MUSIC THEORY FUNDAMENTALS - Chapter 8, Exercise 3 Solution
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Why a Half Cadence Sounds Incomplete

PROBLEM
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Explain, in your own words, why a half cadence (ending on V) sounds
"incomplete" to a listener while a perfect authentic cadence (V-I, both
root position) sounds fully resolved - referring to Chapter 7's own real
scale-degree names where useful.

SOLUTION
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Chapter 7 named the tonic (scale degree 1) the real "tonal center" and
"note of final resolution" - by definition, the one note a piece's own
harmony is always oriented toward and expected to eventually settle on.
The dominant (scale degree 5) was named "second in importance to the
tonic" - a real, strong pull toward the tonic, but genuinely still not
the tonic itself.

A perfect authentic cadence ends by actually arriving at that real
point of final resolution - the tonic chord, in root position, with the
tonic note on top. There is nowhere harmonically "further" to resolve;
the ear's own expectation, built up by the preceding dominant, is fully
satisfied.

A half cadence, by contrast, ends by stopping on the dominant itself -
the chord whose entire real character (per Chapter 7's own naming) is to
create a pull TOWARD the tonic, not to be a resting point on its own.
Ending there leaves that real pull unresolved and audibly "hanging,"
since the ear has been primed to expect one more real step that the
phrase simply doesn't take.

ANSWER: A half cadence sounds incomplete because it ends on the
dominant - a chord defined by its own pull toward the tonic - without
ever actually taking that final step. A perfect authentic cadence sounds
resolved because it does take that step, arriving at the tonic itself,
the real point of final resolution Chapter 7 already named it.

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WHY THIS WORKS AS AN ANSWER

This directly reuses Chapter 7's own real etymological definitions of
"tonic" and "dominant," showing that the felt difference between these
two cadence types isn't arbitrary - it follows directly from the real,
named function each scale degree already carries before any specific
cadence is even built from it.