# The origin of constants in mel-scale formula

To convert frequency to mel, we usually use formula:

$$\mathrm{mel}(f) = 1127 \ln \left(1 + \frac{f}{700}\right)$$

I wonder where are $1127$ and $700$ came from?

I've read the paper which Wikipedia think is the original of MFCC but I still didn't get the origin of those values. I hope somebody here can help me to understand the origin of those values.

The different mel scale formulas are supposed to approximate the human ear's critical bandwidths. Usually the formulas have this form:

$$m=C\log\left(1+\frac{f}{f_0}\right)\tag{1}$$

Below the frequency $f_0$ the mel scale changes approximately linearly with frequency, whereas above $f_0$ it changes logarithmically. This is the result of measurements. The difference between most formulas is the choice of the corner frequency $f_0$, which is usually chosen somewhere between $600\,\text{Hz}$ and $1000\,\text{Hz}$. So the number $700$ in your formula is the corner frequency $f_0$ where the scale changes from linear to logarithmic. The constant $C$ in $(1)$ is normally chosen such that $1000\,\text{Hz}$ correspond to $1000$ mel:

$$C=\frac{1000}{\log(1+1000/f_0)}\tag{2}$$

If we take the natural logarithm and with $f_0=700$ we get from $(2)$ $C=1127$. If instead we use the logarithm with base $10$, we obtain the other well-known constant $C=2595$.

• wow thank you for your explanation, but I have one question, why we choose between 600-1000hz? – malioboro Jan 7 '18 at 3:14
• @malioboro: That's just what came out of listening experiments. – Matt L. Jan 7 '18 at 7:53
• oh I see.. okay, thank you for your nice explanation – malioboro Jan 7 '18 at 8:15