Wideband Audio Waveform Evaluation Networks (WAWEnets) are convolutional neural networks that operate directly on wideband audio waveforms in order to produce evaluations of those waveforms. In the present work these evaluations give qualities of telecommunications speech (e.g., noisiness, intelligibility, overall speech quality). WAWEnets are no-reference networks because they do not require “reference” (original or undistorted) versions of the waveforms they evaluate. Our initial WAWEnet publication introduced four WAWEnets and each emulated the output of an established full-reference speech quality or intelligibility estimation algorithm. We have updated the WAWEnet architecture to be more efficient and effective. Here we present a single WAWEnet that closely tracks seven different quality and intelligibility values. We create a second network that additionally tracks four subjective speech quality dimensions. We offer a third network that focuses on just subjective quality scores and achieves very high levels of agreement. This work has leveraged 334 hours of speech in 13 languages, over two million full-reference target values and over 93,000 subjective mean opinion scores. We also interpret the operation of WAWEnets and identify the key to their operation using the language of signal processing: ReLUs strategically move spectral information from non-DC components into the DC component. The DC values of 96 output signals define a vector in a 96-D latent space and this vector is then mapped to a quality or intelligibility value for the input waveform.