🎶
Just Intonation Calculator
Compare Just Intonation (pure ratio) intervals side-by-side with 12-tone Equal Temperament. Hear the difference with A/B audio, build JI chords, visualize beating frequencies, and explore musical commas.
Hz
A4 = 440.0 Hz
A4
440.000 Hz
JI vs 12-TET Interval Comparison
| Interval | Ratio | JI Freq (Hz) | 12-TET Freq (Hz) | Difference (cents) | Beating (Hz) | A/B Play | Chord |
|---|
Chord Builder
Quick Presets:
Selected intervals: None
Musical Comma Calculations
Syntonic Comma
81/80
The difference between a Pythagorean major third (81/64, built from four perfect fifths) and a pure major third (5/4). This comma is the fundamental discrepancy that makes just intonation difficult to use across all keys. In 12-TET, this comma is tempered out entirely.
Pythagorean Comma
531441/524288
The gap between 12 stacked perfect fifths (3/2)12 and 7 octaves (27). If you tune 12 perfect fifths upward from any note, you arrive at a pitch slightly sharp of where you started. This ~23.46 cent discrepancy is what 12-TET distributes equally among all intervals.
Diesis
128/125
The difference between three pure major thirds (5/4)3 and one octave. Three stacked major thirds in JI fall short of a perfect octave by this amount (~41.06 cents), demonstrating why pure intervals cannot perfectly fill an octave.
Formulas Used
Just Intonation Frequency
fJI = froot × (p / q)
Where p/q is the ratio of the interval (e.g., 3/2 for a perfect fifth). JI intervals are derived from small whole-number ratios, producing pure, beatless consonances.
12-TET Frequency
fTET = froot × 2n/12
Where n is the number of semitones above the root. Equal temperament divides each octave into 12 geometrically equal steps of 21/12.
Cents Difference
Δc = 1200 × log2(fJI / fTET)
Cents measure the logarithmic difference between two frequencies. 100 cents equals one equal-tempered semitone. Positive values mean JI is sharper; negative means JI is flatter.
Beating Frequency
fbeat = |fJI − fTET|
When two close frequencies sound together, you hear a pulsation called "beating" at a rate equal to the absolute difference of the two frequencies. Zero beating means perfect unison.
How to Use This Calculator
- Set your root note and A4 reference — Choose any root note and octave from the dropdowns. Adjust the A4 reference frequency if you use a tuning standard other than 440 Hz. All calculations update instantly.
- Compare JI and 12-TET intervals — The comparison table shows every chromatic interval with its just ratio, both JI and 12-TET frequencies, the cent deviation, and beating frequency. Use the A/B play buttons to hear each interval in both tuning systems back-to-back.
- Build and compare chords — Check the "Chord" boxes next to intervals or use quick presets (Major, Minor, Dom7, etc.) to select intervals. Then play the chord in JI or 12-TET, or use A/B Chord Compare to hear them sequentially.
- Explore musical commas — The comma section visualizes the fundamental discrepancies in tuning theory. Click "Hear the Comma" to listen to the tiny pitch difference that each comma represents.
- Export your data — Copy the table to clipboard or export as CSV for use in spreadsheets, DAWs, or research papers.
Frequently Asked Questions
What is Just Intonation?
Just Intonation (JI) is a tuning system where intervals are defined by exact small whole-number frequency ratios. For example, a perfect fifth is exactly 3/2 (1.5x the root frequency) and a major third is exactly 5/4 (1.25x). These pure ratios produce consonant, beatless intervals because the overtones of the two notes align perfectly. JI has been used in vocal music, string ensembles, and barbershop quartets for centuries.
Why does Just Intonation sound different from Equal Temperament?
In 12-TET, every semitone has the same ratio (2^(1/12) ≈ 1.05946), which means no interval except the octave is a pure ratio. A 12-TET major third is about 14 cents sharper than a pure 5/4 ratio, causing audible beating when sustained. JI intervals sound "purer" and more restful because their harmonics align, but JI cannot maintain these pure intervals across all keys simultaneously — this is the trade-off that equal temperament resolves.
What is a "cent" in music?
A cent is a logarithmic unit for measuring musical intervals. One octave equals 1200 cents, and one equal-tempered semitone equals exactly 100 cents. The formula is: cents = 1200 × log2(f2/f1). Cents allow precise comparison of intervals across different tuning systems. Most trained musicians can perceive differences of about 5-10 cents; differences below 2 cents are generally imperceptible.
What is a beating frequency?
When two tones of slightly different frequency sound simultaneously, they produce a periodic fluctuation in volume called "beating." The beat frequency equals the absolute difference between the two frequencies. For example, if a JI perfect fifth is 660.000 Hz and the 12-TET version is 659.255 Hz, the beating rate is |660.000 − 659.255| = 0.745 Hz, or about one pulse every 1.3 seconds. Piano tuners use beating to achieve precise equal temperament.
What is the syntonic comma?
The syntonic comma (81/80, about 21.5 cents) is the difference between a Pythagorean major third (81/64, reached by stacking four perfect fifths) and a pure major third (5/4). It represents the fundamental conflict between tuning by fifths and tuning by thirds. In meantone temperament, this comma is distributed among the fifths to keep thirds pure. In equal temperament, it is absorbed into every interval.
Can I use Just Intonation in modern music production?
Yes, though it requires careful planning. JI works beautifully for music that stays in one key or uses drones (Indian classical music, some ambient/electronic genres, barbershop quartets). Many DAWs support custom tuning tables (Scala format), and software synthesizers can be retuned to JI. The limitation is that modulating to distant keys will produce "wolf" intervals — severely out-of-tune versions of normally consonant intervals. This calculator helps you plan which intervals will be pure in your chosen key.