How Fast is that Car or Airplane Traveling? Use musical skills to figure it out

As musicians we are skilled at hearing intervals - if we hear 2 notes, separately,  we can tell you how far apart they are and we can name the interval ( 5th, minor 3rd, half step etc. )  At least we should be able to do that!  With that skill comes great power ( not so much responsibility)  - it will enable us to semi-accurately  judge the speed of a passing car or airplane!    At the very least, it will impress your friends.

To review the Doppler effect from Junior High days, recall that the pitch of an oncoming object will suddenly decrease as it begins to go away from us. As the object is heading towards us, the sound is effectively "pushed" along by an amount equal to the speed of the object.  The sound does not actually travel any faster, but the individual sound waves are created to be closer together.  As the object retreats, the opposite happen and the pitch goes down as the object passes and the sound waves that are created are farther apart.  

This is all pretty straightforward, but the part they never teach you is that the amount that the pitch drops can tell you the actual speed of, in this case, the car.  This can work for airplanes as well but the math is a little more complicated because of the fact that we are pretty far removed from the flight path of the airplane  - there's a way to deal with this,  but LATER.   For now,  cars...

Recall, from a previous post on "temperament" that I listed the frequency ratios of all the common musical intervals. Knowing the speed of sound (1126 feet/sec) and the musical interval that we hear as the car goes by, we have enough to calculate the speed of the car.  We assume, in this example, that we are very near the path of the car but not so close as to create a medical problem.
  To review the musical-interval frequency-ratios:
     octave  2:1     =   2
        fifth    3:2        =  1.5
          aug 4th               =     1.41
            4th   4:3             =           1.33
                 major 3rd    5:4          =    1.25
                        minor  3rd    6:5             =  1.2
                           whole tone    1.12/1            = 1.12
                               half step           1.059/1           =    1.059

Let's ASSUME for an example,  that we hear the pitch drop by a WHOLE-STEP as the car passes.
 Let S = the speed of the Car (mph)       Speed of sound = 768mph  or 1126 fps
The Ratio:
Speed of Sound + Speed of Car      equals   1.12     ( the whole-step ratio, R)
Speed of Sound - Speed of Car                      

or

768mph + S      =      1.12  
768mph -S                  1

Solve for S , recalling some of that basic algebra,   S= 43.5 mph
   HAPPILY, the calculation can be condensed down to this simple formula if you wish do it yourself:
                            S = 768(R-1)/(R+1) 
 where S = SPEED OF CAR
            R = RATIO OF THE PERCEIVED MUSICAL INTERVAL CREATED BY THE DOPPLER EFFECT (listed in chart above)

Here are the computed results for the various intervals.
      pitch-drop interval                 speed of car mph
     HALF STEP                                 22.0
     WHOLE STEP                             43.5
     MINOR THIRD                           69.8
     MAJOR THIRD                           85.33
     FOURTH                                     108.8
     AUGMENTED 4TH                   130.7
     FIFTH                                         153.6
     OCTAVE                                      256

Clearly this will work better on a race track than on a neighborhood street.  Measuring the speed of cars is inherently difficult -
  1.  Usually there are more cars than just one and the sounds are confusing
  2.  Cars have an annoying tendency to speed up and slow down.  Go figure.
  3.  Cars don't produce a clear single pitch - the sound of the tire friction on the road overwhelms engine pitches.
  4.  People don't like you standing in their front yard waiting for cars to go by.
  5.  Motorcycles have a clear pitch but they're prone to wild accelerations.

That is why I prefer....
         AIRPLANES
The thing about airplanes is that they are high up in the air, but the plane is traveling TOWARD its destination, not toward me. Well, it is traveling toward me but not as fast, and the rate that it's traveling toward me is constantly CHANGING!   This suggests pulling out that High School Calculus book, but not to worry, we can do it with simple Junior High Trigonometry.  There is one point in the airplane's relationship to me where the airplane is traveling exactly HALF as fast toward me as it is toward its destination.  ( Remember that 1, 2 , sq root of 3  triangle;  30, 60, 90? ) That point, when the plane is seen at a 60 degree angle to the horizon,  is diagrammed below.   We could calculate a speed ratio for any angle using trig, but this angle seems to work well for a specific-case calculation.  It would be pretty hard to hear the plane anyway if the angle were much less than 60 degrees,  because it would be so far away, and this 60 degree angle does make the math pretty simple.

We just do the same calculation as before, solving for S, then multiply S by 2 to get the actual speed of the plane. 

See the chart below for the AIRPLANE version of intervals and speeds.  It would be necessary to have some way to judge when the plane is at the 60 degree point - a piece of cardboard that you level with the horizon, for example
Doppler Pitch Shift                       Speed of Airplane
     HALF STEP                                 44.0
     WHOLE STEP                            87.0
     MINOR THIRD                           139.6
     MAJOR THIRD                          170.7
     FOURTH                                     217.6
     AUGMENTED 4TH                    261.3
     FIFTH                                          307.2
     OCTAVE                                      512

This method should work even if the plane were not passing directly overhead.  Just visualize a tilted geometric plane  that cuts through the observer and the flight path - estimate when the plane hits the 60 degree points ( that's the hard part).

   Airplanes produce a pitch that is easy to hear, there is usually only one plane at a time, there's no risk of being run over, they tend to travel at a consistent speed, and the range of Doppler pitche-changes fits nicely into the musical scale within an octave. So much easier than cars.

Tilted Geometric Plane