MUSIC THEORY FUNDAMENTALS - Chapter 1, Exercise 1 Solution
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Calculating Octaves from a Frequency Ratio

PROBLEM
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A note is being played at 220 Hz. Using the real definition of the octave
as a 2:1 frequency ratio, calculate the frequency of the note exactly one
octave above it, and the frequency of the note exactly one octave below it.

SOLUTION
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An octave above a given frequency is found by multiplying by 2 (the real
2:1 ratio, higher note over lower note):

    octave above = 220 Hz x 2 = 440 Hz

An octave below a given frequency is found by dividing by 2 (the same 2:1
ratio, applied in the opposite direction):

    octave below = 220 Hz / 2 = 110 Hz

ANSWER: One octave above 220 Hz is 440 Hz. One octave below 220 Hz is
110 Hz.

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WHY THIS WORKS AS AN ANSWER

The octave's defining 2:1 ratio means that moving "up" an octave always
means doubling the frequency, and moving "down" an octave always means
halving it - regardless of which specific note you start from. This is
exactly why 220 Hz, 440 Hz, and 110 Hz are all recognized as the "same"
note, A, just in different octaves: A2 (110 Hz), A3 (220 Hz), and A4
(440 Hz) are three real, standard octave positions of the same pitch
class, each one double the frequency of the one below it. A4 = 440 Hz is
the real modern reference pitch this chapter covered.