12. I place a finger 1/3rd of the way down a guitar string to dampen it, and pluck it. Then
I move my finger to 3/4ths of the way down, and pluck the string again. What is the
frequency ratio of the first note to the second note:
a. 1:1
b. 1:4
c. 1:3
d. 3:4

Placing a finger lightly on a guitar string at specific node positions isolates harmonics, producing distinct pitches. At 1/3 down the string, the third harmonic dominates with frequency 3 times the fundamental; at 3/4 down, the fourth harmonic sounds at 4 times the fundamental. The frequency ratio of the first note (1/3 position) to the second (3/4 position) is thus 3:4.​

Harmonic Physics Basics

Guitar string frequency depends on length, tension, and mass density, with standing waves forming modes where nodes are fixed points. Lightly touching a node suppresses lower modes while allowing higher harmonics to ring, as the finger blocks motion without fully stopping the string. The fundamental frequency $f$ relates to higher modes by integer multiples: nth harmonic at $nf$.​

Position Analysis

At 1/3 string length (near 7th fret), the finger sits at a node for the third harmonic ($\lambda /3$ divisions), damping the fundamental and second harmonic. This yields frequency $3f$. At 3/4 length (near 4th/5th fret area), it aligns with the fourth harmonic node ($\lambda /4$ divisions), producing $4f$.​

Option Breakdown

• a. 1:1: Matches identical frequencies, but 3f versus 4f differs; incorrect.​

• b. 1:4: Implies second note four times higher, reversing the 3:4 order; incorrect.

• c. 1:3: Suggests first note one-third the second, but ratio is 3/4, not 1/3; incorrect.

• d. 3:4: Correct, as first (3f) to second (4f) gives 3:4.​

Answer and Key Takeaway

The correct choice is d. 3:4. This demonstrates how node positions dictate harmonic selection on guitar strings.​