifths 324 as our friend 8081, the syntonic comma see page 10. In mean tone tuning, popular during the Renaissance, this misfit was ironed out and the fifths were flattened very slightly, to 514 or 1.4953, falling out of tune to please the difficult thirds and sixths.