Frequency spiral
About this app
Explore intervals on a logarithmic frequency spiral. Listen to frequency ratios and compare just intonation with equal temperament.
Frequency, octaves and tuning
Pitch depends on frequency logarithmically: one octave up means double the frequency, whatever the register. Just intervals have simple frequency ratios: the fifth 3:2, the fourth 4:3, the major third 5:4. Equal temperament instead divides the octave into twelve equal semitones with a factor of 2^(1/12). Every key then sounds alike, but most intervals deviate slightly from the just ratio. The deviation is measured in cents, with 100 cents to an equal-tempered semitone.
What the app shows
Frequency rises along a 3D spiral; one full turn is one octave, so notes with the same name line up vertically. The twelve semitones sit at fixed angles. You choose a lower note and an interval from unison to octave; the readout gives the semitone count, the just ratio and both frequencies in Hz. In the tuning panel you switch between equal and just tuning and read both values in cents and their difference.
Listen closely
Play the major third in both tunings: the equal-tempered third is about 14 cents wider than the just one, and played together you hear clear beating. For the fifth the difference is only about 2 cents, and for the octave it is zero. Then compare the minor third and the sixths, and notice that the spiral shows the same angular distance for the same interval wherever you start.
Frequently asked questions
- What is the difference between just intonation and equal temperament?
- Just intonation tunes intervals to simple whole-number frequency ratios, such as 3:2 for the fifth and 5:4 for the major third, which makes them sound very smooth. Equal temperament divides the octave into twelve equal semitones. That lets you play in every key alike, at the cost of small deviations from the pure intervals.
- How far off is the equal-tempered major third?
- The equal-tempered major third measures 400 cents, the just major third with ratio 5:4 about 386 cents. The tempered third is therefore around 14 cents wide, which you can hear as beating when both notes sound together.
- Why is pitch perception logarithmic?
- Equal musical intervals correspond to equal frequency ratios, not equal differences. From 110 Hz to 220 Hz and from 440 Hz to 880 Hz is an octave each time, although the differences are 110 and 440 Hz. On a logarithmic scale like the spiral, equal intervals therefore have equal spacing everywhere.
- What does this app show?
- A 3D frequency spiral where each turn is one octave. You choose a note and an interval, listen to it and compare just and equal tuning, including the difference in cents.
Subject: Jazz harmony
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