15. Summary

Musicians even before recorded history have made music and musical instruments that sounded good to them. The oldest flute found is about 35,000 years old and is made of the bone of a giant vulture. Ancient Greek mathematicians realized that the notes on a string that sounded pleasant together have specific length ratios. For a long time after that mathematicians thought they could codify the scales used by musicians into set ratios of frequencies. With the advent of modern music which has uses more than one octave, much more harmony and instruments that have fixed scales that are not easily changed, compromises in the ‘perfect’ ratios had to be made. The slight differences between Pythagorean and equal tempered scales can be heard by a trained ear but are not enough to change the overall feel of a musical piece.

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