VQT

Variable-Q transform: the general form of the CQT with a bandwidth offset gamma that widens the low-frequency filters [schorkhuber2010], shortening their wavelets and improving time resolution where the constant-Q filters grow long. specux.cqt is exactly specux.vqt at gamma=0.

specux.vqt(x, *, sr, hop_length=512, fmin=None, n_bins=84,
           bins_per_octave=12, gamma=None, filter_scale=1.0,
           sparsity=0.01, tuning=0.0, output="magnitude", eps=1e-10,
           backend="auto")

import numpy as np

import specux

x = np.random.randn(8, 4 * 22050).astype(np.float32)
V = specux.vqt(x, sr=22050, n_bins=84)         # gamma=None: ERB default
V.shape        # (8, 84, 173) = (..., n_bins, n_frames)

import torch

import specux

xt = torch.randn(8, 4 * 22050, device="cuda", requires_grad=True)
V = specux.vqt(xt, sr=22050, n_bins=84, gamma=10.0)
V.sum().backward()

import cupy

import specux

xc = cupy.random.standard_normal((8, 4 * 22050), dtype=cupy.float32)
V = specux.vqt(xc, sr=22050, n_bins=84)

The gamma parameter

Each filter’s bandwidth is alpha * f + gamma: a constant-Q term plus a fixed offset in Hz.

gamma=None (default): the offset follows the ERB model of human auditory filters [glasberg1990], so low-frequency bins get bandwidths closer to perception.
gamma=0: the constant-Q limit; identical to specux.cqt bit for bit.
gamma>0: a fixed offset in Hz.

Everything else (parameters, shapes, gradients, backends, and the specux.transforms.VQT Module) matches the CQT page.

The configured form

t = specux.VQT(sr=22050, n_bins=84, gamma=10.0)
V = t(x)                                       # (..., n_bins, n_frames)
assert specux.VQT(**t.to_dict()) == t

References

[schorkhuber2010]: the variable-Q formulation.
[glasberg1990]: the ERB bandwidth model.