One of my favorite oral exam questions to give students in introductory physics classes is to ask them whether marching bands should worry about tuning because of the Doppler Effect (lots of details below but the short version: if there’s relative motion between the sound source and listener, the two disagree about what pitch they hear). It often leads to a great conversation about who’s moving and how and who might hear any tuning problems. Sometimes I get a confident “no” from a student and I know immediately that they’ve got some marching band experience. Mostly they talk about how they’ve never noticed it so it must not be a problem. It’s fun to do some quick calculations to see how fast you have to walk for this to be a problem.

Enter my kid’s fun adventure these past few months. They’ve committed themselves to taking the next step from their high school marching band experience (when they won the state championship last summer!) and to set a goal to get into one of the elite Drum Corps International programs. Long story short: They made it! They tried out for a few corps to get a sense of their culture and are super excited to play with Phantom Regiment this summer.

So, the question we’ve been talking about lately is whether marching corps at the elite level should worry about the Doppler effect. They told me a really interesting detail about the training they go through: in order to optimize the tuning for various chords, some corps ask their members to perfect the process of being able to raise or lower their pitch by eight cents. What are cents? If you consider two consecutive keys on a piano, break up the pitch difference between them into 100 pieces. Each of those is a cent. Therefore eight cents is effectively eight percent of the way to the next note on the piano. When they told me about that I started to wonder if the Doppler shifts you’d get in normal Corps routines might be on par with those eight cents.

Structure of this post (for those that want to skip the derivations 🙂

- Doppler effect discussion including a derivation for the Doppler shift at arbitrary orientations of the source and listener (though always with a stationary listener)
- Pitch/frequency discussion including the Just Noticeable Difference
- Issues around simulating the Doppler shift for brass instruments
- Simulations of brass instruments doing simple maneuvers on the field and how that affects their perceived sound
- Discussion of the ramifications (if any) for Drum Corps

## Doppler Effect

For everything I’ll be talking about in this section, the listener will be stationary and only the source will be moving. There are two reasons for this. First, it’s what happens in Drum Corps. The only listeners who are moving are the players themselves and they’re not the target audience. Second, this derivation is much easier without the listener moving. See if you can spot why.

Below is a sketch of the generic situation. The source is L units “up” and x units to the side of the listener. The source is moving at speed v straight down. Note that this represents all possible orientations (see below for a proof of that).

In order to figure out the frequency that the listener hears, we’ll focus on the period of time between when the first and second wavefronts hit. The first is launched when the two are in the orientation shown above. The second is launched after the source moves down for one period *as measured by the source *as shown below.

The time it takes the first wavefront to get to the listener is given by the distance it has to travel divided by the speed it moves at, which is the speed of sound that we’ll label “c”.

The time it takes for the second wavefront to get to the listener is the sum of the time that the source waits to launch it (also known as the period *as determined by the source*) and the time it takes to travel from that location to the listener:

where is the period according to the source. The difference in those times is the period that the listener would measure:

Typically we calculate the Doppler shift as the ratio of listener frequency to source frequency:

This is ugly! We were hoping for some expression that would be independent of either frequency, since certainly that’s how the Doppler effect is typically derived. The way we get around that is to take the limit when . Why do that? If we expect a common ratio to work it should work for all frequencies, and when the period goes to zero then the source has barely moved. Since most marchers move a lot slower than the speed of sound this seems to be a reasonable thing. Taking that limit forces you to go to the Hospital. Sorry, I meant to say L’Hopital. The upshot is:

By the way, you get the same result if you take only the first term of a Taylor expansion of the full expression above. Also note that if x=0 you get the familiar:

While this seems like quite a constrained situation, it turns out that any (stationary listener) scenario can be described with an appropriate L and x. The reason is that there are three physical points of interest: 1) the original location of the source, 2) the location of the listener, and 3) the location of the source when it launches the second wavefront. Those three points will always be on a plane together and so you can reorient them so that (3) is directly below (1) and then you can measure L and x from that. If you let represent the unit vector from 1 to 2 and if you label the three points you get the following for L and x.

