apotomein

Apotomein is a term used in music theory and acoustics to describe the difference between two musical intervals. Specifically, it refers to the difference between a ratio of integer powers of two and three, and a ratio of integer powers of two and three that is close to it. In simpler terms, it is a small, often imperceptible, difference in pitch that arises when comparing certain intervals derived from different harmonic series.

The concept of apotomein is closely related to the history of tuning systems, particularly in Pythagorean tuning and just intonation. In Pythagorean tuning, intervals are derived from perfect fifths (a frequency ratio of 3:2). When these fifths are stacked, they do not perfectly align to form octaves (2:1). This discrepancy leads to what are known as Pythagorean comma, which is a small difference in pitch. Apotomein can be considered a component or a related concept within these tuning discrepancies.

In ancient Greek music theory, the term apotome was used by philosophers like Euclid. Euclid described the apotome as the difference between a major tone (9:8) and a minor tone (10:9), which is a very small interval. This historical usage highlights the long-standing interest in analyzing and categorizing even the smallest pitch differences in musical intervals.

While the theoretical concept of apotomein exists, its practical audibility is a subject of debate. Many listeners would not be able to distinguish between intervals that differ by an apotomein. However, for highly trained musicians and in the context of microtonal music or historical tuning practices, such small differences can become more significant and are sometimes deliberately employed. Understanding apotomein helps to appreciate the mathematical underpinnings of musical intervals and the complexities of tuning systems.