I am trying to detect the frequency of a bell chime.
I am planning to use the simple electret micrphone to detect the sound
I was looking for the tutorials but there seems to have many varieties of them, one diffrent then the other.
I don't know which is better, so I ask for help

what is the best method to calculate frequency in arduino?

This is a difficult problem, and there are a variety of mathematical techniques for dealing with it.

I don't know much about it myself, but you may need to do a 'fourier trasform' of the input data. There's a thing classed 'fast fourier transform' - FFT, and probably some arduino libraries, but all I know about it is the name.

If you know which particular frequencies the bell might have, then this is a bit easier.

Let's say your bell has a frequency of 100hz.

Prepare a table of sines and cosines. This step must be fast.

Sample your signal. Try not to accidentally take samples at exact multiples of the frequency you are after. Each time you sample it, work out what the sine and cosine of a 100Hz signal would be at the sample time. Multiply the signal by that sine and cosine, and accumulate the two values. One way to do this is to sample for a second, check the results, and discard. Another way is to multiply the accumulated value by something less than 1 each time you take a sample, and add the new value.

In any case, if 'S' is the sum of the sines and 'C' the sum of the cosines, calculate SS+CC. There's no need to take the square root, because we are just looking for a relative level. That value indicates the 'strength' of the detection of a 100Hz signal. When it's big enough (and it will take experiment to find that point), then the sound has been detected.

If you only want to estimate the single loudest frequency of the sound samples, you can use a pitch detection algorithm - those work well with quasiperiodic or oscillating signals such as a simple monophonic doorbell chime. (If your bell has multiple very different frequencies then it's a different issue and you will need a FFT)

There are many algorithms, the easiest ones being based on estimating the period of the signal, then inverts that value to get the frequency.

if you want to identify just 1 given bell and If your bell has a simple stable frequency (monophonic), then a simple approach is be to measure the distance between zero crossing points of the signal (and the rate at which the signal changes from positive to negative or back will correlate to the frequency)

To make it super simple, you capture using your system the sound you want to indentify and takes its fingerprint (measure the the rate at which the signal changes from positive to negative or back) and save this. Then when your system is in listening mode, you keep doing the sampling, calculate the zero crossing distances and rate and compare with the fingerprint you took - , if they match then you have most likely identified your sound (and as noted above by Paul - the sampling frequency should be very different and way faster ideally than the frequency you are trying to recognize)