Cents, powers and logarithms

Cents are based on the mathematical concept of logarithms. Logarithms were introduced by Scottish mathematician John Napier (1550 / 4-1617). The name comes from logos (ratio) and arithmós (number). Logarithms are related to powers.

Two to the third power or \( 2^3 \) equals 1 x 2 x 2 x 2:

\( 2^3 = 1 * 2 * 2 * 2 = 8 \)

Two to the fourth power:

\( 2^4 = 1 * 2 * 2 * 2 * 2 = 16 \)

A confusing example that shows why we should start with 1:

\( 2^0 = 1 \) (1 multiplied by no 2)

After understanding the powers, the following image illustrates its relationship with logarithms:

Two, raised to the third power is equal to 8 ( \( 2 ^ 3 = 8 \) ). The logarithm indicates the power to which we must raise 2 to obtain 8. The second line reads: base 2 logarithm of 8 = 3. In other words: to what power do we have to raise 2 (base 2) to obtain 8? The answer is 3.

In music, we use base 2 logarithms because octaves use this base naturally. To find out the frequency of a note at an octave distance, we must multiply the frequency of the first note by two. Starting from a C2 with 60 Hz. We obtain the frequency of C3 in this way:

\[ C3 = 60 * 2 = 120 \]

To get to C4, we multiply by 2, twice:

\[ C4 = 60 * 2 * 2 = 240 \]

Or we can use powers:

\[ C4 = 60 * 2^2 = 60 * 4 = 240 \]

The exponent used is related to the number of octaves:

Note Octaves Power Frequency
\[ C2 \] 0 0\[ 2^0 = 1 \] 60 Hz. \[ 60 * 2^0 = 60 * 1 = 60 \]
\[ C3 \] +1 1\[ 2^1 = 2 \] 120 Hz.\[ C3 = C2 * 2^1 = 60 * 2 = 120 \]
\[ C4 \] +2 2\[ 2^2 = 4 \] 240 Hz.\[ C4 = C2 * 2^2 = 60 * 4 = 240 \]
\[ C5 \] +3 3\[ 2^3 = 8 \] 480 Hz.\[ C5 = C2 * 2^3 = 60 * 8 = 480 \]

Let's now use logarithms:

\( 480 / 120 = 4 \)

\( log_2(4) = 2 \)

\( 120 * 2^2 = 480 \)

We achieve the same result by subtracting the powers. The power needed to reach C5 from C2 is 3 (see the table above), that of C3 1. If we subtract 3 - 1 we obtain 2. The same result that we obtained with \( log_2 (4) = 2 \).

Unintentionally, we have almost arrived at Ellis's formula to calculate the cents:

we divide the frequencies of the notes and look for the base 2 logarithms of the result.

Only a small detail is missing. We will see that in the next section.

It would help if you went through it many times, with a lot of patience, and you will find your way...

Let's summarize the mathematical concepts learned:

Powers: \( \color{red}2 \color{blue}^3 \color{black}= 1 * 2 * 2 * 2 = \color{green}8 \) Multiply 1 by 2 three times
Logarithms: \( log\color{red}_2 \color{black} ( \color{green}8 \color{black} ) = \color{blue}3 \) At what power should we raise 2 to get 8?

If you can go from powers to logarithms and vice versa, you may consider the concept understood!

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