How do musical frequencies relate to the numbers in this signal decomposition?

[bobsmith76]

My book reads

step 1 = 2 cos(654πt)cos(-130πt)
step 2 = 2 cos(2 * 327 * π * t)cos(130πt)
step 3 = f₂ - f₁ = 392 - 262 = 130 Hz

t = seconds
π = pi

I don't see how they get from step 1 to step 2, nor do I understand how they get from step 2 to step 3.

 

[uart]

I don't see how they get from step 1 to step 2

654 = 2*327, and cos() is an even function.

 

[bobsmith76]

ok, where did the numbers 392 and 262 come from? What happened to pi?

 

[uart]

ok, where did the numbers 392 and 262 come from? What happened to pi?

Frequency (in Hz) is equal to the radian frequency divided by 2 pi.

cos(a-b) + cos(a+b) = 2 cos(a) cos(b)

You are given the "a" and the "b" and you need to find the corresponding "a+b" and "a-b" of the decomposed (into individual cosine wave) signal.

 

[CellCoree]

Musical frequencies are closely related to the numbers in this signal decomposition because they represent the different components or frequencies that make up the overall signal. In this case, the signal is being decomposed into two cosine functions, each with a different frequency.

In step 1, the frequency of the first cosine function is 654πt, which can be simplified to 2 * 327 * π * t. This means that the signal has a frequency of 327 Hz, which is equivalent to a C# note on a musical scale.

In step 2, the frequency of the second cosine function is 130πt, which is the same as the frequency of the first cosine function in step 1. However, the amplitude of the second cosine function is multiplied by 2, resulting in a stronger presence of this frequency in the overall signal.

In step 3, we can see that the difference between the two frequencies (f2 - f1) is 130 Hz. This is the frequency of the note that is being played, which is a G# note on a musical scale. This shows how the different frequencies in the signal relate to the notes on a musical scale.

As for the use of pi (π) in the equations, it is a mathematical constant that is used to represent the relationship between a circle's circumference and its diameter. In this context, it is used to calculate the frequencies of the signal components.