## Saturday, April 07, 2012

### Tying alto sax players to trains to explain the Doppler shift

A physician friend phoned me the other day with an interesting problem. He was discussing the Doppler shift with some physics friends. They wondered if it was possible to hear a chord shift from major to minor due to the Doppler shift.

If you don’t know anything about music theory, a major chord and a minor chord are similar in structure. They contain 2 notes in common. In music, we call these notes the root and the 5th. So, if we start on a C, the 5th note above that in our major and minor scales is a G. The difference happens in the third note above the root C. If it is a major chord, it has an E. If it is a minor chord, it has an E flat.

Clearly we could have a situation where I had two trusty French Horn players in stationary positions on my left and right. They could play the C and the G. I could then tie an alto saxophone player to a train car and compel him to play an E as the train sped away. Now, in my imaginary scenario, this alto sax player can play a concert E in tune (which is far fetched, I know, but this is science so bare with me).

To calculate the Doppler shift as sounds move away, the formula is:

Frequency Observed = Frequency*Velocity of Sound/(Velocity of Sound + Velocity of Object moving away)

In this scenario, let’s imagine our horn players are playing Middle C and the G above it. I need my saxophone player to play and E, but I want to hear an Eb. So that we don’t get too fussy, I’m going to use equal temperament frequencies and assume straight lines and not wind resistance so that the cosine of theta doesn’t get involved in the calculations. So:

My Observed Frequency will be 311.127 (Eb)
My Actual Frequency will be 329.628 (E)
The Velocity of sound is roughly 340m/s

311.127Hz = 329.628Hz*340m/s ÷ (340m/s + x)
311.127 = 112073.579 ÷ (340 + x)
105783.180 + 311.127x = 112073.579
311.127x=6290.399
x = 20.21 m/s (or roughly 45 mph)

So, when the train is moving away at 45 mph, I will hear an E flat even though he is playing an E.

Very neat, but it was not actually the question that my friend asked. He was wondering if there was a scenario where all of them were on the train and moving at the same time. In other words, do frequencies fall uniformly during the Doppler shift. My initial reaction was that you couldn’t do it. I called my friend the research scientist. He agreed that it wouldn’t work. Using our same formula, you can see why. If I take the lowest C on the piano (32.7 Hz) and the highest E flat (2489 Hz) and have them played by a tuba and piccolo player (which I have tied to the alto sax player) and send the train of at 64.29 m/s (a little over 143 mph (we are obviously in Japan or Europe at this point and not on Amtrak)) an interesting thing happens. Using the formula above, the C at 32.7Hz will be observed at 27.5Hz which is the A a minor third below. The piccolo player playing the E flat at 2489Hz will be observed at 2093Hz, the C a minor third below. So, If a major chord slides with the Doppler shift, the frequencies will slide uniformly. It will stay major.

I can conceive of one situation where something like my friend suggested could actually happen. If I were at a football game and two French Horn players were marching directly toward me playing a C and a G while the alto sax player was marching directly away from me playing an F, there would come a point in the Doppler shift where I would hear the minor chord C# E G#. Theoretically, I would hear this as a minor chord even though they were playing a suspended chord. Of course for this scenario to happen, they would have to be marching at about 45 mph.

The idea of two French Horn players marching toward me at 45 mph is the stuff of my worst nightmares, and I’d rather not even imagine such a thing happening.