Musical tuning systems determine the exact frequencies assigned to notes, and small differences in tuning can strongly affect how music sounds. Two important systems are equal temperament and just intonation. Equal temperament divides an octave into 12 equal frequency steps, which makes it practical for playing in every key.
Just intonation instead builds intervals from simple whole number ratios, which can make chords sound especially smooth and resonant.
The main scientific idea behind this comparison is how frequency ratios shape consonance and beating. In equal temperament, most intervals are slightly adjusted so that all semitone steps are equal, allowing instruments like the piano to modulate freely between keys. In just intonation, intervals such as 3:2 or 5:4 match harmonic relationships more closely, often reducing roughness in sustained harmony.
The tradeoff is that a tuning that sounds ideal in one key may not work as well after changing to a distant key.
Understanding Equal Temperament vs Just Intonation
The ear does not judge an interval only by the two notes that are named. It responds to the patterns hidden inside each sound. A plucked string, a violin note, or a human voice produces a fundamental frequency plus higher frequencies called harmonics.
These harmonics occur at whole number multiples of the fundamental. When two notes have a simple frequency relationship, many of their harmonics line up or nearly line up. The combined sound can feel stable and clear.
When harmonics fall very close together without matching, they produce rapid changes in loudness called beats. This roughness is one reason some intervals sound tense.
A major third shows the difference clearly. In a justly tuned major chord, the top note of the third can be set so its frequency is five parts for every four parts of the lower note. If the lower note has a frequency of 264 hertz, the upper note would be 330 hertz.
The equal tempered version is a little higher. The difference is small, but sustained notes can reveal it. The harmonic patterns do not align quite as neatly, so a sensitive listener may hear a gentle shimmer.
This does not mean equal temperament is wrong. It means that it chooses consistency between keys over exact harmonic alignment within one chord.
A major problem appears when musicians try to build every note from perfectly pure intervals. Repeatedly moving up by a pure fifth does not bring a player exactly back to the same pitch class after twelve steps. It ends slightly higher than moving through seven octaves.
This tiny mismatch is called the Pythagorean comma. A tuning system must place that mismatch somewhere. Historical keyboard tunings often made some keys sound very pure while leaving one or more intervals badly out of tune.
Equal temperament spreads the adjustment across all twelve fifths. Each fifth becomes only a little narrower than pure, and no single key receives the full problem.
Fixed-pitch instruments need this compromise most. A piano has strings that are set before a performance, so it cannot retune every chord as the music changes key. Fretted guitars face a similar limit because fret positions are fixed.
Orchestral string players, singers, and trombone players can adjust pitch while performing. In a slow chord, they may move a note slightly to make a third or fifth blend better.
When playing with a piano, they usually stay closer to the piano's tempered pitches. Skilled ensemble playing involves hearing these small differences and choosing the pitch that fits the musical situation.
Students can compare tunings using cents, a unit for very small pitch distances. One octave contains twelve hundred cents, while one equal tempered semitone contains one hundred cents. A just major third is about fourteen cents lower than an equal tempered major third, which is large enough for many people to notice when notes are held.
A just fifth differs by only about two cents, so it is harder to identify alone. When learning this topic, separate pitch from timbre.
Two instruments can play the same tuned note yet sound different because their harmonics have different strengths. Listen for slow beating, chord smoothness, and changes after a melody moves into a new key.
Key Facts
- Octave relation: f2 = 2f1
- Equal temperament semitone ratio: r = 2^(1/12) ≈ 1.05946
- In equal temperament, the nth semitone above a starting note is f = f0 x 2^(n/12)
- Just intonation uses simple ratios such as octave 2:1, perfect fifth 3:2, and major third 5:4
- Equal temperament perfect fifth: 2^(7/12) ≈ 1.4983, compared with just intonation 3/2 = 1.5
- Beating frequency for two close tones is fbeat = |f1 - f2|
Vocabulary
- Equal temperament
- A tuning system that divides the octave into 12 equal frequency ratios so music can be played in any key.
- Just intonation
- A tuning system that chooses note frequencies from simple whole number ratios to make intervals sound pure.
- Interval
- An interval is the pitch difference between two notes, often described by a frequency ratio.
- Consonance
- Consonance is the smooth and stable sound produced when frequencies relate in simple ways.
- Beating
- Beating is the pulsing sound heard when two frequencies are close but not exactly the same.
Common Mistakes to Avoid
- Assuming equal temperament means equal frequency differences between notes, which is wrong because the steps are equal ratios, not equal subtraction amounts.
- Thinking just intonation always sounds better in every situation, which is wrong because its pure intervals in one key can create problems after modulation to other keys.
- Using note number differences instead of frequency ratios to compare intervals, which is wrong because musical intervals depend on multiplicative relationships.
- Forgetting that equal temperament slightly adjusts most intervals, which is wrong because only the octave stays exact while intervals like the fifth and major third are approximations.
Practice Questions
- 1 A note has frequency 220 Hz. In equal temperament, what is the frequency 7 semitones above it? Use f = 220 x 2^(7/12).
- 2 Compare a just intonation major third above 240 Hz with an equal temperament major third above 240 Hz. Use 5/4 for just intonation and 2^(4/12) for equal temperament. Find both frequencies and the difference between them.
- 3 A choir sings a chord in one key using just intonation, then a piano joins using equal temperament. Explain why some intervals may sound more blended than others and why slight beating may appear.