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In Raphael's painting, Pythagoras is shown explaining musical ratios to a pupil. The Pythagoreans were fascinated by the relationship between music and maths. The central belief was that pleasing musical intervals - "harmony" - can be expressed as ratios between whole numbers. Vibrating strings whose length is in the proportion 1:2 produce an octave. The proportion 2:3 produces a perfect fifth, while a perfect fourth arises from the ratio 3:4. You can explore these relationships using lengths of elastic or similar. You can also see whether you agree with Plato that tunes in the Lydian mode (F-F) or the Mixolydian mode (G-G) are undesirable, while those in the Dorian (D-D) or Phrygian (E-E) are worthy of a philosopher's approval. For examples:



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