Just / Pure
Perfectly pure major thirds (5:4) and fifths (3:2) in the home key.
Quick Facts
- Creator
- Gioseffo Zarlino and Renaissance theorists
- Historical Era
- Renaissance / Theory
- Formula Type
- just-ratios
- Key Advantage
- Perfectly pure major thirds (5:4) and fifths (3:2) in the home key.
- Key Limitation
- Fixed tonal center — modulating to other keys produces out-of-tune intervals.
- Typical Use
- A cappella choral music, theoretical analysis, and tuning reference for pure intervals.
Mathematical Basis
Just intonation uses simple whole-number frequency ratios (5:4 major third, 3:2 fifth, 6:5 minor third) to achieve maximum harmonic purity in the home key. Intervals are pure but fixed to one tonal center.
Sound Character
Just intonation produces exceptionally pure consonances in the home key: major thirds (5:4), perfect fifths (3:2), and minor thirds (6:5) are all beatless. The result is a choir-like blend where chords fuse into a unified sound. Modulation causes increasing dissonance as the key moves away from the tonal center.
Just / Pure Frequency Table — All 12 Notes at A4=440Hz
| Note | Equal Temp. (Hz) | Just / Pure (Hz) | Cents from Equal | Tunable Offset |
|---|---|---|---|---|
| C4 | 261.626 | 264.000 | +15.64 | 0 |
| Db4 | 277.183 | 281.600 | +27.37 | 11.731 |
| D4 | 293.665 | 297.000 | +19.55 | 3.91 |
| Eb4 | 311.127 | 316.800 | +31.28 | 15.641 |
| E4 | 329.628 | 329.239 | -2.04 | -13.686 |
| F4 | 349.228 | 352.000 | +13.69 | -1.955 |
| Gb4 | 369.994 | 371.250 | +5.87 | -9.776 |
| G4 | 391.995 | 395.999 | +17.60 | 1.955 |
| Ab4 | 415.305 | 422.400 | +29.33 | 13.686 |
| A4 | 440.000 | 440.000 | 0.00 | -15.641 |
| Bb4 | 466.164 | 469.333 | +11.73 | -3.91 |
| B4 | 493.883 | 495.000 | +3.91 | -11.731 |
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
Just / Pure originates from the Renaissance / Theory era, developed by Gioseffo Zarlino and Renaissance theorists. It was primarily used for A cappella choral music, theoretical analysis, and tuning reference for pure intervals..
Just intonation was championed by Renaissance theorist Gioseffo Zarlino as the ideal for vocal polyphony. It remains the natural tuning of unaccompanied choirs and a cappella ensembles.
Who Uses Just / Pure Today
Just intonation is used by a cappella choirs, barbershop quartets, and string quartets exploring pure intonation. It is the theoretical basis for understanding consonance and the physics of harmonic resonance.
Tune with Just / Pure — Get Tunable.
Tunable supports Just / Pure and 17 other tuning systems including equal temperament, Pythagorean, just intonation, and well temperaments. See exact Hz values in real-time as you play.
