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Harmonograph

13 12 arranging The harmonies the power of silence The simple ratios of the primary overtones and undertones can be plotted on an ancient grid known as a lambdoma opposite top, after the greek letter . Some intervals are the same 84 63 42 21, and if lines are drawn through these it quickly becomes apparent that the identities converge on the silent and mysterious ratio 00, which is outside the diagram . A further contemplative device used by the Pythagoreans was the Tetraktys, a triangle of ten elements arranged in four rows 123410. The basic form is given opposite lower left, the first three rows producing the simple intervals. In another lambdoma opposite lower right, numbers are doubled down the left side and tripled down the right, creating tones horizontally separated from their neighbours by perfect fifths. After the trinity 1, 2 and 3 notice the numbers produced, 4, 6, 8, 9, 12, and then look again at the picture on page 5. Below we see interval positions on a monochord, a simple instrument with a moveable bridge under a string stretched over a sounding board. Pythagorean and medieval tunings, called 3limit, recognised no true intervals except for ratios involving 1, 2 and 3. The lambdoma below right expresses this numerically as any element relates to any neighbour by ratios only involving 1, 2 and 3, so we can move around by octaves and fifths. Squares 422, 932 and cubic volumes 823, 2733 also appear. Add further rows and the numbers for the Pythagorean scale soon appear, 1 98 6481 43 32 2716 169 21. This has four fifths and five fourths but no perfect thirds or sixths. These came later with the diatonic scale and its perfect thirds 654 as polyphony and chords slowly took over from plainchant and drone.
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