Will a marching band on the go sound out of tune to an observer due to the Doppler Effect?

Question

Upon reading Simon Singh's Big Bang, I came upon an explanation of the Doppler Effect (besides the known Formula 1 and ambulance sirens examples):

Trumpeters were split into two groups and asked to play the note E-flat. One group of trumpeters played from an open-top railway carriage on a piece of newly opened track between Utrech and Maarsen, while the other trumpeters remained on the platform. When both groups were stationary then both notes were the same, but when the rail carriage was approaching, then a musically educated ear could detect that the note became higher, and it became even higher as the speed of the carriage increased. When the carriage moved away, the note became deeper. This change in pitch is associated with a change in the wavelength of the sound waves.

_{Chapter 3, The Great Debate, sub chapter World in Motion; page 243.}

I also found a video that is similar to the above experiment:

Both of these experiments have the trumpeter(s) moving relatively fast, but while a marching band is marching, in the usual relatively slow speed, is there any notable difference in the pitch to a musically educated ear caused by the Doppler Effect?

_{I understand that the effect would not be that big, but does it happen at all?}

Answer 1

The pitch difference will be very slight, so some very good ears might notice, but it's likely that many of the instruments will be slightly out of tune to a greater degree than the doppler effect caused at marching speed.

Based on the fomula from the Wikipedia page, and assuming a marching musician moves about .5 meters per second away from the listener, an A 440 note would sound like it is 439.35 Hz (assuming a speed of sound of 340.29 m/s). That's about 2.6 cents flat.

By comparison the G# below A 440 is 415.3 Hz. If I trust my digital guitar tuner (and it's a pretty nice one), the pitch fluctuations in a guitar string played forte as the sound decays can be as much as 5 cents. Also, I think .5 meters per second (1.5 feet per second) might be faster than a marching musician is likely to go, most of the time, and this is assuming that the marcher is moving directly towards or away from the listener. If the marcher is moving at an angle or sideways, the doppler effect will be much reduced.

Edit adding more numbers for marching 2 m/s

Matt Putnam's answer reveals that .5 m/s is actually slow for a marching musician. Taking his number for 2 m/s, and assuming a worst case scenario of one musican marching directly towards, and another directly away from the listener, I'll run the numbers again.

The musican marching away while playing A 440 will be shifted down to 437.43 Hz, which is slightly more than 10 cents flat. The one marching towards the listener will be shifted up to 442.6 Hz, which is slightly more than 10 cents sharp. 20 cents total is approaching half a quarter tone and is noticable. The beat frequency between them is slightly over 5 Hz.

Again, that's worst case scenario. In practice, it's probably not often noticed because of tuning variations between instruments, multiple instruments playing the same parts but having different doppler shifts, plus in a stadium with reverberation and crowd noise it's even harder to hear the doppler shifts in isolation. So yes, it theoretically is noticable, but likely hard to pick out from all the other undesirable things happening to the overall sound.

Answer 2

Yes, this can definitely happen to a noticeable degree.

Let's say there's a block of trumpets facing the audience, and the drill has alternate columns move forwards and backwards. If the step size is 6-to-5 (6 steps to go 5 yards) at a tempo of mm=160, this means that each player is moving at over 2 m/s. The combined relative speed difference of 4 m/s is roughly 1% of the speed of sound, corresponding to a difference in frequency of about 1%. A half step in 12-tone equal temperament is a difference of about 6%. So that means the Doppler effect will cause the notes to be about 1/6 of a half step apart, very noticeable even to untrained ears.

Answer 3

Going by just the title only, the answer is no.

The Doppler shift is relative: it's a factor that is multiplied by the frequency of the tones, shifting them all up or down by the same relative amount. Since our hearing (both volume wise and tone height wise) is logarithmic, the only thing we'd notice is that the whole piece being played is transposed. But the band itself is in tune.

As an example, consider two instruments playing two A's: one at the canonical 440 Hz, one an octave lower, at 220 Hz. If the musicians are sitting in an open car travelling at 72 km/h (20 ms/) towards the listener, the Doppler factor is 330/310 = 1.064516 (330 m/s for the speed of sound). The 440 A would now sound at 468.4 Hz, which is near an A-sharp. The 220 A now sounds at 234.1935, which is also near an A-sharp, just (or still) an octave lower. The "piece" is transposed by half a note, but is otherwise in tune.

If you happen to have perfect pitch, then yes, it will sound out of tune. But from a musical perspective, it sounds fine.

This is also the reason that any piece of music played through a radio (say, from a travelling car) towards or away from you, still sounds fine. It's only when the car gets close, and the relative speed of the car with respect to the observe changes, it sounds off.

If the band would be close, things become a bit hairier: the relative/projected speed with respect to the observer of various instruments in a large band would each be different. (You may want to make a sketch of this.)
Consider a long line of band members, passing at a reasonable speed: the first ones are already past you, producing a lower note for your hearing, while the last ones are still coming towards you, producing a higher note. The person just passing right in front of you has a relative speed of 0 m/s (in the direction that the sounds between her and you travels), and the note from that instrument is at the correct absolute pitch. But the band as a whole will sound distorted and out of tune.

Answer 4

A band sounds musically "out of tune" if the intervals between the notes played by different members are out of tune.

Wikipedia's article about the Doppler effect states that all frequencies become multiplied by the same factor. This factor is approximately (1 - Δv/c), where c is the speed of sound and Δv is the rate of change in distance between the source and listener. So if the A above middle C shifts from concert 440 Hz to 443 Hz or 437 Hz, a pitch an octave below it shifts from 220 Hz to 218.5 Hz or 221.5 Hz, and a pitch a fifth above it shifts from 660 Hz (or thereabouts) to 664.5 Hz or 655.5 Hz. They're not in tune with concert pitch, but as long as the entire band is moving toward or away from the listener, they remain in tune with one another.

But where a marching band differs from someone carrying a boom box is that not all members necessarily have the same Δv. In a parade, as the band approaches you, everyone's Δv is negative, causing the whole band to be more or less uniformly sharp. But as it passes your seat, the musicians at the front of the line are marching away from you, and those at the back are marching toward you. This means they have different Δv, causing those at the front to sound flat and those at the back sharp until the whole band passes. The musicians, however, hear the same Δv because their relative distance to each other does not change.

When a high school or university marching band performs on a football field, it makes drill formations while performing a medley incorporating the school's fight song. When the members change formations, some members are marching toward you and others away as they proceed through formations. This causes their Δv to change, which knocks members approaching a section of the stadium out of tune with members marching in the opposite direction.

From a musical perspective, this can be covered up with drum solos. In a parade, the band can stop momentarily and play the refrain before heading into a drum solo while passing the TV cameras, and the TV crew can use a directional microphone to try to pick up the next band with its uniform Δv as the previous band passes. When writing for a marching band, a composer may give a drum solo during the largest formation changes, or give notes only to a section moving in one direction. However, the bell-front brass instruments used in marching bands minimize this somewhat, as their highly directional sound output makes the sharp approaching instruments louder than the flat receding ones.