# 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.

## 1 Answer

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
• is there any refferences where I can read more about this? – malioboro Jan 11 '18 at 18:33
• The paper you linked to in your question is actually a standard reference. – Matt L. Jan 11 '18 at 21:05