Pythagorean Just
Standard Pythagorean tuning with pure fifths — the textbook chain from Eb to G#.
Quick Facts
- Creator
- Ancient Greek theory (attributed to Pythagoras)
- Historical Era
- Ancient / Medieval
- Formula Type
- pythagorean-stacking
- Key Advantage
- Standard Pythagorean tuning with pure fifths — the textbook chain from Eb to G#.
- Key Limitation
- Wolf fifth between G# and Eb; major thirds very sharp (+22 cents).
- Typical Use
- Medieval music, theory study, and the standard Pythagorean reference tuning.
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 Just Frequency Table — All 12 Notes at A4=440Hz
| Note | Equal Temp. (Hz) | Pythagorean Just (Hz) | Cents from Equal | Tunable Offset |
|---|---|---|---|---|
| C4 | 261.626 | 260.741 | -5.87 | 0 |
| Db4 | 277.183 | 278.438 | +7.82 | 13.685 |
| D4 | 293.665 | 293.334 | -1.96 | 3.91 |
| Eb4 | 311.127 | 309.026 | -11.73 | -5.865 |
| E4 | 329.628 | 330.000 | +1.96 | 7.82 |
| F4 | 349.228 | 347.654 | -7.82 | -1.955 |
| Gb4 | 369.994 | 371.250 | +5.87 | 11.73 |
| G4 | 391.995 | 391.111 | -3.91 | 1.955 |
| Ab4 | 415.305 | 412.035 | -13.69 | -7.82 |
| A4 | 440.000 | 440.000 | 0.00 | 5.865 |
| Bb4 | 466.164 | 463.539 | -9.78 | -3.91 |
| B4 | 493.883 | 495.000 | +3.91 | 9.775 |
Frequencies in Hz at A4=440Hz. Positive cents = sharper than equal temperament. Negative = flatter. Formula: f = f_equal × 2(cents/1200)
Tunable Offset shows the cent values in Tunable's custom temperament format (C=0 reference). To enter this temperament manually: Settings → Tuner → Alternate Temperament → Add → enter these 12 values.
Historical Context
Pythagorean Just originates from the Ancient / Medieval era, developed by Ancient Greek theory (attributed to Pythagoras). It was primarily used for Medieval music, theory study, and the standard Pythagorean reference tuning..
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 Just 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 Just — Get Tunable.
Tunable supports Pythagorean Just and 17 other tuning systems including equal temperament, Pythagorean, just intonation, and well temperaments. See exact Hz values in real-time as you play.
