How does temperature affect tuning?

Question

Off the back of comments on one of the answers to this question: How do room layouts affect tuning?

So, I know the short answer is "It depends on the instrument", but ideally I'd like to know a few rules and enough reasoning behind them so that I can understand the physics and theory to generalise further. I'm talking "short term" temperature changes, and the scenario is: you've rehearsed in a cold and empty auditorium (some people wore gloves), now the audience has show up and the room is noticeably warmer (they're fanning themselves with their programmes - it's hot!). What characteristics of your instrument dictate whether you need to sharpen up or flatten down and why?

Context: I'm a flute/piccolo player and have played mostly in wind bands, sometimes in orchestras. I know that my instrument is generally flat when cold and that the band usually "warms up" before tuning. I also know that if we haven't been playing for a while in a cold room, some wind players might blow warm air through their instruments before playing.

Previously, I'd assumed it was to do with the instrument dimensions. I assumed that for a flute, warm metal expands inwards (as well as outwards), the tube narrows and the pitch goes up. For some reason I'd never considered that the heat would cause it to lengthen, too, and that this increase in length would cause a (fractional!) decrease in pitch. Turns out I was wrong about the metal. This calculator suggests that the tube would actually only get bigger, so my theory about the metal expanding and somehow sharpening the pitch is wrong.

Comments on this question: How do room layouts affect tuning? imply that what is really affecting the tuning is the speed of sound changing in warm air. This makes sense since:

v (speed) = f (frequency) times w (wavelength)

If speed goes up, either frequency or wavelength (or both?) must also go up. And warm air with less molecule density will propagate sound faster. And air is probably easier to warm than metal (although air in a warm metal tube will stay warmer than air in a cold metal tube). But I am also aware of the counter example in the comments where a loudspeaker set to produce a 440Hz signal will produce that sound regardless of the surrounding air/medium. And I'm also aware (from the comments) that string instruments going flat in warmer temperatures makes sense - the string material lengthens, tension decreases, flatter notes.

So, part of my question is really is there some relationship between the material that vibrates to cause the sound and temperature and pitch? Is speed of sound in warm air only relevant when the sound is caused by a vibrating column of air? Or is it still relevant for the violins, just that the dominant factor there is the loss of tension?

Answer 1

Let's compare the effects "flute becomes longer due to increasing temperature" and "density of air and thus speed of sound changes with increasing temperature" numerically:

Let's assume we have a wooden flute (or organ pipe) of 1 m resonator length = 1 m wave length. At 20 °C the speed of sound in air is 343,43 m/s, so the basic frequency of the flute is 343,43 Hz.

Now let's increase the temperature from 20 °C to 25 °C, a not-so-unrealistic scenario:

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Thermal expansion of the flute material

For the change in length of the flute, the formula is , meaning

Change in length is approximately equal to coefficient of thermal expansion times length at the beginning times change in temperature.

The coefficient is for (oak) wood. The length at the beginning was 1 m, the change in temperature is 5 °C = 5 K.

This gives us a change in (wave) length of 0,00004 m (or 0,04 mm).

Assuming the change in length were the only factor, this would cause a drop in frequency from 343,43 Hz to 343,416 Hz calculated using this very useful tool from Sengpiel Audio, a drop of 0,014 Hz (0.07 cent).

Increased speed of sound

Using the same tool, we calculate that a rise of 5 K leads to an increase in speed of sound to 346,35 m/s and thus to a new frequency of 346,35 Hz, a rise of 2,92 Hz (14.7 cent).

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So the influence of the air density change is about 200 times higher than the influence of the thermal expansion of the wood.

This may be a very rough calculation, but it points in the right direction.

Answer 2

There are a couple of different mechanisms that cause instruments to change pitch when the temperature changes. Which one applies depends on how the instrument makes sound happen.

When sound comes from a vibrating string

At sea level the speed of sound is vw=(331 m/s)√(T/273K), where T is the temperature in Kelvin. So if you're at sea level and it's 70ºF, the speed of sound is 343.65 meters per second, and when the temperature drops to 55ºF the speed of sound drops to 338.75 m/s.

The wavelength is the speed of sound divided by the frequency. So if you've got an instrument like a piano that's creating a vibration, that vibration has a frequency - temperature isn't going to change it. And that means since the speed of sound changed, the wavelength must change. You still have the same amount of sound energy being produced, so the instrument actually loses a little volume, but the pitch stays the same.

Except the pitch doesn't stay the same - because the temperature causes the parts of the piano to expand or contract, and they do it at different rates (called coefficients of expansion). That can make the strings tighter or looser, which will change the tuning.

When the sound comes from a vibrating air column

This one is kind of counter-intuitive. Wind and brass instruments don't make sounds like pianos, even if they're starting with a vibrating part like a reed. They're making the air vibrate inside a tube. The tube fixes the length of the wave - so the math works out differently.

Wavelength is still the speed of sound divided by frequency. So if you play an A440 at 70ºF, and your instrument is in tune, you've got wavelength = 343.65 (m/s)/440 Hz, or 0.78 m. And if you're in tune at 55ºF your wavelength = 338.75 (m/s)/440 Hz, or 0.77 m.

But your wavelength can't change! It's fixed by the length of your instrument's pipe and the location of the toneholes. So you still have the same wavelength: 0.78 m, and you have the same speed of sound, 338.75 m/s. And that means you're producing a frequency to balance the equation: 434 Hz. You're flat.

Once the air inside your instrument warms up, you'll get the right frequency. And at that point the speed of sound won't affect your tuning, because temperature will change the wavelength, but not the frequency, just like it will for the piano.

Answer 3

In addition to excellent Johannes answer:

Air humidity can also change speed of sound by 0.1–0.6% and this corresponds to several cents, which might be audible.

https://en.wikipedia.org/wiki/Speed_of_sound#Dependence_on_the_properties_of_the_medium

All these consideration might get more complicated since the musician blows warm and humid air into the instrument, and also warms it up by touching.

Answer 4

Anyone who owns a guitar will notice differences based on humidity. In high humidity, the wood of the guitar expands, causing the strings to become tighter and the pitch of the notes to be sharp. When the humidity decreases, the pitch of the strings tend to become flatter.

This principle, then, should be the same for heat. Increasing heat causes solids to expand, and decreasing causes them to contract.

I found a study done by someone online. (not sure who, or for what purpose of presentation). This person hypothesized the opposite and found he was wrong.

Answer 5

Metals expand when they are warm, and contract when they are cold. Since the pitch of a flute or any other pipe instrument depends on the length of the tube, the pitch will change with temperature. But the pitch also depends on the density of the air in the tube, which also varies with temperature, so don't assume that the pitch will be lower in warm weather.

Stringed wooden instruments, are going to be much more complicated. Wood may expand with heat, but it will shrink if it dries out. Metal strings will expand and contract with temperature. Gut strings will change with humidity, and temperature as well.

I have a wooden harp with both wire and nylon strings. I tend to find that if the wire strings have gone sharp, then the nylon ones are usually flat, and vice-versa.