Ptolemaic tuning

Ptolemaic tuning, often referred to as just intonation, is a musical tuning system where intervals are precisely calculated using pure natural thirds and fifths. This system, originating from ancient Greek musical theory, aimed for perfect consonance based on simple frequency ratios.123

However, just intonation presents significant challenges for polyphonic music due to the inherent mathematical complexities of dividing the octave into consonant intervals. While it sounds pure for melodies, it creates dissonances when multiple notes are played simultaneously. Consequently, this system was largely replaced by tempered tuning systems, such as meantone temperament and later equal temperament, which offered greater flexibility for harmony. The Ptolemaic system is distinct from the Ptolemaic system in astronomy.214