Why do major thirds sound sharp in Pythagorean tuning?

In Pythagorean tuning, major thirds sound sharp because the system is constructed by prioritizing perfectly consonant fifths. By deriving all intervals from these natural fifths, the resulting major thirds and major sixths become too wide by approximately 22 cents, or by a ratio of 81:80.1

This specific discrepancy is known as the comma of Didymus, or syntonic comma. Because Pythagorean tuning relies on these pure fifths, it fails to align with the natural thirds found in the harmonic series, leading to intervals that can sound severely dissonant when played simultaneously. While this system is considered satisfactory for melodic lines, the accumulation of this comma makes the major thirds noticeably sharp compared to just intonation, where thirds are derived to be acoustically pure.21