Finding Frequency Fourier Style

Last week, I had a look at the FFT module in Maxima and it was fun times. I was a little disappointed at the interpolated result for the frequency when analyzing the results of the FFT of the 440 Hz waveform. Mind you, if the application is to find a nearest note frequency and, being a human with an at least average sense of musical perception, you already know you're hearing notes that sound like 12 semitones per octave (exclusive counting, i.e., C up to B), then you can coerce the results to the nearest value of  \((440) 2^{i/12}\), for \(i\) over integers. If you don't intend to get any deeper than determining the major pitches, then this is fine. (Could get more tricky if the whole thing is internally shift tuned—everything a half of a semitone flat, but internally tuned, but then maybe you could create a filter to catch that case and accordingly shift your frequency base from 440 to something else that models the sound better.)

We got close enough last time for goal 1 (find the basic notes), but still, I wondered whether I could get closer to the true frequency than last time. I also thought I should use a different example frequency so nobody gets the wrong idea that this is somehow only good for integer frequencies. (Integers are discrete, but discreteness goes beyond just integers.)

So, let's use \((440)  2^{1/12}\) or approximately 466.16 Hz.

load(fft); wave(amp, freq, t) := amp*sin(2*float(%pi)*freq*t); sampleRate: 44100.0$ timeStep: 1/sampleRate$ N: 2^14; inputFFT: makelist(wave(40, 440*2^(1/12), t*timeStep), t, 1, N)$ transformed: real_fft(inputFFT)$ wxplot2d([discrete, makelist(i*sampleRate/N, i, 1, 200), firstn(map(lambda([u], abs(u)), transformed), 200)]);

Fig. 1. Frequency of 466.16 Hz.

wxplot2d([discrete, makelist((i-1)*sampleRate/N, i, 170, 180), map(lambda([u], abs(realpart(u))), firstn(rest(transformed, 169),11))]);

Fig. 2. Zooming in a little closer on the points.

So, let's take these same points and try a cubic spline.

load("interpol")$ f: cspline(map(lambda([f,a], [f, a]), makelist((i-1)*sampleRate/N, i, 170, 180), map(lambda([u], abs(realpart(u))), firstn(rest(transformed, 169),11)))); wxplot2d(f, [x, 450, 480]);

Fig. 3. Cubic spline view of things. Still looks fake. But less fake in the middle.

The cubic spline is helpfully printed out with characteristic functions which are mainly readable. I pick out the one that has the interval of concern, namely, 0.5264689995542815*x^3-738.1894409400452*x^2+345013.3968874384*x-5.374983574143041*10^7. We can use differentiation to find the location of the maximum value.

findMax: 0.5264689995542815*x^3-738.1894409400452*x^2+345013.3968874384*x-5.374983574143041*10^7; diff(findMax, x); float(solve(diff(findMax, x)=0, [x]));  

Which gives roots of [x=465.6995765525049,x=469.0682718425062]. Clearly 465.7 is the one we want. We're not any closer than we were last time using linear interpolation, although the general appearance of the graph looks more realistic and we did include another bit of the Maxima library.