2.5 Summary

• Digital audio is usually synthesized and manipulated in memory as an array of floats (unquantized) in nominal range \([-1, 1]\), paired with a sample rate \(f_s\).

• Synthesis is the inverse of recording: dream up a continuous function \(x(t)\) and evaluate it at sample times \(t_n = n / f_s\).

• Loops are a ubiquitous primitive in computer music programming, because synthesis involves sampling functions at many points in time.

• Vectorized computation replaces explicit Python loops with whole-array operations. It is equivalent but faster (precompiled inner loops) and more readable (one expression instead of many).

• NumPy is the standard vectorization library. The core operations: array creation (np.array, np.zeros), slicing, element-wise arithmetic, multi-dimensional arrays, and broadcasting.

• Stereo audio is a 2D array of shape (num_samples, num_channels). To downmix to mono, take array.mean(axis=1).

• Pyquist is a small wrapper around NumPy that bundles samples + sample rate into a single pq.Audio object, plus I/O / playback / plotting helpers.

• Adding two pq.Audio objects mixes them. Both array-style slicing (audio[a:b], in samples) and .segment(offset=, duration=) (in seconds) return a new pq.Audio with the sample rate carried along.