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What is the difference between equal temperament and just intonation?

Just intonation tunes intervals to simple whole-number ratios, like 3:2 for a fifth and 5:4 for a major third. Equal temperament makes every semitone the twelfth root of 2, so every key works the same. The cost is that most intervals land a few cents off pure, and the piano's thirds beat audibly.

Minor 3rd: piano 300¢, pure 315.6¢The piano is 15.6 cents narrower than pure6:55:44:33:25:37:42:10¢1200¢ = octaveticks: piano keys every 100¢ · dots: pure ratios · dashed: this piano interval11.9 beats per second near 1320 Hz
Piano minus pure
-15.6 ¢
Beats per second on a piano
11.9 Hz
Upper note, piano
262 Hz
Upper note, pure
264 Hz

Challenge: Find an interval where the piano is within 2.5 cents of pure (not the octave).

Play

Pick the major 3rd and press "Back and forth." Listen for the shimmer that comes and goes.

Challenge: Find an interval where the piano is within 2.5 cents of pure (not the octave). The box under the picture turns green when you get it.

Stuck? Pick one of the examples from the “Try an example” menu, or press “New example.”

Understand

2s/12 vs. pq,beats=∣q fup−p flow∣2^{s/12}\ \text{vs.}\ \frac{p}{q},\qquad \text{beats} = \lvert q\,f_{up} - p\,f_{low} \rvert

A piano splits the octave into 12 equal ratios, so moving up ss keys multiplies the frequency by 2s/122^{s/12}. A pure interval is a small whole-number ratio such as 5/45/4. Compare them on a log scale, cents:

cents=1200log⁡2f2f1\text{cents} = 1200\log_2\frac{f_2}{f_1}

The top ruler shows the piano's keys every 100 cents and the pure ratios as dots. Because they miss, harmonic qq of the upper note and harmonic pp of the lower note almost meet, and their loudness pulses at the difference. The lower graph shows that pulse over one second.

Use

Every input has a unit menu, so you can type values in the units you already have. Results follow your units.

Show the work

  1. The piano: 12 equal ratios per octave, so each step is the 12th root of 22^{3/12} = 1.1892\ \text{vs.}\ \frac{6}{5} = 1.2
  2. In cents (a log scale, 1200 per octave)100 \times 3 = 300\ \text{vs.}\ 1200\log_2\frac{6}{5} = 315.64
  3. Harmonic q of the upper note nearly meets harmonic p of the lower note. The difference is the beat rate\lvert 5 \times 261.63 - 6 \times 220 \rvert = 11.9\ \text{beats/s}

Export

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Pick a note and an interval, then compare the piano and pure tunings by ear.

  • Real piano strings are slightly stiff, so their upper harmonics run a little sharp. Tuners stretch the octaves to match, which shifts these numbers a bit.

For learning and estimation. Verify with applicable codes, standards, and a qualified professional before using in design, construction, or safety-critical work.

Cheat card

piano step=21/12≈1.05946\text{piano step} = 2^{1/12} \approx 1.05946
cents=1200log⁡2f2f1\text{cents} = 1200\log_2\frac{f_2}{f_1}
beats=∣q fup−p flow∣\text{beats} = \lvert q\,f_{up} - p\,f_{low} \rvert
SymbolMeaningUnit
p:qp:qpure interval ratio
flow,fupf_{low}, f_{up}the two notesHz
  • Fifths and fourths are only 2 cents off, so they barely beat. Thirds and sixths are 14 to 16 cents off and shimmer.
  • Beats speed up as you go higher, because the same cents gap is more hertz.
  • Singers, string players, and brass sections drift toward pure intervals when they hold a chord. That's why an a cappella chord can ring brighter than a piano.

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Where it’s used

  • Music & Sound
    Piano tuners count beats per second between notes to set equal temperament by ear.
  • Physics
    Two nearly equal frequencies add into a tone whose loudness pulses at their difference.
  • Music
    Guitarists notice that a major chord tuned to sound sweet in one key sounds sour in another.

Questions people ask

Why don't pianos use pure intervals?

Pure intervals don't stack into a closed system. Twelve pure fifths overshoot seven octaves by about a quarter of a semitone. Equal temperament spreads that error evenly so every key works.

What are beats?

When two harmonics are almost the same frequency, they drift in and out of step. You hear the loudness pulse at the difference between them.

Is equal temperament out of tune?

Every interval except the octave is slightly off pure, by design. Most listeners grew up with it and hear it as normal.

Updated : First published.

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Sample LOG-0006 · rev 1 · model LOG-M06 · Level 4 · advanced algebra · updated 2026-11-11