How does the pitch of a wind instrument vary over time due to the effects of varying temperature, humidity, CO2 concentration and condensation in the bore? This page gives a musician-friendly summary of a scientific study we conducted to answer these questions. More details are given in the paper.
Although we used human players to 'warm up' the instrument, we measured the resonance frequencies of the instrument alone, without a player. One reason, of course, is that good human players can and do compensate for the instrument and tend to play
in tune independently of the instrument. Another is that instruments are a bit more reproducible than people. The trombone was a suitable choice for detailed study because it is a medium sized brass instrument, and brass mouthpieces have the advantage that we could attach our measurement instruments to the mouthpiece within a couple of seconds of the musicians ceasing to play. This cannot easily be done with woodwinds. We comment below about the likely effects of different instrument sizes and materials.
Here, the solid curves show
change in the frequencies of the playing resonances plotted against time of playing (on a log scale). At right, these changes are converted into the changes of pitch that would result (in the case of no compensation from the player). The dashed curves show +/– one standard variation, and give an indication of the range of values for different players and measurements. (The red curves are the averages for the the minima in the acoustic impedance spectrum (flutes play at minima); the black curves the maxima (all other wind instruments play at maxima). There is no reason why maxima and minima should vary in different ways.)
First, we should note that the thermal expansion or contraction of the instrument itself is completely negligible in comparison with changes in the gas inside: metals expand by only about .001% to .002% per °C. All else equal, the pitch frequency is proportional to the speed of sound in the gas in the bore, and the speed of sound depends on the temperature and the density of the gas (and to a smaller extent on its heat capacity).
A simplified explanation involves primarily three effects:
1. Exhaled air has an increased concentration of CO2. This molecule is heavier than the average for air (which is roughly 80% N2 and 20% O2). Higher density lowers the speed of sound, and the CO2 in exhaled air is the reason for the downward slope of the graph for the first few seconds. (Exhaled air also has lower O2 concentration, but this has less effect because the mass of O2 is close to the average for air.)
2. With playing, the air inside becomes warmer. The speed of sound in air is proportional to the square root of the absolute temperature. Normal temperatures are roughly 300 K, so it takes about 6°C to change the speed of sound by 1% (K is for kelvin, the units for absolute temperature. A temp difference of 1°C equals a temperature difference of 1 K. Add 273 K to the temp in °C to convert.)
3. Another important thing is the increaing humidity of the air in the instrument: your breath is nearly saturated at 37°C and when it cools in the instrument, water condenses on the metal providing a water reservoir to keep the air nearly saturated in the instrument. Further, saturated warm air has a higher concentration of water vapour than saturated cold air.
The speed of sound is inversely proportional to the square root of the average molecular mass of the air. Water molecules are lighter than nitrogen or oxygen, so humid air is less dense than dry air, all else equal. (Yes, I know that non-scientists talk about humid air being 'heavy', but I think that by this they mean that one sweats less effectively in humid air.) These effects of humidity and temperature add up, so it's easy for them to produce 1% or even 2% or more increase in speed of sound, and therefore in playing frequency. A semitone is only 6%, so even 1% is a lot. Happily for musicians, the effect of CO2 substantially offsets the effect of higher temperature and humidity.
Summarising the calculations: 1% increase in frequency requires either a 6°C increase in temperature, a 3% increase in water concentration or a 6% fall in CO2 concentration, or some linear combination of these (where concentrations are measured as a fraction of all gases present). See the paper for details.
A surprise. These results surprised me when I first saw them: from my experience playing, I hadn't expected the initial fall to be so large. If I assemble the instrument and play, I notice an increase in pitch over time, but don't notice a big initial fall. On reflection, I realised that I didn't think too much about the pitch in the first three seconds of playing a cold dry instrument: I'm checking that everything works. So my perceptual graph would be mainly the right hand side of the graph shown above. More about this below.
Changes in quality and width of resonances
When physicists talk about the quality factor Q of a resonance, we mean the ratio of its frequency to its bandwidth. A high Q resonance has tall maxima and deep minima and sharp peaks and troughs. For musicians, a small leak or roughness in the bore reduces the Q factor. A high Q resonance is generally easier to play, and its pitch is harder to 'bend'.
However, the changes shown in the graph below are small compared with those from a noticeable leak and probably difficult to detect subjectively by playing.
As you can see, we found
that the Q factor decreased a few percent in the first few seconds, and then slowly decreased further. (The magnitudes of the peaks in acoustic impedance decreased, those of the minima increased, and the bandwidth of both increased.) Again, the fall in the first few seconds is probably not noticeable, because, for me at least, the embouchure and sensory feedback system are adapting over that time.
The obvious cause of a lower Q is acoustic losses in the system. We propose that water condensed on the walls is responsible. A tiny amount of condensation and evaporation in each cycle of a standing wave takes energy out of the wave and increases the losses. The extent of wet walls gradually increases as humid air continues to flow into a cool instrument, giving a probably monotonic decrease in the Q factor.
What about other instruments?
There are a number of reasons that make it difficult to do measurements like these, so we have not done them for other instruments. We'd expect qualitatively similar results for nearly all wind instruments played by humans but there would quantitative differences, which we'll now discuss.
First, woodwinds have the interesting difference that, for most of their length, the bore is connected to the outside air via tone holes, whose state (open or closed) depends on the music. So the lower bore would in general contain a mix of breath and environmental air. Except for the flute family, woodwinds have reeds. The pitch depends on the mechanical properties of the reed, and these vary with humidity and time of playing. (Players usually wet the reeds well before playing.)
In some instruments, including the piccolo and the flute, the air in the bore is replaced rapidly and the instrument heats up reasonably quickly. A tuba takes a long time to fill with breath and the temperature of most of the bore remains close to ambiant. In all these three, the rate of exhalation is high, so we'd expect relatively low CO2 concentration (players breathe to replenish air, not oxygen). At the other extreme, the oboe has a very low flow rate, so the CO2 concentration can rise considerably if the player plays a long phrase. The warming up of wooden instruments is complicated because wood is not a good thermal conductor, so the bore can warm up faster than the outside of the instrument.
Other instruments and more measurements? As we mention above, brass instruments have the advantage for these measurements that it is relatively easy to take the instrument from the player, attach it to the impedance spectrometer and start a measurement in a second or three. That is rather harder with the more difficult mouthpiece geometries of woodwinds. We haven't done trumpets or tubas largely because it's a lot of work to get in a bunch of volunteers and make the measurements. Why are measurements challenging? Well, it's not hard to measure impedance spectra precisely if one has plenty of time to attach the instrument, check the seals, and measure over a several seconds. One problem for measurements like ours is the (musician's) Uncertainty Principle: a fast measurement gives poor frequency resolution, and a long one gives poor time resolution. For example, how long do you think you'd need to measure 100 Hz with a precision of 0.1%? And how would this affect the extrapolation back to time zero? Using higher resonances than the fundamental helps, but there are limits because the higher resonances are weaker. See the paper for details.
Measuring played notes is possible, of course, but musicians (especially good ones) tend to compensate for changes in the instrument. If you're thinking of doing measurements, you are welcome to ask us questions or even to send us your planned method for comments.