To get a sense of how the Doppler shift changes as x goes away from zero (which is how it’s usually derived in introductory physics), here’s a plot of the perceived frequency when the source frequency is 440 Hz (common tuning note) with L=10 meters and x ranges from zero to 10 meters with the source moving at 1% of the speed of sound:

Another way of looking at is is to plot what you hear as a sound sources approaches and then passes you but misses you by, say 1 meter. This time I’ve set the speed to be 55mph.

## Hearing pitch changes

I know, I know, you don’t care about all that derivation. You just want to know if people marching around causes enough of a problem for elite drum corps. Ok, fine, but now we have to talk about how much pitch shifting you, or more likely the DCI judges, can handle before things are either noticeable or bad (or perhaps those two are the same). Mathematically as soon as a pure tone gets mixed with one even one cent off you’ll hear beats, which is often the hallmark of things being out of tune, but even then you’d have to know if the two tones are being played long enough for you to even notice (one cent off at 440 Hz is 440.25 so the beat period would be 4 seconds). Secondly, there aren’t any pure tones in Drum Corps (no flutes allowed) so there’s always some things that aren’t perfectly in tune.

That takes me to another favorite thing to teach: You can’t possibly get all 12 notes in an octave (7 white keys and 5 black keys on the piano) to be in tune all at once. Speaking an an amateur piano tuner, that really sucks! What’s going on here? Western music has coalesced around 12-tone scales for centuries and now you’re learning that it’s an impossible task. Yep, them’s the breaks. The way I usually teach it is that if you declare a frequency ratio of 3 to 2 as pleasant sounding (it’s what we call a perfect 5th in music, so C and G together or, for us brass-leaning folks, Bflat and F), then you should be able to go all the way around the circle of fifths by raising the frequency by a factor of 1.5 (3 to 2 remember) and eventually come back to your original note, though of course several octaves higher. Ok, let’s do it. We’ll start at A 440 and then keep multiplying by 1.5 12 times to all the way around the circle of fifths and back to an A. The order is A, E, B, G flat, D flat, A flat, E flat, B flat, F, C, G, D, A. What do you get? . Ok, big deal. But how many octaves up is that? . Uh oh, that’s not a integer! See the problem? If it was a perfect 7 octaves we’d be in business, but it’s not.

So how does the music industry deal with that mistake? Well, there’s lots of ways. Each is called a “temperament” and there’s been lots of shifts to which one is dominant over the centuries. Some, like the “Pythagorean Temperament” sounds great for 5 or 6 of the notes but really can sound like crap for the rarely used accidentals in the key of the song. If you’re going to just use those 5 or 6, that’s the temperament for you. However, given how most instruments these days can play in any key (one key’s accidental is another key’s super important 5th) and given how composers often make use of key changes, the “equal Temperament” is basically the dominant player these days. Have a electronic tuner in your pocket right now? It uses the equal temperament. “Equal” sounds great, but what it really means is that only the As are perfect and the problem described above is spread around equally to all the other notes. That’s right, if you use a digital tuner, only your As are perfect! Don’t believe me? Well the E on a digital tuner is set to 659.255 Hz. That’s not the “perfect” 440×1.5=660 you were expecting, is it? The other temperaments spread that pain in different ways, some concentrating it on a single interval that people have called the “wolf fifth” because it sounds so horrible if you use it.

Ok, long tangent done, back to what “sounds bad” when it comes to pitch problems. There’s a few ways to describe this. One is the beats described above. If you’re tuning with a buddy who’s playing the same instrument, you can hear that you’re close to being in tune when you hear your combined sound to be throbbing (loud, soft, loud etc). That rate at which the beats happen tells you the difference in your frequencies, so if you can make that rate be zero, you’re in tune. That works best when you’re just a few Hertz off from each other, otherwise the beating is so fast you can’t really hear it.

Another way is to listen to one instrument and then another (so they’re not playing together) and determine who is sharper (or flatter) than the other. Humans can do this, but only down to a frequency separation of about 8 cents (remember those from above?). That’s what’s known as a Just Noticeable Difference. In other words, if two instruments are only off by 4 cents, say, you can’t tell them apart if they play one at a time, but you can hear the beats if they play together.

