Tuning a Major Third in a Wind Ensemble Chorale
A 36-member high school wind ensemble tunes to A440 before rehearsal. The final chord of its chorale is an F-major triad: tuba on F2, trombone on A3, and clarinet on C4. A tuner shows the tuba at 0 cents, the clarinet at +2 cents, and the trombone at +12 cents relative to equal-tempered A3. The conductor stops and says, 'The third is sharp; tune it down.' Several students object that the tuner proves the trombone is nearly in tune. The ensemble has a festival evaluation in three weeks, and the director wants the students to understand why a visually 'in-tune' chord can still sound cloudy. The group has 50-minute rehearsals and access to a drone, a piano, and individual tuners, but the director wants to reduce dependence on the visual display.
1. Explain why the tuner reading and the conductor's comment can both be correct.
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The tuner measures distance from equal temperament, not from the ideal tuning of a sustained major triad. In equal temperament, the major third is 400 cents. In just intonation, the pure 5:4 major third is about 386 cents, roughly 14 cents narrower. If the trombone reads +12 cents above equal-tempered A3 while the tuba is at 0 cents, the third is about 412 cents above the root, which is roughly 26 cents wider than the pure third. The tuner is therefore accurate as an equal-temperament meter, and the conductor is accurate as an ensemble-tuning judgment: the note is close to the tuner's target but too high for a pure third above F.
2. Calculate an approximate cent adjustment for the trombone so the chord better matches just intonation, and describe how you would verify it by ear.
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The tuba's F is the reference at 0 cents. Equal-tempered A3 is 400 cents above F; the trombone is reading +12 cents, so it is about 412 cents above F. A just major third is about 386 cents above the root. The difference is 412 − 386 = 26 cents, so the trombone should come down roughly 26 cents, or about a quarter of a semitone, from its current pitch. To verify by ear, have the tuba sustain F2 and the trombone play A3 softly. Lower the trombone gradually until the slow beating between the frequencies disappears and the third sounds settled rather than bright. Then add the clarinet on C4 and check that the fifth and octave reinforce without reintroducing roughness.
3. Design a 10-minute rehearsal sequence to tune the final chord without relying only on visual tuners.
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Minute 1: establish the tuba's F2 drone and have the ensemble hum F. Minutes 2–3: tune the fifth first, adding the clarinet on C4 and asking students to listen for the absence of beating; the just fifth is only about 2 cents wider than equal temperament, so it should settle quickly. Minutes 4–6: add the trombone alone and have it slide downward until the major third locks; then alternate between the sharp and settled versions so students hear the difference. Minutes 7–8: build the full chord from the root upward, with each section entering only after the previous interval is clean. Minutes 9–10: transfer the same listening to the final two measures of the chorale, asking students to identify whether the third is low, centered, or high without looking at a tuner.