Just Intonation (Major) vs. Pythagorean Tuning
Compare the tuning characteristics of Just Intonation (Major) and Pythagorean Tuning — cent deviations per note, practical guidance, and historical context.
At a Glance
| Feature | Just Intonation (Major) | Pythagorean Tuning |
|---|---|---|
| Category | just-intonation | pythagorean |
| Formula Type | just-ratios | pythagorean-stacking |
| Historical Era | Renaissance / Theory | Ancient / Medieval |
| Key Advantage | Perfectly pure major thirds (5:4) and fifths (3:2) in the home key. | Pure perfect fifths (3:2) — ideal for medieval polyphony and chant. |
| Key Limitation | Fixed tonal center — modulating to other keys produces out-of-tune intervals. | Major thirds are very sharp (+22 cents) and the wolf fifth makes distant keys harsh. |
| Typical Use | A cappella choral music, theoretical analysis, and tuning reference for pure intervals. | Medieval music, chant, early polyphony, and Pythagorean theory study. |
Cent Deviations: All 12 Notes vs. Equal Temperament
Positive cents = sharper than equal temperament. Negative = flatter. Difference column shows Pythagorean Tuning minus Just Intonation (Major): positive means Pythagorean Tuning is sharper.
| Note | Just Intonation (Major) (¢) | Pythagorean Tuning (¢) | Difference (¢) |
|---|---|---|---|
| C4 | 0.00 | -5.87 | -5.87 |
| Db4 | -11.73 | +7.82 | +19.55 |
| D4 | +3.91 | -1.96 | -5.87 |
| Eb4 | +15.64 | -11.73 | -27.37 |
| E4 | -13.69 | +1.96 | +15.65 |
| F4 | +1.96 | -7.82 | -9.78 |
| Gb4 | -9.78 | +5.87 | +15.65 |
| G4 | +1.96 | -3.91 | -5.87 |
| Ab4 | -15.64 | +9.78 | +25.42 |
| A4 | -15.64 | 0.00 | +15.64 |
| Bb4 | +17.60 | -9.78 | -27.38 |
| B4 | -11.73 | +3.91 | +15.64 |
When to Choose Each
Choose Just Intonation (Major) when:
Choose Just Intonation (Major) for a cappella choirs, string quartets, and any ensemble exploring pure intonation in the home key. Best suited to music that stays near one tonal center rather than modulating widely.
Choose Pythagorean Tuning when:
Choose Pythagorean tuning for medieval ensemble music, plainchant, early polyphony, and Pythagorean melodic writing where pure open fifths are the primary consonances. String players often gravitate naturally toward this intonation for melodic lines.
Historical Context
Just Intonation (using 5:4 major thirds and pure ratios) and Pythagorean tuning (derived from stacked pure fifths, 3:2) represent two competing theoretical traditions: the renaissance just-ratio system championed by Zarlino versus the Pythagorean tradition of medieval theorists. They agree on the perfect fifth (3:2) but diverge sharply on the major third (pure 0 cents vs. +22 cents sharp).
- Just Intonation (Major)
- Developed by Gioseffo Zarlino and Renaissance theorists — Renaissance / Theory era
- Pythagorean Tuning
- Developed by Ancient Greek theory (attributed to Pythagoras) — Ancient / Medieval era
Compare Temperaments in Tunable — Get Tunable.
Tunable supports Just Intonation (Major), Pythagorean Tuning, and 14 other tuning systems. Hear the difference in real-time as you play.
