generalized intervals

Generalized intervals extend the concept of traditional musical intervals (the distance between two notes) to more complex contexts, such as different tuning systems, microtonal music, or abstract mathematical settings. Unlike simple intervals like a third or fifth, generalized intervals can describe any measurable distance between pitches, including fractions of a whole step or unconventional scales. They provide a flexible framework for analyzing and comparing musical relationships beyond classical Western tuning, capturing the nuances of pitch differences in various musical styles and theoretical models. Essentially, they allow for a precise and universal way to discuss how various pitches relate to each other.