πΆ Tuning Temperament Calculator
Compare frequencies across equal temperament, just intonation, and Pythagorean tuning.
Frequency Comparison Table (one octave from root)
How to Use This Calculator
Select a concert pitch (A4 tuning) and a root note. The table shows every note in the chromatic scale from that root, with exact frequencies in equal temperament, just intonation, and Pythagorean tuning. The cents deviation columns show how far each alternative system drifts from equal temperament for each individual note.
Select your concert pitch. The standard is A4 = 440 Hz. Select 415 Hz for Baroque pitch or 432 Hz if you work in that alternative tuning convention.
Choose the root note. This sets which note the just intonation and Pythagorean ratios are built from. The tuning sounds best in chords and scales near this root in just intonation.
Read the deviation columns in cents. Green or near-zero values mean the tuning systems agree. Yellow means a noticeable difference (5 to 15 cents). Red means a large difference that will be audible as beating between instruments.
Pay special attention to the major 3rd row. Just intonation's major 3rd (5/4 ratio) is 13.7 cents flatter than equal temperament. This is why a choir singing a cappella naturally goes flat if they follow just intonation thirds without compensating.
Temperament Frequency Formulas
In 12-tone equal temperament, every semitone is exactly the same size: the 12th root of 2 (approximately 1.0595). This means no interval except the octave is perfectly pure, but all keys sound equally good or equally imperfect. Just intonation uses pure integer frequency ratios that produce beatless, acoustically perfect intervals within one key but become increasingly out of tune in other keys. Pythagorean tuning stacks pure perfect 5ths (3:2 ratio) to derive all notes, giving clean 5ths but very sharp major thirds.
Real-World Examples
When You Need This
String quartets and vocal ensembles that perform without fixed-pitch instruments naturally drift toward just intonation when playing chords, because the pure tuning sounds more resonant and satisfying in the moment. Knowing the cents deviations from equal temperament helps coaches and conductors explain specifically why a chord sounds out of tune when a player holds equal-temperament pitch while the rest of the ensemble shifts toward just. The major 3rd is the most common culprit because the 13.7-cent difference is clearly audible.
Composers and producers writing for historical instruments or authentic Baroque performance need to know that a piece in C major tuned to meantone or Pythagorean temperament will sound different in E major or Bb major. Distant keys have wolf intervals, which are pairs of notes that clash severely because the tuning system has no room left to accommodate them. Using this calculator to check which intervals are furthest from equal temperament in Pythagorean tuning helps you avoid writing cadences that sound dissonant on period instruments tuned that way.
Frequently Asked Questions
What is equal temperament (12-TET)?
12-TET divides the octave into 12 equal semitones with a ratio of 2^(1/12) β 1.0595. It allows playing in all 12 keys with the same instrument without retuning. Slight compromises on pure harmonic ratios.
What is just intonation?
Just intonation uses pure integer ratios (3/2 for a fifth, 5/4 for a major third). It sounds acoustically pure in the key it's tuned to, but chords in other keys become increasingly out of tune.
What is Pythagorean tuning?
Pythagorean tuning stacks pure fifths (3/2 ratio) to derive all notes. Perfect fifths are pure, but thirds are quite sharp (β22 cents off). Common in medieval music theory.
Why does a piano go slightly out of tune?
Pianos use equal temperament, which means all intervals except the octave are slightly "wrong" relative to pure ratios. Additionally, stretched octaves are used on piano because long strings vibrate in slightly inharmonic partials.
What is a cent in tuning?
A cent is 1/1200 of an octave, or 1/100 of a semitone. Just intonation's major third (5/4) is 386.3 cents, while equal temperament's major third is exactly 400 cents β a difference of 13.7 cents.