# How to Calculate the Difference of Pitch Standards in Cents

###### tags: `Music Theory` `Math` `Tuning`

:::info
Under the most common tuning standard today,
the **Equal temperament** (12-TET),
the difference of frequency between any half step is
$\log_{12}2=2^{1/12}\approx1.05946$.
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What's a half step?

The interval(Difference in frequencies) difference
between any two adjacent notes on a piano.

For example:

X:1
L:1/2
|c, _d, | c b,|

But that's way too big to describe and compare the difference between tuning systems.
Hence, **cent** was invented.
:::info
The definition of a cent in musical tuning is $\log_{1200} 2=2^{1/1200}\approx1.00057779$

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To calculate the difference between any 2 frequencies, we have to consider
how many times do we have to multiply a certain frequency $A$ to $2^{1/1200}$
to get $B$

Therefore, assume $x$ is the times we have to multiply:

$$A\times (2^{1/1200})^x= B$$

$$(2^{1/1200})^x=\frac{B}{A}$$

so

$$\begin{aligned}x &=log_{\left(2^{1/1200}\right)}\frac{B}
{A} \\ &= 1200\times log_{2}\frac{B}{A} \end{aligned}$$

$x$ will be the interval in cents we are looking for.

Some calculators only have $log_{10}x$ function.
Then, enter

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$$x=1200 \times (log\frac{B}{A}) \div log2$$
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