Hybrid Multi-Rate Constant-Q Transform

André van Schaik
International Centre for Neuromorphic Systems
The University of Manchester
June 2026
with Claude Code Opus 4.8

A from-scratch, fully invertible Constant-Q Transform (CQT) developed step by step in a single Jupyter notebook (hybrid_cqt.ipynb). It is written to be didactic: every stage — the analysis kernels, the efficient multi-rate algorithm, the matching inverse, and a set of test signals — is explained in the markdown alongside the code.

Why a CQT?

Music is logarithmic in pitch, so a transform should space its bins geometrically (per octave) and give each bin a bandwidth proportional to its centre frequency — a constant-Q filter bank. This is also, roughly, how the cochlea analyses sound. A naïve CQT is expensive, though: the lowest bin needs a very long window, which would set the size of one huge FFT run at every hop.

How it works

• Multi-rate. Halving the sample rate and the frequency leaves the required window length unchanged, so each octave is analysed at its own decimated rate. Every octave then shares one small N_fft and one set of kernels.
• Per octave. Frame the (decimated) signal, take a single batch FFT, and matrix-multiply by pre-computed frequency-domain kernels. Cost is O(M · B · N log N): many small FFTs instead of one large one.
• Invertible. The kernels are normalised so their squared responses sum to 1, making the filter bank a Parseval (tight) frame. Its inverse is then just the adjoint Kᴴ (a transpose — no pseudoinverse or SVD), followed by overlap-add and upsampling.

What's in the notebook

• HybridCQT (forward) and HybridICQT (inverse), sharing the same kernel matrix.
• A visualisation of the filter bank: kernel overlap and the flat (tight-frame) coverage.
• Six worked examples — pure sine, chord, log chirp, major scale, inharmonic bell, and band-limited noise — each with audio and a spectrogram, chosen to probe a different property.
• Round-trip reconstruction with SNR, and forward / round-trip performance benchmarks.

Setup

python3 -m venv .venv
source .venv/bin/activate
pip install -r requirements.txt

Usage

Open hybrid_cqt.ipynb in Jupyter or VS Code and run the cells in order, top to bottom.

Default parameters

Parameter	Value	Description
FS	44100	Sample rate (Hz)
F_MIN	65.41	Lowest bin centre — C2
N_OCTAVES	7	C2 → C9 (≈ 8.4 kHz)
BINS_PER_OCTAVE	24	Quarter-tone resolution (2 bins per semitone)
HOP_LENGTH	64	≈ 1.45 ms; gives the top octave an 8× overlap

The small hop is chosen for the inverse (the overlap-add needs generous overlap). For analysis-only use it can be enlarged toward N_fft/2 for faster processing at coarser time resolution — see section 2.5 of the notebook.