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I was doing experetiments in Audacity,creating sine waves and trying to find out what is the smallest amount of freqency change possible,how it looks like...

I found out that the super small changes in freqency act kind of like amplitude modulation,thats very interesting,what really blew my mind was how small changes resulted in different wave.

for example,I made 1 second max amplitude 12000hz sine,and then same but 12000,0000001,I inverted the second one and combined them together so they phase cancel and only thing that remains is difference between the two.In this case first difference was found at sample number 4

that would suggest freqency resolution of one millionth of hertz per 4 samples,since 12000 TO 24000hz is one octave,thats freqency resolution of 12 billion per octave

I noticed this resolution is tied to lenght and loudness(amplitud) of the signal(sinewave),making it longer gives higher resolution,also making it louder gives higher res too... at -6db,(16384amplitude instead of 32768 maximum ) this ability to get some kind of change was lowered,small change in freqency that would yeald change at max volume did nothing at lowered amplitude

I now better understand why longer FFT window lenght gives better freqency resolution

what is your opinion on this,I would like to understand this better,I have feeling my calculations are wrong,so what is real resolution of wav at what samplerate,bit-depth,signal lenght and amplitude,per octave....

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    $\begingroup$ Frequency resolution is dependant on the signal-to-noise ratio. In zero noise (including quantization noise), one only needs 3 or 4 un-aliiased samples to completely determine all the parameters of any perfect sine wave, including frequency. $\endgroup$
    – hotpaw2
    Commented Oct 7, 2016 at 18:34

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There are two ways to look at this. Bottom up: go through all the bit combinations and write down what frequencies they give. Top down: Forget about quantization (to 16 bits) and consider the samples as real numbers. Then later introduce quantization as error added to the samples.

You can have a look at the problem from both angles, but I think the top-down approach is more fruitful here, because it allows you to grasp the concepts without letting the quantization confuse you. Discrete Fourier Transform preserves the number of degrees of freedom. Transforming a certain number of real samples will give the same number of real numbers to describe the signal in the frequency domain.

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