The pitch of a musical note is determined by its frequency in hertz (Hz). The tr

Question

The pitch of a musical note is determined by its frequency in hertz (Hz). The trigonometric function p(t) = a sin(2n/r) approximates the sound of a note with volume a, frequency/, and a duration of t seconds. For the purposes of the following problems, we will assume all pitches are played at a volume where a = 1. Within a major scale, the frequencies of the notes maintain certain ratios to the first note of the scale. The first note to the third note have a frequency ratio of 4:5, the first to fifth a ratio of 2:3, the first to seventh a ratio of 8:15, and the first to the octave (eighth) a ratio of 1:2. Below, you will see the graph of the G below middle C at 196 Hz in black, alongside another pitch in grey as well as the eight notes of an octave of the G major scale with their pitch names.

Explanation / Answer

Consider the black wave (given as G below middle C). Note that one complete oscillation (between 2 crests) occurs in a time period equivalent to 6 squares.

The grey wave has 5 complete oscillations in 24 squares. This means that it's time period is 24/5 = 4.8 squares. Note that the frequency will be higher than the black wave, as the time period is less.

The ratio of these two is 6/4.8, or 5/4. Hence, the frequency of the grey wave is (5/4)*196 = 245 Hz.

Also, as given in the question, the first note to the third note has a frequency ratio of 4:5. Hence, the grey wave corresponds to B below middle C.

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