Tuning systems
About this app
Just intonation (integer ratios, fifth 3:2) vs. equal temperament (2^(1/12)): the deviation in cents visible, the beating of two tones audible.
Two ways to tune
Just intonation builds intervals from whole-number frequency ratios: fifth 3:2, fourth 4:3, major third 5:4. Such notes blend very smoothly. Equal temperament divides the octave into twelve equal semitones with a factor of 2^(1/12) ≈ 1.0595. Every interval is then the same size everywhere, and you can modulate into any key. The cost is that almost all intervals deviate slightly from the pure ideal. This is measured in cents: a semitone is 100 cents, an octave 1200.
What the app shows
For a root (C, F, A or B) and an interval from unison to octave, the app shows the frequency of the upper note in just and in equal temperament, the deviation in cents and the beat frequency. The waveforms of both notes are drawn on top of each other, with an adjustable number of visible periods. For a fifth above A4 = 440 Hz you get 660 Hz against 659.26 Hz, about 2 cents and a beat of 0.74 Hz.
Listen and compare
Play an interval in just and in equal tuning one after the other. The beating demo sounds both upper notes together, and the volume pulses at the difference frequency. Compare the fifth with the major third: the equal-tempered third is about 14 cents wider than the just one, and its beating is much faster.
Frequently asked questions
- What is the difference between just intonation and equal temperament?
- In just intonation intervals have simple whole-number frequency ratios such as 3:2 or 5:4 and sound without beating. In equal temperament all twelve semitones are the same size, with a factor of 2^(1/12). All keys then sound alike, but most intervals are slightly out of tune.
- How large is the difference for fifths and thirds?
- The equal-tempered fifth is about 2 cents narrower than the just fifth 3:2, which is barely audible. The equal-tempered major third is about 14 cents wider than the just third 5:4, and the minor third about 16 cents narrower than 6:5. With thirds the difference is clearly audible.
- What is a cent in music?
- A cent is one hundredth of an equal-tempered semitone, so an octave has 1200 cents. The scale is logarithmic: a frequency ratio r equals 1200 · log₂(r) cents. This allows small differences between tunings to be compared.
- What can I hear in the Tuning Systems app?
- Every interval above a chosen root in just and equal tuning, one after the other or as beating when both upper notes sound together. The app also shows frequencies, ratios and the deviation in cents.
Subject: Music (Gymnasium)
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