From "TrackMyTour" cycling blog.
Originally posted: Wed 9 Aug 2023
The Alaska Highway

I've been hearing a drop of about a minor third in the pitch of each car that passes. What's a minor third? It's "Hey Jude": hum that opening lyric without words - sounds sort of like cars passing on the highway? Try sliding from the higher note down to the lower note. Would you believe that you can use just that melody to find the car's speed?

The notes aren't even the same with every car - some play it higher, some lower, and there are different sorts of vehicles sounds - wind noise, road noise, engines, etc. But listening for the notes, among all the different sounds, I keep hearing the same interval as each car passes. That quick drop in pitch just as it goes by - they all sort of sing the same little two-note tune. I wonder, what does that consistent interval represent? Let's see.

Doppler-shift pitch formula:

Fc: observed frequency of approaching object
Fg: observed frequency of receding object
Fs: actual frequency
V: velocity of sound
Vs: Velocity of source
Fc = Fs × V / (V - Vs)
Fg = Fs × V / (V + Vs)

In formula terms, pitch of the car approaching (Fc) to the pitch as it recedes (Fg) - that drop happens on different notes, but musically there's something constant - the minor third. What's going on here?

We have formulas for the doppler-shifts. Is there a formula for a minor third, and can we connect them somehow?

When we hear a musical interval - we're hearing a ratio of the frequencies of the two notes. The upper tone isn't the lower tone plus some number of Hertz, it's the lower tone times a set ratio. For instance, when you hear an octave interval, the frequencies might be apart by 100hz or 1,000hz, but the top one is always 2× the bottom one. That's what makes it an octave.

Octaves would be easier math, but we have a minor third, 3 semitones. 12 semitones in an octave makes this ¼ octave. The math trick is to use just ¼ of an octave's "doubling power". Instead of times 2, the ratio is times 2^¼

Quick proof of why 2^¼: To create a full octave by stacking four ¼ octaves, you'd multiply by that ratio 4 times. 2^¼ × 2^¼ × 2^¼ × 2^¼ = 2¹ = 2

A ¼ octave interval means the higher note is 2^¼ times the frequency of the lower one. 2^¼ = 1.189... When the cars pass and the pitch drops a minor third, then the "coming" pitch is about 1.189 × the "going" pitch. That's the formula for a minor third. Call that the car's Passing Tones.

Fc/ Fg = 1.189 (for minor 3rd)

This formula divides the coming frequency by the going frequency. The actual frequency values cancel out in this division, leaving us with the relationship: the higher note is 1.189 times the lower one - whatever that was. It shows that we don't need to know what the actual pitches are. Good thing, because I don't have perfect pitch. I can't tell you if that RV was whistling a B-flat, but I do hear relative pitch and can tell you the intervals.

And with just the Passing Tones we know their velocity. How Cool. 😎

Velocity of sound is about 767 mph. (varies with temp, humidity, pressure). Now we can plug that all back into the doppler formulas.

Fc/ Fg = 1.189
(767 + Vs) / (767 - Vs) = 1.189
767 + Vs = 1.189(767 - Vs)
767 + Vs = 912 - 1.189Vs
Vs = 145 - 1.189Vs
2.189Vs = 145
Vs = 145/2.189

Vs = 66 mph

It checks out.

Solving for Vs above, gives a general Passing Tones formula:

Vs = 767 × ( r - 1) / (r + 1)

Where r is the ratio of the higher frequency / lower frequency. If you know the ratio - ie, the interval, you can find the speed.

Simplifications:

• This is of course for a stationary listener; that speed is relative to your ear, not relative to the ground. My cycling speed on bike tour is darn close to "stationary", compared to highway traffic. That said, cars coming towards me would have slightly wider intervals than those coming up behind me.
• We presume car travels a constant speed.
• It also fudges some geometry - do you see where? It presumes cars travel in a line directly towards you, and away from you. Do not stand in the middle of highways.

I heard Passing Tones of a major third once today - the opening Da-Da-Da-DUM of Beethoven's 5th. A major 3rd is 4 semitones = 1/3 octave. The ratio is 2^⅓ = 1.26.

Vs = 767 × ( 1.26 - 1) / (1.26 + 1)
Vs = 767 × 0.26/2.26
Vs = 88.2mph

Likewise, a Major 2nd, or whole tone (ratio is 1.123) gives 44 mph.

Who needs a radar gun? Just listen if the cars hum "Hey Jude" when they pass by.

Appendix A:

The Passing Tones can be anywhere on the scale. Their musical interval tells us the speed.

Q & A

1. If you're in the car, can you use this trick to know how fast you're going?

Yes - if you pass by something making a constant tone. That simply changes the reference frame.

2. Are there other kinds of scales with different intervals?

Yes. I used the Equal Temperament scale, which has a ratio of 2^1/12 between each semitone. In Just Temperament, a minor third for instance has a frequency ratio of 1.2, instead of 1.189. This would give a result of ~70 mph. In reality the measurement wasn't precise enough for that to matter.