Questions: Just Intonation and Harmonic-Series-Based Composition
3 questions to test your understanding
Score: 0 / 3
Question 1 Multiple Choice
What frequency ratio defines a perfect fifth in just intonation?
A5:4
B3:2
C4:3
D9:8
A just perfect fifth is a 3:2 ratio — the second and third harmonics of the overtone series. 5:4 is the just major third, 4:3 is the just perfect fourth, and 9:8 is the just major second. The 3:2 ratio produces zero beats between the two tones, giving the pure, locked-in sound characteristic of just intonation.
Question 2 True / False
Just intonation is universally superior to equal temperament because most of its intervals are mathematically pure ratios.
TTrue
FFalse
Answer: False
Just intonation produces acoustically pure intervals within a fixed key but creates the 'comma problem' when modulating: stacking pure fifths or thirds generates small pitch discrepancies (commas) that make distant keys sound out of tune. Equal temperament distributes these discrepancies evenly so every key is equally usable — a genuine trade-off, not a deficiency.
Question 3 Short Answer
What is a 'comma' in the context of just intonation, and why does it matter for practical composition?
Think about your answer, then reveal below.
Model answer: A comma is a small interval (typically 20–25 cents) that arises from the discrepancy between different just-intonation paths to the same pitch — for example, stacking 12 pure fifths (3:2 each) overshoots 7 octaves by about 23 cents (the Pythagorean comma). This matters because it makes fixed-pitch just-intonation instruments unplayable across multiple keys without retuning.
Commas expose the inherent arithmetic incompatibility of stacking simple ratios. Just intonation composers must either restrict themselves to a limited harmonic region, design instruments with extra pitches per octave, or write for flexible-pitch ensembles (voices, strings) that can adjust on the fly.