The speed of sound in air (in meters per second) is 331.5 + 0.6 T, where "T" represents the temperature in Celsius degrees.
At room temperature (20 degrees Celsius) that comes out to 343.5 m/sec. (Actually, the equation above is an approximation. The speed of sound in a gas is proportional to the square root of its absolute (Kelvin) temperature, so the graph of sound speed vs. temperature is parabolic. But in the narrow range of temperatures where we live and work that relation is very nearly linear.)
For a one-degree increase in temperature the speed of sound increases by 0.6 meters per second.
In percentage language that's 0.6 ÷ 343.5 = 0.00175 = 0.175%.
The frequency of a standing wave in a wind instrument is proportional to the speed of sound. Therefore the fractional or percentage change in frequency due to a temperature change will be equal to the corresponding fractional or percentage change in sound speed. (The effect of thermal expansion of the brass is insignificant compared to that of the air.)
This means the frequency of a note played on a wind instrument will want to increase by about 0.175% for each Celsius degree of temperature increase.
A semitone is a 5.95% change in frequency. Since 0.175 divided by 5.95 comes out to be 0.03, we expect a pitch change of about 3 cents for each one-degree (C) change in temperature. (Remember, a "cent" in this context is one hundredth of a semitone, just as a semitone is one twelfth of an octave.)
A Celsius degree is 1.8 Fahrenheit degrees. Increasing room temperature by one Celsius degree is like changing it from 68 to almost 70 degrees Fahrenheit, which feels like going from "a bit cool" to "kind of warm". But a 3-cent change in pitch is not enough to make you want to adjust a tuning slide.
As explained earlier, the second-valve tubing length is about 0.0595 times the overall effective length of the instrument. 0.175% of the overall effective length is about 3% of the second-valve tubing length. On a BBb tuba the second valve tubing is about 6 inches (15 cm.) on each side. Three percent of that length is 0.18" or 0.45 cm. That is the amount of main tuning slide adjustment theoretically needed on a BBb tuba to compensate for a one-degree (C) change in temperature.
To make this more practical, let's consider a change in temperature of ten Fahrenheit degrees: (10 deg F)(5/9) = 5.55 deg C.
5.55 × 0.18 inch = about one inch. . . . . OR, . . . . 5.55 × 0.45 cm = about 2.5 cm.
That is the amount of main tuning slide adjustment needed theoretically to compensate for a ten-degree (F) change in temperature on a BBb tuba. On a CC tuba the adjustment would be about 0.88 times that amount, or about 7/8". On a euphonium or trombone it would be about half an inch.
As you play on a cold instrument it will gradually warm up, of course, so listening and adjusting are always necessary.
Next: Why is the Fifth Partial So Flat?
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