FFT - Revisiting Symmetry

There was something missing in my last post that might have created part of the problem with the results. Although I observe what looked like symmetry, I didn't constrain any of my interpolated results based on that assumption of symmetry. So, I thought to myself, there ought to be a mathematical way to tell my equations that they should assume a symmetrical situation, the way I am expecting it to look. I'm not really sure this is broadly applicable, but my intuition about a sine wave is that it should have some kind of symmetry in the resulting appearance. This seems to be borne out by scattered examples of continuous Fourier transforms.

I'm not yet sure how to use symmetry, to find the value, but I can see a way to evaluate the accuracy of a given guess. Mind you, I only know how to evaluate it by appearance. Let's introduce a little formula I will call the flip formula.

$$F(x, C) = 2 C - x$$

Suppose you choose \(C = 440\), meaning that you are going to treat 440 as the center. So, for example, \(F(439, 440) = 441\) and \(F(441, 440) = 439\). Let's apply the flip function to the data in one of our previous examples and see what it looks like when we assume the center is 440. 

First, here is the graph we had with just a few points near the peak:

Fig. 1. Just the results of the FFT.

Flip(x, freq) := 2*freq - x$ thePoints: [[427.972412109375,1.436293850207828],[430.6640625,1.84470452365816],[433.355712890625,2.584065181382392],[436.04736328125,4.330481956818618],[438.739013671875,13.53281618900214],[441.4306640625,11.89160409385532],[444.122314453125,4.114533847954862],[446.81396484375,2.48170175135288],[449.505615234375,1.773621776595303],[452.197265625,1.378080393261782],[454.888916015625,1.125573209744724]]$ flippedPoints: map(lambda([u], [Flip(u[1], 440),u[2]]), thePoints)$ allPoints: sort(append(thePoints, flippedPoints), lambda([a, b], a[1] <= b[1]))$ wxplot2d([discrete, map(lambda([u], u[1]), allPoints), map(lambda([u], u[2]), allPoints)]); 

 Let's have a look at what it looks like when we use 439.5, 439.8, and 440.

Fig. 2. "Guessing" frequency 439.5 Hz. Sure doesn't look smooth.

Fig. 3. "Guessing" frequency 439.8 Hz. Looks better.

Fig. 4. "Guessing" frequency 440 Hz. Looks about as smooth as we're going to get.

There are some considerable drawbacks to this approach. The biggest one is that I had to look at the results. The second drawback is that I don't have a number to put to these to decide which is better. And this is transformed from one of the simplest types of wave, a sine wave. To evaluate the soundness of the guess, I would need some formula to fit the points to.