Tuning System Calculator
Compare note frequencies across equal temperament, just intonation, and Pythagorean tuning systems.
About this calculator
This calculator always computes and displays all three tuning systems at once for the Note and Octave you choose -- Highlight System is a cosmetic label only and never changes any of the three frequencies. Equal Temperament is the modern standard: every semitone is exactly 2^(1/12) times the one below it, which makes all twelve keys equally in tune (and equally slightly out of pure tune) so instruments can modulate freely between keys. Just Intonation instead uses small-integer frequency ratios (like 3/2 for a perfect fifth) that make certain intervals ring with zero beating, at the cost of those ratios only being pure relative to the tonic -- modulate away from it and the purity breaks down.
Pythagorean tuning builds every interval by stacking pure 3/2 perfect fifths, which keeps fifths and fourths extremely clean but produces thirds that drift noticeably sharp or flat from equal temperament, historically called the "Pythagorean comma." Cents Diff reports how far Just Intonation and Pythagorean tuning sit from Equal Temperament in cents (1/1200th of an octave, roughly the smallest pitch difference most listeners can detect), so you can see at a glance which intervals a fretted or fixed-pitch instrument is compromising on. Octave and Reference A4 Frequency scale all three systems together and proportionally -- they don't change which system sounds "more in tune," only the absolute pitch everything is measured from. What this doesn't model: any temperament beyond these three (meantone, well temperaments like Werckmeister, or non-Western tuning systems), and it assumes A4 is the tuning anchor for all three rather than letting Just Intonation or Pythagorean anchor to a different tonic.
Inputs
Results
Equal Temperament
440 Hz
How to Use This Calculator
- Enter the Reference A4 Frequency — 440 Hz is concert pitch, though some ensembles tune to 442 Hz or higher.
- Choose the Note and Octave you want the frequency of.
- Set Highlight System if you want to note which tuning system you're comparing against — it doesn't change any of the three results, all of which are always calculated.
- Review the Equal Temperament (Hz) result, the tuning system used by virtually all modern fixed-pitch instruments.
- Compare Just Intonation (Hz) and Pythagorean (Hz) and their Cents Diff against equal temperament to see how far historical tuning systems drift from modern pitch.
How the result changes with Octave
| Octave | Equal Temperament |
|---|---|
| 2 | 110 Hz |
| 3 | 220 Hz |
| 6 | 1,760 Hz |
| 8 | 7,040 Hz |
What each input means
- Reference A4 Frequency
- Concert pitch reference frequency. Standard is 440 Hz.
- Note
- The note to calculate, given as semitones above A. 0=A, 3=C, 7=E.
- Octave
- Octave number. A4 = 440 Hz is octave 4.
- Highlight System
- Cosmetic only -- all three systems are always calculated and shown regardless of this choice.
How this is calculated
Worked example, using the default values
- Identify Input Parameters4 parametersReference A4 Frequency = 440, Semitones from A = 0, Octave = 4, Highlight System = 1 = 4 input(s) provided
- Calculate Equal TemperamentEqual Temperament440 = 440
- Calculate Just IntonationJust Intonation440 = 440
- Calculate PythagoreanPythagorean440 = 440
Engine last updated . Checked against 4 independently-derived tests — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Does changing Highlight System actually change any of the frequencies shown?
No. All three systems -- Equal Temperament, Just Intonation, and Pythagorean -- are calculated and displayed every time regardless of what Highlight System is set to. It exists only to note which system you're most interested in comparing against for your own reference; changing it between Equal Temperament, Just Intonation, and Pythagorean produces the exact same three output numbers every time.
Why do Just Intonation and Pythagorean tuning give different frequencies for the same note?
They build their ratios from different starting principles. Just Intonation uses small-integer ratios chosen to make specific intervals (thirds, fifths, sixths) as pure and beat-free as possible relative to the tonic. Pythagorean tuning instead stacks pure 3/2 perfect fifths repeatedly to generate every note in the scale, which keeps fifths pure but pushes thirds noticeably away from the small-integer ratios Just Intonation targets -- the two systems optimize for different intervals sounding pure.
What does a Cents Diff of a few cents actually mean in practice?
A cent is 1/1200th of an octave, and most listeners can detect a pitch difference somewhere in the 5-10 cent range in isolation, less on a held, sustained tone with another reference pitch to compare against. A Cents Diff of 10-20 cents (common between Pythagorean thirds and Equal Temperament) is audible as a distinctly different color to the interval, which is why fretted and fixed-pitch instruments settled on equal temperament as a practical compromise across every key.
Why does raising the octave change the frequency so much?
Every system here doubles frequency for each octave you go up, since an octave is defined as a 2:1 frequency ratio in all three tuning systems shown. Octave has the largest measured effect on the output frequency of any input in this calculator because it's an exponential scale factor, not an additive one -- moving from octave 4 to octave 5 doubles the frequency outright, while changing Note only shifts it by the much smaller ratios within a single octave.
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