Take the length of a trombone to be 275 cm in the first position. How far does the slide have to be moved to lower the pitch one semitone? [Remember that the length is increased by twice this amount.]
To find out how far the slide has to be moved to lower the pitch by one semitone, we first need to know the ratio of frequencies between two adjacent semitones. In equal temperament, which is commonly used in Western music, each semitone represents a frequency ratio of 2^(1/12) ≈ 1.0595.
Now, let's break down the problem. We know that when the slide is in the first position, the length of the trombone is 275 cm. When the slide is extended to lower the pitch, the total length of the air column increases by twice the amount the slide is moved.
Let's represent the distance the slide needs to be moved as x cm. Therefore, the increased length of the air column will be 2x cm.
To find the frequency ratio when the length is increased by 2x, we can use the formula:
Frequency ratio = (Length + 2x) / Length
Substituting the given values:
Frequency ratio = (275 + 2x) / 275
Now, we know that the frequency ratio should be ≈1.0595 because we want to lower the pitch by one semitone. So we can set up the equation and solve for x:
1.0595 = (275 + 2x) / 275
To isolate x, let's first multiply both sides of the equation by 275 to get rid of the fraction:
275 * 1.0595 = 275 + 2x
Now, let's solve for x:
291.8625 = 275 + 2x
Subtracting 275 from both sides:
291.8625 - 275 = 2x
16.8625 = 2x
Finally, divide both sides by 2 to solve for x:
x = 16.8625 / 2
x ≈ 8.43 cm
Therefore, to lower the pitch by one semitone, the slide needs to be moved approximately 8.43 cm.