11.0 Changing playback speed

Recap (resampling, from Chapter 7). Resampling maps one sample vector to another, generally of a different length,

\[\mathbf{x} = [x[0], \ldots, x[N-1]] \;\longrightarrow\; \mathbf{y} = [y[0], \ldots, y[M-1]],\]

by reading the original at fractional positions \(p\) via interpolation, \(y[m] = \text{Interpolate}(\mathbf{x},\, p)\). Changing the sample rate from \(f_s^1\) to \(f_s^2\) uses \(p = m \cdot f_s^1/f_s^2\) and a new length \(M = N\, f_s^2/f_s^1\).

We have actually seen resampling in one other guise already. When wavetable synthesis reads a table faster or slower to change its pitch, that is resampling. The same idea lets us change the speed of a recording, and with it, its pitch.

Here the goal is to change a clip’s duration from \(T^1\) to \(T^2\) while keeping the sample rate fixed. The new length is

\[M = N \cdot \frac{T^2}{T^1},\]

and we read the original at interpolated positions exactly as before, now with the ratio \(T^1/T^2\):

\[y[m] = \text{Interpolate}\!\left(\mathbf{x}, \; p = m \cdot \frac{T^1}{T^2}\right).\]

Stretching or squeezing the signal in time shifts every frequency it contains by the factor \(T^1/T^2\). Playing a clip at twice the speed halves its duration and raises every frequency by an octave, chipmunk-style.

Notice that changing the sample rate and changing the speed are fundamentally the same operation. The only difference is the ratio used to convert between sample indices, and whether we play the result back at a new sample rate or the original one. In Pyquist, we can change speed by reinterpreting the sample rate and then resampling back:

ratio = 2.0                                       # 2x speed, up an octave
sped_up = pq.Audio(audio.samples, int(audio.sample_rate * ratio))
sped_up = sped_up.resample(audio.sample_rate)

Original speed

Half speed (down an octave)

Double speed (up an octave)

The same recording played at three speeds. Changing speed also changes pitch, because stretching the signal in time scales all of its frequencies.