MUSIC THEORY FUNDAMENTALS - Chapter 5, Exercise 3 Solution
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Confirming the 4th-to-6th-Harmonic Ratio Is a Perfect Fifth

PROBLEM
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Using this chapter's own real 4:5:6 harmonic-ratio explanation, show the
arithmetic confirming that the interval between the 4th and 6th
harmonics really is a perfect fifth (3:2 ratio), not some other interval.

SOLUTION
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The 4th harmonic of a fundamental frequency f is at 4f. The 6th harmonic
is at 6f. The ratio between them, higher over lower:

    ratio = 6f / 4f = 6/4

Simplify the fraction 6/4 by dividing both numbers by their greatest
common factor, 2:

    6/4 = (6 divide by 2) / (4 divide by 2) = 3/2

ANSWER: The ratio between the 6th and 4th harmonics simplifies exactly
to 3:2 - the same real ratio Chapter 1 established as the perfect
fifth's own defining frequency ratio. This confirms the interval from
the 4th harmonic to the 6th harmonic really is a perfect fifth.

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WHY THIS WORKS AS AN ANSWER

This is the same reduction technique used throughout this course
whenever comparing harmonics: express both frequencies as multiples of
the shared fundamental (4f and 6f), form the ratio, and simplify. The
fact that 6:4 reduces to exactly 3:2 - and not to some other, more
complex fraction - is precisely why the major triad's outer two notes
(root to fifth) sound as consonant and stable as the fifth interval
itself does on its own, since they share the exact same underlying
ratio.