Why twelve notes? An economic model of scale size and evidence on chromatic pitch-class use in Western classical music, 1485–1963

Abstract Why do musical systems organise pitch into finite sets rather than draw without limit on the pitch continuum? We model scale-system size as an economic choice that balances harmonic possibilities against instrument-construction and learning costs. An acoustic calibration indicates that total harmonic value grows faster than linearly but more slowly than the number of possible interval pairs. When learning costs eventually rise more steeply than harmonic value, the model yields a finite optimum even as physical construction costs approach zero. We evaluate the model’s comparative patterns using 623 Western classical MIDI compositions by 48 composers spanning 1485–1963. Pitch-class entropy rises over historical time and within composers’ careers; its association with national income is positive but fragile. Use of the twelve encoded semitone classes broadens before 1900 and then saturates, while dissonance continues to rise and usage evenness can keep changing within the fixed palette. Because MIDI represents pitch within twelve semitone classes, these results measure chromatic pitch use rather than historical scale-system size itself. Evidence from instrument history and electronic music complements the corpus analysis. Together, the model and evidence show how technological and cognitive costs can shape the evolution of musical form.