Source edition
Euclid. Musici scriptores Graeci. Jan, Karl von, editor. Leipzig: Teubner, 1895.
Source data
Open Greek and Latin · CC BY-SA 4.0
Cloned and adapted by Humanitext, with ongoing edits.
Summary
This ancient Greek treatise on music theory presents a mathematical approach to defining intervals as numerical ratios and constructing a musical scale on a monochord (canon). It begins by establishing that pitch is determined by the frequency of movement and defines consonances as multiple or epimoric (superparticular) ratios. Through a series of geometric-style propositions, the work mathematically demonstrates the relations of major intervals, showing that the octave is a duple ratio (2:1), the fifth is 3:2, the fourth is 4:3, and the whole tone is 9:8, while proving that six whole tones are slightly larger than an octave. Finally, it provides practical instructions for dividing the monochord by geometric proportion to mark the fixed and movable notes of the "immutable system" (systema ametabolon). The work achieves a rigorous mathematical foundation for the sensory harmony of music.