Audio and speech data are increasingly used in machine learning applications such as speech recognition, speaker identification, and mental health monitoring. However, the passive collection of this data by audio listening devices raises significant privacy concerns. Fully homomorphic encryption (FHE) offers a promising solution by enabling computations on encrypted data and preserving user privacy. Despite its potential, prior attempts to apply FHE to audio processing have faced challenges, particularly in securely computing time frequency representations, a critical step in many audio tasks. Here, we addressed this gap by introducing a fully secure pipeline that computes, with FHE and quantized neural network operations, four fundamental time-frequency representations: Short-Time Fourier Transform (STFT), Mel filterbanks, Mel-frequency cepstral coefficients (MFCCs), and gammatone filters. Our methods also support the private computation of audio descriptors and convolutional neural network (CNN) classifiers. Besides, we proposed approximate STFT algorithms that lighten computation and bit use for statistical and machine learning analyses. We ran experiments on the VocalSet and OxVoc datasets demonstrating the fully private computation of our approach. We showed significant performance improvements with STFT approximation in private statistical analysis of audio markers, and for vocal exercise classification with CNNs. Our results reveal that our approximations substantially reduce error rates compared to conventional STFT implementations in FHE. We also demonstrated a fully private classification based on the raw audio for gender and vocal exercise classification. Finally, we provided a practical heuristic for parameter selection, making quantized approximate signal processing accessible to researchers and practitioners aiming to protect sensitive audio data.