The Throb in the E-flat Chord: Tuning Systems Meets a Piano at 442
A high school string orchestra of 34 players (12 first violins, 10 second violins, 6 violas, 4 cellos, 2 basses) is preparing a slow, chorale-like movement for a joint concert with the wind ensemble in the auditorium. In the final eight bars the director hears a slow, regular throb in a sustained E-flat major chord. The cellos hold the third of the chord, G3, stopped on the D string; the basses sustain the root; the first violins hold the fifth. Every player's tuner reads A=440, and every open string is in tune to it. The auditorium piano was tuned last week to A=442, and the piano plays on two numbers of the program. A strong student says the cellos are simply out of tune and should adjust their thirds until the throb disappears. The director has two rehearsals before the concert.
1. Explain what is producing the throb and how you would confirm the diagnosis, distinguishing it from a genuine tuning or technique problem.
Read the worked answer
The throb is beating - the interference between two frequencies that are close but not identical, here generated by coincident partials: the fifth partial of the E-flat root against the fourth partial of the G. A pure (just) major third is a 5:4 ratio and sits about 14 cents narrower than an equal-tempered major third. When the cellos tune the G to a tuner, they place it roughly 14 cents sharp of the pure third, and against a sustained root and fifth that produces an audible roughness. The cellos are doing exactly what the tuner asks; this is a tuning-system problem, not a technique problem.
Confirm it by isolation:
- Cellos hold G alone - no beats.
- Add the basses' E-flat - beats appear.
- Add the first violins' B-flat - beats worsen or spread.
That sequence points at the third, not at the players. Then rule out technique: if the stopped G were genuinely out of tune, it would be off by far more than 14 cents and would also be out against the open D string. Check the stopped G against the open D. If it is clean, the instrument and the finger are fine and the problem is the interval.
2. What reference pitch should the orchestra use for the concert, and how should the director manage the change with only two rehearsals left?
Read the worked answer
The reference has to be the piano's A=442, because the piano cannot move and it is the only fixed-pitch instrument on the program. 442 is about 8 cents above 440, so the orchestra must move, not the piano. Note that this compounds the first problem: the pure third sits about 14 cents below equal temperament at whichever reference you choose, so a player who has been tuning to 440 and then bends the third may end up nearly a quarter tone from any tuner reading. That has to be named out loud, or students will "correct" the third back up.
Two rehearsals:
- Set the frame first. Have every student change their tuner to 442 and explain why. Take the tuning note from the piano at the start and at the end of each piece - a long slow movement drifts down toward the old reference, and that drift is the usual failure. Tune open strings to 442, then a slow E-flat scale to establish the key.
- Then tune the interval. In the 442 frame, have the cellos lower the third by ear until the throb disappears, and have them hold it while the violins add the fifth. Students must hear that the resulting note reads "flat" on the tuner and is nevertheless correct.
- Dress-rehearse the collision. Run the two piano numbers first in the final rehearsal, then check the chorale movement's E-flat chord directly against the piano.
3. A parent volunteer shows a cellist a phone app reporting the note is "15 cents flat." How should the director respond without undermining the student's ear training?
Read the worked answer
Do not dismiss the app. The number is real; the question is what it measures and what it does not.
Say the distinction plainly: an app compares one note to an equal-tempered standard. The ear compares one note to the other notes sounding at the same time. Both readings are correct and they answer different questions. A tuner cannot tell you whether two notes are in tune with each other.
A parent-friendly phrasing: "A tuner can tell you if a single note matches a standard. It can't tell you if a chord is in tune. In a sustained major chord the third has to sit about 14 cents low, or you hear the beating - that's what the app is showing." If the parent wants to check, ask them to listen for the throb with their own ears: it disappears at the pure third and returns at the tuner's version.
For the student, the important move is not "ignore the tuner" but when to use which tool: tuner for open strings, unaccompanied passages, and matching a fixed-pitch instrument; ear for intervals inside a chord. Have her demonstrate both for the parent - tune the G to the app, let the throb appear, then lower it by ear and let the chord lock. That reframes the app as a partial tool rather than an authority, and it protects the ear training the program is trying to build.