What was by far the most frustrating thing about this project was how long it took me before I figured out that the output of the spectrogram needs to be complex-conjugated before inputted into istft(). I was getting very strange distortions that depended on window size, because the time axis for each individual window was flipped. istft() still distorts a little depending on window size, but now the output waveform is at least recognizable as the input waveform. To reduce distortion and aid in pitch differentiation, larger window sizes are used.

wave = generateWave(20,22000,44100,5,0);
wave.wav

displayDFT(wave,44100) The spectrum indicates frequencies existing from 20Hz to 22kHz, as expected.	quickSpectrogram(wave, 4096, 4096, 0); The peak of the magnitude curve for each window is quite narrow and precise for lower frequencies, but becomes ambiguously large for higher frequencies, due to the large window size.
quickSpectrogram(wave, 256, 4096, 0); The opposite effect occurs for a window size that is too small. Peak detection is less precise for lower frequencies.	varWinSpectrogram(wave, 44100, 0,0); The initial low-resolution, non-overlapping spectrogram used to estimate pitch for the variable window spectrogram
varWinSpectrogram(wave, 44100, 0,0); The variable window spectrogram uses larger windows for lower frequencies, and smaller windows for higher frequencies. The peak width is approximately constant throughout the spectrum hpsofoff is false because the wave has no harmonics, and isNoisy is false because the wave is noise-free.	[S F T Y] = quickSpectrogram(wave,1000,1024,0); sc = pitchCorrector(S, F, Y, semitones); sc = conj(sc); iwave = istft(sc,1024,1000,500); quickSpectrogram(iwave,1000,1024,0); The waveform is pitch-corrected to a half-step scale. This is depicted by the discrete, equal-width blocks of pitch in the spectrogram.
[S F T Y] = quickSpectrogram(wave,1000,1024,0); sc = pitchCorrector(S, F, Y, cmajor); sc = conj(sc); iwave = istft(sc,1024,1000,500); quickSpectrogram(iwave,1000,1024,0); The waveform is pitch-corrected to C-major (though wave2 starts on A-440 so C-major will sound like A-minor), however the spacing of pitch blocks and the sound of the waveform both indicate that the wave was corrected to a pentatonic scale instead (the inverse of the C-major scale.) This could indicate a problem with compareToPitches(), though it worked as expected for smaller test cases. This should be further explored.	A closer look at the last image. wave2 files are slower so individual pitches are easier to distinguish waveSemi.wav waveCmajor.wav wave2Semi.wav wave2Cmajor.wav

noisywave = addNoise(wave,0.5);
noisyWave.wav

displayDFT(noisywave,44100)	varWinSpectrogram(noisywave,44100,0,0); With the added noise, varWinSpectrogram() has difficulty calculating correct window sizes.
A close-up of the initial spectrogram reveals a large DC component to the wave, throwing off approximate pitch calculations.	varWinSpectrogram(noisywave,44100,0,1); With the high-pass filter turned on, varWinSpectrogram functions as normal. The rest of the noise does not interfere with pitch detection much. The scale of this image was chosen to emphasize the fact that the low-frequency components of the signal are still intact.
	noisyWaveSemi.wav noisyWave2Semi.wav
[S F T Y] = quickSpectrogram(noisywave,1000,1024,0); sc = pitchCorrector(S, F, Y, semitones); sc = conj(sc); inoisywave = istft(sc,1024,1000,500); quickSpectrogram(inoisywave,1000,1024,1); For some reason, the normal spectrogram shows no information about the pitch-corrected noisy wave, but with HPS turned on, the discrete pitches are just barely visible. The double curve gives off the misleading impression that the original wave contained harmonics, but this is just an artifact due to noise being constant (in a statistical sense) at every frequency. Note that HPS was not used in the first quickSpectrogram() as, again, the wave has no harmonics.

Recorded
glissando.wav

displayDFT(glissando,44100)	varWinSpectrogram(glissando,44100,0,0); Most of the frequency content is located between 0 and 5000Hz. The spectrogram provides a nice view of the voice's harmonics.
varWinSpectrogram(glissando,44100,1,0); The fundamental frequency was already pronounced enough in the first spectrogram, but the HPS accentuates it further.	[S F T Y] = quickSpectrogram(glissando,4000,4096,1); sc = pitchCorrector(S, F, Y, semitones); sc = conj(sc); iglissando = istft(sc,4096,4000,2000); quickSpectrogram(iglissando,4000,4096,0); The harmonics do indeed get scaled by the correct amount.
	voiceSemi.wav voiceCminor.wav
[S F T Y] = quickSpectrogram(glissando,4000,4096,1); sc = pitchCorrector(S, F, Y, cminor); sc = conj(sc); iglissando = istft(sc,4096,4000,2000); quickSpectrogram(iglissando,4000,4096,0); Wave2 starts on A-440, but will be pitch corrected to A-flat or B-flat, so that C-minor...will not sound like C-minor. However, once again, it seems as though the signal is being pitch corrected to the scale that is the inverse of C-minor :(

The paper by Laroche and Dolson cited in the Background page mentions that phase rotation is also required when shifting frequencies. There was not enough time for experimentation with phase during this project, but doing so could explain the distortion heard in modified sound files. The unaligned pitch spectrum would cause beating due to alternating constructive and destructive interference when summing component waveform. Second, in all cases, pitch correction for low frequencies is very unclear, and this is because the smaller the frequency, the less likely its peak will be shifted to the correct location in pitchCorrection(), due to substantial rounding error. Some kind of interpolation would be necessary to correctly shift the low-frequency pitches. I did not have enough time to design an iSTFT for a variable window-size spectrogram, meaning that the varWinSpectrogram() did not play a part in the final pitch correction. This should not be difficult as all that is necessary is to take a different iSTFT for each bin in the varWinSpectrogram. VarWinSpectrogram would most likely have to be modified to also return winsize, the array of window sizes, and the constant maxwin, so that the extended iSTFT could use this information to properly resynthesize the original signal. Hopefully this would increase pitch detection accuracy, and by doing so, improve the pitch correction algorithm. Lastly, in order to make this pitch-corrector more realistic, the borders between pitches would have to be smoothed out. Once the pitch corrector finds the next exact pitch, it would have to plot a smoothing function between the next pitch and the last exact pitch and correct frequencies to the intermediate smoothing values. The challenge would be in its timing, in case the smoothing process time takes longer than the time the signal stays at a certain frequency. One way to solve this would be to us a proportional-integral-derivative controller (PID loop) which could also be used for real time pitch correcting. This would allow for more natural-sound pitch correcting, as long as the parameters were such that there was little overshoot.