Doppler Cents Puzzler

[V T]

Hi Mark,

Here is my commentary:

>The fact that the tone goes flat 10 cents when going
>away merely confirms that the difference between the
>stopped truck and the moving cycle produces a 10 cent
>differential in pitch.

True.  He didn't even slow down!  Incredible.

>So I considered the pitch coming from the stopped
>truck to be 1, and the sound to be travelling at 1100
>feet per second. In order to reach a pitch of 2, the 
>cycle would have to be moving at the speed of sound
>toward the truck, to achieve a total of 2200 feet per
>second closing speed.

Correct.

>With that thought in mind, I just calculated that a
10 >cent increase in pitch equalled 10/1200 of the
speed >of sound, so:

This is where the trouble started.  It is true that
the octave consists of 1200 cents, but the frequency
increments (in Hz) between those 1200 tones are not
linearly divided into equal amounts.  Note the
following:

- One semi-tone is equal to a frequency ratio of
1.05946 times, but

- One cent is equal to frequency ratio of 1.0005778
times.

Note that 1.0005778 is not equal to 1.05946/100, just
as one semi-tone is not equal to a pitch increment of
1+(1/12)=1.08333 times.

You have to multiply 1.0005778 one hundred times _with
itself_ (raise it to the hundredth power) to get
1.05946.

So, the principle is that each higher pitch is equal
to the previous pitch multiplied by a constant.  That
constant will depend on the pitch increment you
choose.  If you want to step in semi-tone increments,
the constant that defines two adjacent steps will be
2^(1/12).  If you want to step in one cent increments,
that constant will be 2^(1/1200).  If you want to step
in octave increments, the constant will be 2^(1/1). I
am using "^" as the symbol for "raised to the the
power of ...".

Some examples:

One cent sharp from A440 is:

440*1.0005778 = 440.254228Hz

Note that the 1 cent increment up from 440 Hz is equal
to 0.254228 Hz

Two cents sharp from A440 is equal to:

440*1.0005778*1.0005778 = 440.508602 Hz

The difference in pitch between the last two tones is
0.2543745 Hz, which is slightly different from
0.254228.  It doesn't seem like much, but if you make
that small error 1200 times in a row, it turns into an
ice cream disaster.

>10 cents higher than nominal pitch =
>10/1200 * (speed of sound in air)
>or 1/120 * (1100 ft/sec) = 9.1666 ft/sec (speed of
>bicycle)
>9.166 ft/sec * 3600 secs/hour = 33,000 ft/hour
>33,000 / 5280 (ft/mi) = 6.25 mph

>However, you arrived at 2 meters/second, which equals
>7200 meters/hour, 
>which translates to 4.47 miles per hour. So would you
>tell me how you got there? Thanks!
>
>-Mark Schecter

The other small difference which is not really
relevant is that I used 345m/sec for the velocity of
sound which is slightly off from 1100 ft/sec.

Vladan

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