Why does pitch change with temperature? September 08 2014
The age-old question! Why does my horn go sharp when it gets hot and flat when it's cold? Contrary to many band room conversations, it has very little to do with the expansion or contraction of the metal of the instrument. In fact, the expansion of the instrument would (very slightly) counteract the effect.
Short answer
Because physics.
Long answer
Because sound is a pressure wave, it requires molecules to bump into each other. The "frequency" we use to describe this wave is, in a sense, a measurement of how quickly these molecules can bump into each other. This rate depends on the relationship between the pressure and temperature of the fluid through which the sound is traveling. There are many other factors that affect the properties of a fluid, but temperature significantly affects the speed of sound through a gas. To start, we'll use an example of a band tuning on stage during a sound check with the hall's AC running. The find the velocity of the speed of sound in this air, we have to compare it to air at 0 degrees centigrade:
(Velocity in hall) = (velocity at 0C) + 0.606*(temperature of hall)
V1 = 313.3 meters/second + 0.606*(20 centigrade) [68 degrees F]
V1 = 325.42 m/s
Now let's say the audience of 1000 people is seated in the hall and they're really excited and breathing heavily and sweating and heating the place up. We can figure out the new speed of sound:
V2 = 313.3 meters/second + 0.606*(25 centigrade) [77 degrees F]
V2 = 328.45 m/s
So what does this mean? Sure, the sound travels faster, but how does that translate into frequency. We have a handy equation for that, too:
Wavelength (W) = Velocity (V) * Period (T)
The wavelength of the wave is, simply, the length a single cycle occupies. The period is how many seconds between each full cycle of the wave. Using this equation for our scenario can show us either how much longer our horn needs to be, or how much the frequency would increase. Let's do the change in pitch first, if we were to leave our horn the same length.
W1 = V1 * T1
W2 = V2 * T2
For our first case, the lengths of the horn/wave are the same.
W1 = W2
V1 * T1 = V2 * T2
T2 = (V1/V2) * T1
You can now see that the period of the wave in the hot hall is based on a ratio of the speeds of sound in the two situations. To get the frequency, we merely invert the equation -- instead of wanting the number of seconds the wave takes, we want how many waves per second there are. (T = 1/f)
f2 = (V2/V1) * f1
f2 = (328.45/325.42) * 466.2 hertz [tuning Bb]
f2 = 1.009311 * 466.2 hz
f2 = 470.5 hertz
That's roughly 15% closer to a B natural than the original tuning pitch! Unfortunately, this change in velocity of the sound doesn't affect all instruments equally. Some are affected more severely by the actual change in shape of the instrument.
If we flip it around, and ask how much we'd have to lengthen our horn to adjust for the heat, we merely rearrange the equation and find the ratio setting the frequency (or period) equal to each other.
W = V/f
f = V/W
V1/W1 = V2/W2
W2/W1 = V2/V1
Notice here that the increase in length is the same ratio as the increase in velocity of the speed of sound! This makes the math easy since we've already calculated that:
W2/W1 = 1.009311
As a percentage, this means you would have to adjust your tubing length by 0.9311%... That's almost an inch on a standard trombone and even more on a tuba. But what gives? One rarely has to pull out that much to adjust. The secret lies in what you're putting into the horn. Hot air. When you think about it, the most simple circumstance would be playing your horn when it's 98.6 degrees out. We all know this is uncomfortable, but it would be the closest you'd get to having the conditions of the horn unaffected by your breath. At normal room temperatures, you have to take into account the fact that you're resonating the horn at some temperature gradient between your breath and the room. This dampens the affect of the room's temperature on the pitch. As you also may have noticed, you might tend to go sharper when you're in direct sunlight, even if the ambient temperature isn't that bad. The radiant heat from direct sunlight would tend to bump up the temperature inside the horn more than the convection of the air over the surface of the metal inside a room.
It is important to note that you can force your horn to play any pitch with your lips, but it will only resonate at those for which the horn is the correct length.
-- Neil
