Compare Equal Temperament, Just Intonation, Pythagorean tuning, and Meantone temperament. See how they differ in cents for each note and understand when each is used.
Each semitone is exactly 100 cents — a mathematically equal division. Every key sounds equally "in tune" (or equally "out of tune"). Invented for keyboard instruments, became universal in the 18th century. Slightly out of tune intervals compared to pure harmonics but allows transposition to any key.
Intervals based on simple whole-number frequency ratios (perfect 5th = 3:2 = 702 cents). Sounds incredibly pure in one key — intervals based on the harmonic series. Cannot modulate freely — other keys sound out of tune. Used in choir singing, early music, barbershop harmony.
Built by stacking pure perfect 5ths (3:2 ratio). Perfect 5ths are pure; major thirds are very sharp (+22 cents). Ancient system — used in medieval music where parallel 5ths were common. Harsh major thirds make modern chord-based music sound poor.
Compromise: makes major thirds pure (5:4 = 386 cents) by slightly narrowing the 5ths. Renaissance and Baroque standard for keyboard instruments. Works beautifully in certain keys; others (remote keys) have a notoriously harsh "wolf interval."
A family of systems (Werkmeister, Kirnberger, etc.) that provide usable tuning in all keys while preserving some key colour. Bach's Well-Tempered Clavier was likely written for a well-tempered system — each key has a distinct character. Not equal temperament.
Nature's own tuning: integer multiples of a fundamental frequency. 1f, 2f, 3f, 4f... Overtones of any pitched sound. Just intonation derives from this. Electronic synthesis can produce pure harmonic series; acoustic instruments slightly deviate from it.