Question

The lowest note on a guitar is typically tuned to an E2. The next lowest string is tuned to an A2 at 110 Hz, which is a fourth (5 half steps) higher than the E2. The next string is tuned to D3, which is a fourth higher than A2.

(a) What are the frequencies of the lowest three open strings on a guitar tuned this way?

(b) If I play the open E2 on the guitar, and then push the A2 string down at the second fret, to increase its frequency by two half steps, the two strings will make a perfect fifth interval (7 half steps). If I do this and then play the two strings, what is the dominant beat frequency I’m most likely to hear?

(c) If I play the open E2 and the open A2, the interval I get is a fourth. If I play these two opens strings together, what is the dominant beat frequency I’m most likely to hear? Hint: a “just” fourth, for which there is no beating, involves a frequency ratio of 4/3.

Answer 1

a)The fourth interval has a ratio 4/3

So, D3 / A2 = 4/3

D3/110 = 4/3

D3= 4/3 x 110 = 146.67 Hz

Now, Next string is tuned to G3

G3/D3 = 4/3

G3 = 4/3 x 146.67 = 195.56 Hz

Now next string is tuned to B3 by the third interval(5/4 ratio) in standard guitar tuning. But in our case, it is tuned to C4

C4/G3 = 4/3

C4 = 4/3 X 195.56 = 260.74 Hz

Next string is tuned to F4

F4/C4 = 4/3

F4 = 4/3 x 260.74 = 347.65 Hz ( In standard guitar tuning it is tuned to E4).

(b)

Now, A2/E2 = 4/3

E2 = 3/4 x 110 = 82.5 Hz

Second fret on A2 string is B2.

And Perfect fifth has a ratio 3/2

So, B2/E2 = 3/2

B2 = 3/2 x 82.5 = 123.75 Hz

Beat frequency = difference in frequencies = 123.75 - 82.5 = 41.25 Hz

(c) Now beat frequency = A2 -E2 = 110 - 82.5 = 27.5 Hz