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Harmonograph

41 40 Tuning Troubles the Pythagorean comma Leaving the harmonograph drawings and returning to the principles of music, you may have noted that musical intervals do not always agree with one another. A famous example of this is the relationship between the octave and the perfect fifth 32. In the central picture opposite, a note is sounded in the middle at 0, and moved up by perfect fifths to give the sequence C, G, D, A, E, etc. numbered opposite, each turn of the spiral representing a perfect octave. After twelve fifths we have gone up seven octaves, but the picture shows that we have overshot the final octave slightly, and gone sharp. This is because 3212 129.75, whereas 27128. The difference is known as the Pythagorean comma, proportionally 1.013643, approximately 7473. If you kept on spiralling you would eventually discover, as the Chinese did long ago, that 53 perfect fifths or L almost exactly equal 31 octaves. The first five fifths produce the pattern of the black notes on a piano, the Eastern pentatonic scale see page 50. The smaller pictures opposite show repeated progressions of the major third 54, the minor third 65, the fourth 43 and the whole tone 98 all compared to an invariant octave. Its strange. With all this harmonious interplay of numbers you would have expected the whole system to be a precisely coherent whole. It isnt. There are echoes here from the scientific view of a world formed by broken symmetry, subject to quantum uncertainty and so far defying a precise comprehensive theory of everything. Is this why the near miss is so often more beautiful than perfection Perfect Fourths Major Wholetones Perfect Fifths Major Thirds Minor Thirds Pythagoras Comma
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