Questions: Tuning and Intonation Assessment by Ear

Question 1

A string quartet holds a sustained major third chord. One player slightly flattens her upper note toward the pure 5:4 ratio, away from equal temperament. What happens to the sound?

• A. The chord sounds more out of tune, because she is deviating from the equal-temperament standard
• B. The beating decreases, because her note is moving closer to frequency alignment with the lower pitch
• C. The beating increases, because any deviation from equal temperament introduces interference
• D. The sound is unchanged, as beating is determined by room acoustics rather than the players

Answer: (B)

The equal-tempered major third is 14 cents sharp relative to the pure 5:4 ratio. Beating occurs when two frequencies are nearly but not exactly aligned — the more misaligned, the faster the beating. By flattening toward 5:4, the player is moving toward alignment, reducing the beating rate. The common misconception (option A) assumes equal temperament is the standard of 'in tune,' but beating is an acoustic fact about frequency alignment, not a judgment about temperament.

Question 2

You hear rapid, harsh beating in a sustained major third played on an organ. This is most consistent with which tuning system?

• A. Just intonation, which tunes major thirds to a pure 5:4 ratio
• B. Equal temperament, which tunes major thirds 14 cents sharp relative to pure
• C. Pythagorean tuning, which stacks pure fifths and produces major thirds that are noticeably sharp
• D. Meantone temperament, which narrows major thirds toward the pure 5:4 ratio

Answer: (C)

Pythagorean tuning generates major thirds via a chain of pure fifths (3:2), producing a major third of 81:64 — about 22 cents sharper than the pure 5:4 ratio of 80:64. This causes rapid, harsh beating in sustained chords. By contrast, just intonation's major third is beatless, equal temperament's is slightly impure (14 cents sharp, slower beating), and meantone deliberately narrows the third toward pure.

Question 3

In equal temperament, the perfect fifth is tuned closer to its pure ratio than the major third is to its pure ratio.

• T. True
• F. False

Answer: True

True — the equal-tempered perfect fifth is only about 2 cents flat of the pure 3:2 ratio, which is nearly imperceptible. The equal-tempered major third, however, is about 14 cents sharp of the pure 5:4 ratio — audibly impure in sustained contexts. This asymmetry is why string players and singers often adjust thirds more aggressively than fifths when playing in ensemble.

Question 4

Equal temperament is used on most Western instruments because it produces the purest possible intervals.

• T. True
• F. False

Answer: False

False — equal temperament deliberately compromises interval purity in order to allow transposition to any key without re-tuning. Nearly every interval in equal temperament (except the octave) is slightly misaligned from its pure ratio. The system is used on fixed-pitch instruments (pianos, fretted guitars) for practical reasons. String players and singers do not use equal temperament; they adjust intonation contextually toward pure ratios in sustained harmonies.

Question 5

Why does equal temperament sacrifice pure interval ratios, and under what musical circumstances might a performer choose to deviate from it?

Model answer: Equal temperament divides the octave into 12 equal semitones so that all keys are equally usable and transposition is possible without re-tuning. This requires slight misalignment of almost every interval from its pure frequency ratio. Performers deviate from equal temperament in sustained harmonies — particularly major thirds and pure fifths — where the beating of tempered intervals is audible. String players, singers, and wind players commonly tune toward just intonation in slow, sustained passages and return to equal temperament for chromatic or quickly moving lines.

The core insight is that there is no single 'correct' tuning — different musical contexts optimize for different priorities. Equal temperament optimizes for transposability; just intonation optimizes for harmonic purity in a given key; Pythagorean tuning optimizes for pure fifths. The skill in ensemble intonation is understanding which intervals matter most in context and adjusting accordingly.