MUSIC THEORY FUNDAMENTALS - Chapter 4, Exercise 3 Solution
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Why Both the Perfect Fourth and Perfect Fifth Are Consonant

PROBLEM
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Explain, using Chapter 1's own overtone-series material, why the perfect
fourth (4:3 ratio) and perfect fifth (3:2 ratio) are both considered
highly consonant, even though the perfect fourth's ratio is slightly
less simple than the fifth's.

SOLUTION
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Chapter 1 established that consonance comes from how simple (how small
the integers are in) a frequency ratio is - the smaller the numbers, the
more the two notes' own overtone series overlap, and the more consonant
they sound. The perfect fifth's 3:2 ratio and the perfect fourth's 4:3
ratio are both genuinely simple by that same real standard, even though
3:2 is technically simpler than 4:3 (smaller numbers overall).

What matters for the "perfect" classification isn't that an interval has
THE single simplest possible ratio - it's that it's simple relative to
the major/minor intervals it's being contrasted against. Compare 4:3 (the
fourth) against, for example, a major third's real 5:4 ratio or a major
second's real 9:8 ratio: 4:3 is still meaningfully simpler than either of
those. Both the fourth and fifth sit in a genuinely different, simpler
tier of ratio than any major or minor interval does, which is exactly why
both - not just the fifth alone - earn the "perfect" classification.

ANSWER: Both the perfect fourth and perfect fifth are highly consonant
because both correspond to real, low-integer frequency ratios (4:3 and
3:2) that are simpler than any major or minor interval's own ratio - the
fourth doesn't need to match the fifth's exact level of simplicity to
still qualify as genuinely, comparatively simple, and therefore "perfect."

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WHY THIS WORKS AS AN ANSWER

This directly extends Chapter 1's own core claim - simple integer ratios
produce consonance - by showing that "perfect" is a real category with
some internal range, not a single fixed simplicity threshold. Chapter 4's
own semitone-count table already places the fourth and fifth in the same
"perfect" category together, distinct from every major/minor interval;
this exercise explains the real acoustic reasoning behind why that
grouping makes sense despite the two ratios not being identically simple.