Stack twelve perfect fifths and you don't land back on the starting note — you overshoot by 23.5 cents. This mathematical impossibility forced centuries of musical compromises. Explore them here.
Temperament note Equal temperament Wolf interval Pure interval
Temperament System
Fifth Size
Comma
0.0 ¢
Wolf Fifth
—
Purest Third
—
Tonic
C
Playback
Temperament Comparison
Deviation from equal temperament for each note. Positive = sharp, negative = flat.
Note
Pythagorean
Just
1/4 Meantone
Equal
About Temperament Compass
Why this project?
The mathematical impossibility of perfect musical tuning is one of the most elegant problems at the intersection of mathematics, physics, and art. No power of 2 can ever equal a power of 3, which means the circle of fifths can never close perfectly. This single fact shaped centuries of music history, instrument design, and even the emotional character of different musical keys. Despite the wealth of writing on the topic, interactive tools for actually seeing and hearing these differences are rare.
What is it?
Temperament Compass visualizes different musical tuning systems on a circular compass of fifths. Each note is positioned around the circle according to its position in the chain of fifths. You can switch between historical temperaments (Pythagorean, Just Intonation, 1/4-comma Meantone, Equal Temperament) and watch how the notes shift position. Click any note to hear it played with Web Audio, and compare the wolf fifth — the most out-of-tune interval — across systems.
How does it work?
The compass is built from the chain of fifths: C → G → D → A → E → B → F♯ → C♯ → G♯ → D♯ → A♯ → F. In equal temperament, each fifth is exactly 700 cents (1200 × log₂(2^(7/12))). In Pythagorean tuning, fifths are pure at 701.96 cents (1200 × log₂(3/2)), which creates a 23.46 cent Pythagorean comma. In 1/4-comma meantone, fifths are narrowed to 696.58 cents to make major thirds pure. The compass plots each note's cumulative deviation from equal temperament as a radial distance, making the mathematical structure visible.
How to use it
Select a temperament from the panel on the right to see its compass visualization.
Adjust the fifth slider to explore custom temperaments between 680–720 cents.
Click note chips at the bottom to play individual notes and hear their tuning.
Press "Play Scale" to hear all twelve notes ascending in the selected temperament.
Press "Wolf Fifth" to hear the most dissonant interval in the current temperament.
Hover over the compass to see note names and cents deviation for each position.
Scroll the comparison table at the bottom to see deviations across all temperaments.