This is a preview. Log in through your library . Abstract Perceived octaves are stretched, requiring frequency-dependent ratios that generally exceed 2:1. Small intervals seem to be compressed ...

Pythagoras (569-490 BC) established that the octave (the ratio 1:2 of a string's length) was the fundamental musical interval and saw it as a sign that nature itself, including the planetary orbits ...

Have you ever looked at a piano keyboard and wondered why the notes of an octave were divided up into seven white keys and five black ones? After all, the sounds that lie between one C and another ...