Pythagorean Tuning
Pure perfect fifths (3:2) — ideal for medieval polyphony and chant.
Quick Facts
- Creator
- Ancient Greek theory (attributed to Pythagoras)
- Historical Era
- Ancient / Medieval
- Formula Type
- pythagorean-stacking
- Key Advantage
- Pure perfect fifths (3:2) — ideal for medieval polyphony and chant.
- Key Limitation
- Major thirds are very sharp (+22 cents) and the wolf fifth makes distant keys harsh.
- Typical Use
- Medieval music, chant, early polyphony, and Pythagorean theory study.
Mathematical Basis
Pythagorean tuning stacks pure perfect fifths (ratio 3:2, 702 cents) to derive all 12 pitch classes. The Pythagorean comma (531441:524288, ~23.46 cents) is left unresolved, creating a wolf fifth.
Sound Character
Pythagorean tuning features pure, resonant open fifths that ring with exceptional clarity. Major thirds are very sharp (+22 cents compared to just intonation), giving melodies a bright, tense quality. It is ideal for medieval parallel organum, plainchant, and early polyphony where fifths and octaves are the primary consonances.
Pythagorean Tuning Frequency Table — All 12 Notes at A4=440Hz
| Note | Equal Temp. (Hz) | Pythagorean Tuning (Hz) | Cents from Equal |
|---|---|---|---|
| C4 | 261.626 | 260.740 | -5.87 |
| Db4 | 277.183 | 278.438 | +7.82 |
| D4 | 293.665 | 293.333 | -1.96 |
| Eb4 | 311.127 | 309.026 | -11.73 |
| E4 | 329.628 | 330.001 | +1.96 |
| F4 | 349.228 | 347.654 | -7.82 |
| Gb4 | 369.994 | 371.251 | +5.87 |
| G4 | 391.995 | 391.111 | -3.91 |
| Ab4 | 415.305 | 417.658 | +9.78 |
| A4 | 440.000 | 440.000 | 0.00 |
| Bb4 | 466.164 | 463.538 | -9.78 |
| B4 | 493.883 | 495.000 | +3.91 |
Frequencies in Hz at A4=440Hz. Positive cents = sharper than equal temperament. Negative = flatter. Formula: f = f_equal × 2(cents/1200)
Historical Context
Pythagorean Tuning originates from the Ancient / Medieval era, developed by Ancient Greek theory (attributed to Pythagoras). It was primarily used for Medieval music, chant, early polyphony, and Pythagorean theory study..
Pythagorean tuning was standard throughout the Medieval and early Renaissance periods, favored by theorists from Boethius to medieval polyphonists. Composers such as Perotin and Machaut worked within this framework.
Who Uses Pythagorean Tuning Today
Pythagorean tuning is used today by early music ensembles, medieval music reconstruction groups, and in academic theory courses teaching interval derivation. String players naturally gravitate toward Pythagorean intonation for melodic lines.
Tune with Pythagorean Tuning — Get Tunable.
Tunable supports Pythagorean Tuning and 15 other tuning systems including equal temperament, Pythagorean, just intonation, and well temperaments. See exact Hz values in real-time as you play.