Harmonics

The term "harmonics" can refer to two distinct concepts: in acoustics, it describes tones that are exact multiples of a fundamental frequency, creating a series of overtones. In mathematics, a harmonic sequence is a series of numbers whose reciprocals form an arithmetic sequence.12

The ancient Greek philosopher Ptolemy explored musical theory in his work Harmonics. He sought a balance between empirical observation and the mathematical, often mystical, approaches to music, such as those of the Pythagoreans. The Pythagoreans, in the 6th century BCE, were among the first to investigate the mathematical relationships within musical intervals, discovering that simple numerical ratios (like 1:2 for an octave, 2:3 for a fifth, and 3:4 for a fourth) correspond to consonant sounds. Ptolemy's work in this area contributed to the understanding of musical intervals and tuning systems.34526