What is really the Mel-filterbank? After applying STFT to a signal, you typically need to convert the frequencies to the Mel scale. Typically the frequency values range from $0$ to $N/2$, where $N$ is the width of each sliding window.
I have read so many articles about Mel Spectrogram. Basically they all say to convert frequencies to Mel Scale in the following way:

Partition the Hz scale (i.e. partition a sequence of numbers ranging from $0$ to $N/2$). In this way we will have a number of bins. This number is for example indicated in librosa by &quot;n_mel&quot;. These bins are called frequency bands.

Covert each bins into Mel Scale using the formula $2595*log(1+f/700).$

Application of triangular filters for each bins to capture the energy at each frequency band and approximate the spectrum shape.


It seems that when this is buried in the articles, nobody really knows why these operations are carried out. I find it difficult to fully understand:

Why divide the Hz scale into bins?

What exactly is meant by a frequency band?

Why do we apply triangular filters?

What exactly is a triangular filter? How is it defined mathematically and from what concept?


Can anyone explain exactly these things to me or recommend a book where these steps are explained in detail? There seems to be no book that explains these concepts exactly.
This was my first google hit: https://medium.com/analytics-vidhya/understanding-the-mel-spectrogram-fca2afa2ce53
It seems to explain both how and why?
Mel frequency binning is afaik only used when human perception is involved. Either directly, or for signals that have probably been adapted to human hearing (such as human speech). It is a way to represent signal information in a way that mimics «low level auditory perception», or could be seen as removing perceptually redundant information.
I have not understood why it is that we generally have the same temporal reolution in narrow low frequency bands, as in wide high frequency bands. Historically, it may have been due to limited processing power and the efficiency of a single STFT windowing process. And/or it could reflect the limited neuron firing rate after the acoustic-mechanical filtering carried out by our ear.
But from a dsp perspective, I would expect a variable time/frequency resolution, where high &amp; wide bands gets more temporal resolution.