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Equal Temperament vs Just Intonation
Tuning Systems and Frequency Ratios
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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.
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.