Answer:
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
It is given that the circle of fourth is generated by starting at any note and stepping upward by intervals of a fourth(five half-steps)
[tex]Q = Q_0 \times f ^0\\Q = Q_0 \times 1.05946^n[/tex]
Put n = 5 in the above formula, we get
[tex]Q = Q_0 ( 1.05946 ^n)\\Q = Q_0 ( 1.05946 ^5)\\Q = Q_0 (1.335) cps[/tex]
So, by 1.335 factor the frequency of a tone increased if it is raised by a fourth.
Total number of half steps in one octave = 12 half- steps
So, total number of fourths required to complete the entire circle of fourths = 12 fourths
Total half steps in complete circle of fourths will be 12×5 = 60 half-steps
Now, the number of octaves is [tex]\frac{60}{12} =5[/tex] octaves.
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