#### Cents, powers and logarithms

Cents are based on the mathematical concept of logarithms. Logarithms were introduced by the 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:

$$2^4 = 1 * 2 * 2 * 2 * 2 = 16$$

This is a tricky example that shows why we should start with 1:

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

The following figure illustrates the relationship between powers and logarithms:

Two, raised to the third power equals 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, what power do we have to raise 2 (base 2) to get 8? The answer is 3.

In music we use base 2 logarithms because octaves inherently use this base. To find the frequency of a note an octave away, we need to multiply the frequency of the first note by two. Let us take C2 at 60 Hz. This gives us the frequency of C3:

$C3 = 60 * 2 = 120$

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.$C5 = C2 * 2^3 = 60 * 8 = 480$

Now let's use logarithms:

• Let's use the frequencies of C5 (480 Hz.) and C3 (120 Hz.). The notes are at two-octave distance.
• We divide the frequencies and the result is 4:

$$480 / 120 = 4$$

• What is the logarithm base 2 of 4, or what power must we raise 2 to get 4? The answer is 2, just as the number of octaves.

$$log_2(4) = 2$$

• If we multiply C2's frequency $$2^2$$ we get C4's frequency:

$$120 * 2^2 = 480$$

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

Inadvertently, we are almost at Ellis' formula for calculating cents:

We divide the note frequencies and find the base 2 logarithms of the result.

There is only one small detail missing. We will see this in the next section.

It can all be confusing. It is like a labyrinth of mirrors in which it is easy to get lost. You have to go through it many times, with a lot of patience, and little by little you will find your way...

Let's summarize the mathematical concepts learned:

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

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