## Simulating Doppler Shifted Brass

I wanted to be able to not just calculate how much the Doppler shift affects Drum Corps but also try to simulate the sounds a little. My first try was to see how to make a midi sound file be off by a little. Unfortunately Mathematica’s “SoundNote” function that can access midi samples only lets you play pitches on the equal temperament (but see below to see how I got around that). So my second try was to simulate a single instrument (a tuba, since that’s what my kid plays) with just a small collection of pure tones. I used the amplitudes and frequencies from this page to produce this sound of a single tuba:

The nice thing about just a small number of pure tones is that it’s easy, then, to apply the relevant Doppler shift. It definitely sounds like a low brass instrument, but I was hoping for better.

Luckily there’s a somewhat new function in Mathematica that gives me access to the raw time data for the midi samples. Using that, I can simulate the Doppler effect by resampling that time series and then playing it back at the original sample rate. I do that by interpolating the original data so that I can make an estimate of the pressure wave’s value in between the data points that were actually collected. Here’s what an unshifted low B flat from a tuba sounds like from the midi collection:

I guess it sounds a little better. Ok, so now I have all the tools to make some simulations.

## Drum Corps simulations

Here’s the situation I’m going to simulate:

Instead of having them play a series of notes, I’ll just simulate them playing a simple B-flat major chord with the contras on the low B-flat, the baritones on the D, the mellowphones (think marching French horns) on the F, and the trumpets on the high B-flat. I’m only using 12 players (3 each of the various horns all shuffled together spanning 50 yards). Here’s what the simulation sounds like when they’re standing still 10 meters away:

Here’s them 10 meters away but jazz running towards us (they cover 5 yards every 6 steps at 160 beats per minute so 6 to 5 160).

Here’s the same but 1 meter away (there’s a big sound difference here because the ones in the middle dominate the sound).

And here’s the prior three all strung together one after another:

Finally, in case none of those sound bad to you, here’s the 10 meters away version with them running at three times the speed.

To get a sense of how many cents sharp the players are in these simulations, here is a plot of the cents for everyone, first at 10 meters:

and then 1 m away:

What’s interesting about these two is that if there had been a player in the middle, their cents would have been the same for both, namely 10.3. But the biggest change as the horn line gets closer is that the ones towards the end have very small shifts.

## Drum Corps Ramifications

So, can you hear the difference? Certainly the corps think that somewhere around 8 cents is worth worrying about, as my kid has been asked to try to reliably make that sort of change. But if they’re marching around, the Doppler effect brings about that level of pitch shift into play, so if you don’t correct for it, all the other things you’re doing at that level are washing out.

Of course, a lot of the big hits you think of in drum corps happen when they’re standing still. Certainly the youtube traffic seems to head toward clips like this one of them just warming up standing still:

(I love that video. I get goose bumps every time I listen to it).

So, if on the order of 8 cents is meaningful to you, it might be worth it to look at your drill (that’s their marching diagram/orders) and determine if there are times when making a light adjustment might make sense. I’m happy to help (but likely only for Phantom 🙂

## What do you think?

- I really liked this, especially the part where you . . . My question is . . .
- I think this is dumb, especially the part where you . . . My snarky comment is . . .
- I know for a fact that some judges move around while judging and so you’re a liar when you say that the target audience doesn’t move.
- Why do you say any three points can share a plane? I’m sure there’s some crazy non-Euclidean geometry where that’s not true but I’m too furious with you right now to prove it.
- Flutes don’t produce a single pure tone, you anti-woodwind-ite.
- My Just Noticeable Difference for quality blog posts is telling me that this sucks.
- I thought you said your kid got into Phantom Regiment. Why are you showing a Carolina Crown video?
- Why didn’t you talk about the cents difference it takes to go from equal tuning to perfect 5:4 and 3:2 tuning which is surely what these corps do?
- Once again you’ve forsaken Python to do all this in
*Mathematica*. Loser.

This is amazing. You should publish it somewhere.

and lose the opportunity to communicate with my readers (like this)? 🙂 I often think about how my blogging has gotten me to think more deeply about these calculations than my true publications ever have.

Not “either/or”; BOTH! You should blog about it (definitely), but then you should find a venue to (traditionally) publish it.

Made a youtube vid that put into practice the continuous calculation of off axis Doppler for a the B-flat hornline chord used in this post: https://www.youtube.com/watch?v=pbPph12l2-8