A4 in Pythagorean Tuning

A4 is 440.000 Hz in all tuning systems that use A=440 as their reference. In Pythagorean Tuning, the surrounding chromatic notes are tuned according to a chain of pure perfect fifths (3:2 ratio) stacked above and below a reference pitch, producing a 12-note chromatic scale.

This system was used for medieval polyphony and string ensemble open-string tuning.

Chromatic Scale at A4=440 Hz in Pythagorean Tuning

The table below shows all 12 chromatic notes at octave 4. A4 is the tuning reference — its frequency is 440.000 Hz in all temperaments at this concert pitch. The other notes show how Pythagorean Tuning tunes each interval relative to A4.

Positive cents = sharper than equal temperament. Negative = flatter. 100 cents = 1 semitone.

Pythagorean Tuning: Mathematical Formula

Pythagorean tuning builds each pitch by stacking pure 3:2 fifths — a ratio verified by the Greek mathematician Pythagoras as producing the most consonant interval after the octave. Each fifth measures exactly 701.955 cents, slightly wider than the 700-cent equal-tempered fifth. To close the 12-note octave, the remaining interval — the "Pythagorean comma" (23.46 cents) — is left unresolved, appearing as a narrow diminished sixth between the end-points of the chain. The result is pure fourths and fifths throughout, but major thirds that beat audibly at 81/64 (407.82 cents) rather than the pure 5:4 (386.31 cents).

Formula type: Cent offsets from equal temperament

How Pythagorean Tuning Sounds

Pythagorean tuning produces a sound often described as bright and transparent. Perfect fifths and fourths ring with exceptional purity, giving open-string string music a resonant, crystalline quality. Major thirds are noticeably wide and dissonant by modern standards — they have a characteristic tension that was considered acceptable in medieval polyphony, where thirds were classified as imperfect consonances. The system rewards music that emphasizes fourths, fifths, and octaves while avoiding prolonged major or minor thirds.

Historical Context

Pythagorean tuning traces its roots to ancient Greece, where Pythagoras and his followers codified the mathematical relationships between string lengths and musical intervals. The system dominated Western music theory through the medieval period, shaping the compositional style of Gregorian chant and early polyphony. Composers such as Leonin and Perotin in the Notre-Dame school worked within a Pythagorean framework where the perfect fifth was the supreme consonance. As Renaissance harmony increasingly favored pure major thirds, Pythagorean tuning gradually gave way to meantone systems, but it remained a reference for music theorists and string ensemble practice well into the Baroque era.

Other Tuning Systems for A4

See A4 in all temperaments →

For a full deep dive into Pythagorean Tuning, see the Tunable guide to Pythagorean Tuning.