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Just Intonation Calculator
This just intonation calculator compares pure-ratio JI 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. JI ratios are rooted in the harmonic series — the natural overtone sequence that acoustic instruments produce — which is why pure intervals sound consonant: the partials of the two notes lock together without beating.
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 found in the harmonic series, producing pure, beatless consonances when the partials of the two pitches align.
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.
Why Are Pianos Not Perfectly in Tune? Equal Temperament vs Just Intonation
Equal temperament (12-TET) divides the octave into 12 geometrically equal semitones, each with a frequency ratio of 21/12 ≈ 1.0595. This means every key on a piano is equally playable, but no interval except the octave is a pure whole-number ratio. Just intonation, by contrast, defines intervals by small integer ratios: the perfect fifth is exactly 3/2, the major third is exactly 5/4. Those pure intervals produce beatless consonances because the overtones of the two notes align. Understanding the contrast between these tuning systems is fundamental to both historical and modern music theory. You can explore how far any interval deviates from pure using the interval calculator or see the full semitone layout on the equal temperament chart.
The practical cost of purity is key-dependency: a JI major scale tuned for C will have out-of-tune intervals when you modulate to G or D. To measure the gap, musicians use cents (100 cents = one equal-tempered semitone). A JI major third (5/4) sits about 14 cents flat of the 12-TET major third, and a JI perfect fifth (3/2) sits about 2 cents sharp of the 12-TET fifth. This interval comparison shows differences small enough that most listeners notice only on sustained chords or slow vibrato-free tones. This is why pianos, which are fixed-pitch instruments shared across all keys, are tuned to equal temperament rather than just intonation — and why a chromatic tuner calibrated to 12-TET will show a slight offset when an a cappella choir locks a pure major third. The just intonation frequencies in the table above give you the exact Hz values for every interval in any chosen key.
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 based on ratios using only 2, 3, and 5 is called 5-limit tuning; extending to ratios involving 7 produces 7-limit JI, which adds the harmonic seventh. JI has been used in vocal music, string ensembles, and barbershop quartets for centuries, and continues to be explored in microtonality and alternative tuning research.
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 in Scala (.scl) format, and software synthesizers such as Surge XT and Vital can load these files to retune every note. The limitation is that modulating to distant keys will produce wolf intervals — severely out-of-tune versions of normally consonant intervals that arise when the same note serves two different harmonic functions. This calculator helps you plan which intervals will be pure in your chosen key. You can cross-reference the cent deviations here with an equal temperament chart to understand exactly how far each note drifts between JI and 12-TET.
What is a wolf interval in Just Intonation?
A wolf interval is a badly out-of-tune version of a normally consonant interval that appears in fixed-pitch JI tunings. It arises because JI tunes certain notes for pure consonance in the home key, and those same notes then form impure intervals in other keys. The most notorious is the wolf fifth — when you tune a scale to have pure 3/2 fifths, the fifth that "wraps around" the tuning (typically G#/Ab to Eb) can be as far as 36 cents away from 3/2, sounding harsh and beating. Historical keyboard builders addressed this by splitting keys or using unequal temperaments such as meantone, which distribute the error more evenly.
How does Just Intonation relate to the harmonic series?
Just intonation intervals are drawn directly from the overtone series (harmonic series) that naturally emerges from vibrating strings, columns of air, and singing voices. The second harmonic is the octave (2/1), the third is a perfect twelfth or 3/2 above the octave (3/2), the fifth harmonic gives the pure major third (5/4), and the seventh gives the harmonic seventh (7/4). Because acoustic instruments produce these partials, two notes in a JI interval share overlapping harmonics, reducing beating and creating the sense of consonance. The harmonic series calculator lets you explore these relationships interactively.
What other tuning systems use just intonation principles?
Pythagorean tuning uses only perfect-fifth ratios (3/2) to derive all intervals, giving pure fifths but a wide major third (81/64). Meantone temperament (especially quarter-comma meantone) narrows the fifths slightly to produce a near-pure major third (5/4) at the cost of a wolf fifth. Well temperament (the tuning Bach may have intended for the Well-Tempered Clavier) uses unequal semitones so that every key is playable but each has a distinct harmonic character. Extended JI adds higher-prime ratios (7-limit, 11-limit) to incorporate intervals not available in standard 5-limit tuning. All these systems represent different trade-offs between purity and transposability — trade-offs that 12-TET resolves by accepting moderate impurity everywhere.
How do I export this calculator's data to use in a DAW?
Use the Export CSV button to download the full comparison table with JI frequencies, 12-TET frequencies, cent deviations, and beating rates for your chosen root and A4 reference. You can open this in a spreadsheet to build a custom tuning table, or manually enter cent offsets into your DAW's per-note pitch-correction or MIDI pitch-bend settings. For DAWs with Scala format support (Ableton Live via Max4Live, Reaper, Bitwig Studio), you would construct an .scl file listing the cent values of each step; this calculator gives you exactly those cent values in the "Difference (cents)" column. Some users also use this data with a musical interval calculator to double-check interval relationships before finalizing a tuning set.