The pitch of a whistle is determined by its length, and the speed of sound (340 m/s).
To determine how long to make a whistle, you need to know the frequency of the desired pitch:
Based on A 440 (A5), you can determine the frequency of C5 (three half steps higher than A5) by multiplying it by 2^(3/12). 3/12 is 0.25, and 2^0.25 is 1.189. 1.189 * 440 = 523. Higher octaves are multiples of 2: 1046, 2092, 4084. ( units: Hz, which is also 1/s)
The length of the whistle is simply the speed of sound divided by the frequency:
C5 : 0.65
C6 : 0.33
C7 : 0.16
C8 : 0.08
It’s this last one that interests me — because an oboe reed is approximately that length, and it sounds “C” when blown by itself.
But curiously, an oboe reed isn’t 8 cm long — they’re usually 6.9-7.1 cm. What would account for this difference?
It turns out that the value for the speed of sound I used was for dry air at sea level at 60 degrees (F).
However, an oboe reed is blown, so the air moving through it is 98 degrees, and 100% humid. Calculating the correct speed of sound requires knowing the density of the air, and its adiabatic constant. Fortunately, someone has already calculated this. They give these values:
30°C 351.51
40°C 359.17
Linear interpolation for 37° gives us close approximation for the speed of sound through human breath: 357 m/s
Using this revised value, we find that our ideal oboe reed would be 8.5 cm. This is ever further from the actual length of an oboe reed!
So where does the additional 1.6 cm come from? If the pitch of the instrument was determined entirely by the length of the reed, then it would be impossible to influence the pitch of the instrument by how you play (e.g. your embouchure). I assure you, the pitch of the oboe is very easy to influence! Therefore it is clear that the musician is part of the instrument — Thus the extra 1.6 cm are coming from the resonance of the players mouth.