MUSIC THEORY FUNDAMENTALS - Chapter 1, Exercise 2 Solution
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Calculating a Perfect Fifth from A440

PROBLEM
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Using the real 3:2 frequency ratio for a perfect fifth, calculate the
frequency of the perfect fifth above A440 (440 Hz). Round to the nearest
whole Hz.

SOLUTION
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A perfect fifth above a given frequency is found by multiplying by 3/2
(the real 3:2 ratio, higher note over lower note):

    fifth above = 440 Hz x (3 / 2)
    fifth above = 440 Hz x 1.5
    fifth above = 660 Hz

ANSWER: The perfect fifth above A440 (440 Hz) is 660 Hz.

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WHY THIS WORKS AS AN ANSWER

The 3:2 ratio comes directly from the overtone series this chapter
covered: the 3rd harmonic of a fundamental, divided by its 2nd harmonic,
produces exactly a 3:2 relationship - the real physical basis of the
perfect fifth's own consonance. Multiplying 440 Hz by 3/2 gives 660 Hz
exactly, with no rounding needed in this particular case, since 440 is
evenly divisible by 2. In real Western music, the note a perfect fifth
above A is E - so 660 Hz corresponds to the note E5 under this simple
"just intonation" tuning. (Note: the real equal-tempered tuning system
used on modern pianos adjusts this slightly, to about 659.26 Hz, so that
every key can modulate cleanly - a refinement covered later in this
course.)