Different Systems of Tuning

Pythagorean, equal temperament, and just intonation are 3 different systems of tuning. Pythagorean and just both are based on the naturally occurring harmonic series. Equal temperament is a compromise and can be derived mathematically. All produce scales with a different third. Pythagorean tuning uses only perfect fifths, just intonation uses perfect fifths and major thirds while equal temperament uses cents and 12 equal notes to an octave.