What *is* pitch-shift?

Question

As I understand it, pitch-shifting (without changing speed) and time-scaling (without changing pitch) are two sides of the same coin, because if we can do one we can get the other through resampling and changing the playback speed of samples.

But when I think about it, I don't really understand what it means to pitch shift. The Wikipedia article is completely unenlightening, because it only describes pragmatic methods. They all intend to implement a "pitch shift", but at no point is it actually defined what that is. Like, I know it when I see it, I'm not stupid, but is that really the best we can do?

I struggle to come up with something, because in order for pitch to be, say, doubled, roughly speaking the wavelength of all waveforms "locally" has to be halved and each repeated once, but what is local?

• Is there a mathematical definition of a "pitch shift" operation on a signal, or is this purely a subjective human phenomenon?

Answer 1

« Pitch is a perceptual property of sounds» https://en.m.wikipedia.org/wiki/Pitch_(music)

As pitch is perceptual, pitch-shifting should also be a perceptual thing?

I’d say that pitch shifting is a tool to accomplish something similar to a musician transposing a tune. I.e. the duration of the song and each note is kept constant. The timbre of musical instruments is kept constant. But the root key is changed e.g. From C# to Ab, and the relationship between fundamental notes and harmonics is kept. I.e. a melody jumping up one 5th will still jump up a fifth, and a clarinet tone with a square-like distribution of harmonics will continue to have just that.

Answer 2

There are (at least) 2 classes of time-pitch modification. Time domain and frequency domain. The time domain algorithms are more relevant to your question about locality.

The human ear-brain combination makes an estimate of the periodicity of long enough semi-stationary sounds, and uses that periodicity estimate as part of its construction of the psychological perception of "pitch".

So if you double a semi-stationary sound segment exactly one period in length, or halve a pair of periods, and do so over some portion of a pitched span of sound (which usually requires a certain minimum length), you can often trick a human in to hearing the same sound as being twice as long, or half as long, or by various ratios by doing this over only some fraction of the span.

So the "locality" is to try make a psycho-acoustic estimate what a typical human ear-brain would estimate as the local periodicity (which may be slightly different than mathematical periodicity, due to masking and other perceptual phenomena).

Then if you shorten or length the sound span by some ratio of resampling, the periodicity will change, and the pitch will be perceived as higher or lower.