Acircle of fourths is generated by starting at any note and stepping upward by intervals of a fourth (five half-steps). by what factor is the frequency of a tone increased if it is raised by a fourth? how many fourths are required to complete the entire circle of fourths? how many octaves are covered in a complete circle of fourths?

a) Q₀ × 1.335 cps

b) 12 fourths

c)  5 octaves

Step-by-step explanation:

The circle of fourth is generated by starting at any note and stepping upward by intervals of a fourth(five half-steps)

(a)  By what factor is the frequency of a tone increased if it is raised by a fourth?

consider the following exponential growth formula :

Q = Q₀ × f °

Q = Q₀ × 1.05946 ⁿ

Substituting n = 5

Q = Q₀ × 1.05946 ⁿ

Q = Q₀  × 1.05946⁵

Q = Q₀  × 1.335 cps

Therefore, note that  five half-step higher will be increased by factor 1.335

(b)  How many fourths are required to complete the entire circle of fourths?

SOLUTION

In each step, number of half steps = 5

Total number of half steps in one octave = 12 half- steps

Therefore, total number of fourths required to complete the entire circle of fourths = 12 fourths

(c) Total number of half steps in one octave = 12 half-steps  

Total half steps in complete circle of fourths  

= 12×5  

= 60 half-steps

Calculating number of octaves (dividing it by 12)  

= 60/12  

= 5 octaves

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