Which physical properties of a stimulus can a human attend to? I am not only curious about speech, but for concreteness in illustrating my question, I consider the case of speech perception. Assume a listener is presented with an acoustic waveform. The wave causes deviations from the atmospheric pressure at the listener's eardrum. Thus, the acoustic pressure as a function of time “comes for free”, so to speak. Mechanically so.

From sound pressure we can derive amplitude, and by the fourier transform, amplitude as a function of time is transformed to the frequency domain. That is, amplitude as a function of time becomes amplitude as a function of frequency. We use this transformation because it reveals properties of the wave which are perceptually relevant, e.g. formant frequencies.  Presumably, listeners can attend to properties of the waveform describable under this transformation.

But what about mathematical models of the acoustic wave which are not simply transformations on the wave, but which derive some richer structure from amplitude fluctuations over time. For instance, what about a model which constructs a higher dimensional representation of the wave from the one-dimensional pressure fluctuations? (Imagine that instead of amplitude being a point moving back and forth along the real number line, the wave is represented as a point moving around in, say, 3-dimensional space). Say that such a model revealed a perceptually relevant property P of the waveform. Would the listener be able to “hear” P directly, or would would she need to engage in some information processing task to recover P from pressure fluctuations over time?*

*I realize that human listeners don't actually manipulate equations in their heads. I use the mathematics just to describe the underlying neural computation.
I found a paper called Summary statistics in auditory perception (McDermott, Schemitsch &amp; Simoncelli, 2013) that might be relevant to your question. If you can't access it in full, you can a great description of the paper here. Please note that I'm writing about it from memory and that some details might not be correct. 

The authors had a task where participants had to judge whether two sound excerpts are same or different. These sound excerpts could, further, have similar or dissimilar summary statistics, meaning that some higher-order regularity was superimposed on the initially generated sounds (such as the somewhat regular pattern of the sound of sea waves crashing, or the wind picking up, or a fire crackling). So, the sounds could differ in a local way (which specific frequency was displayed at some moment) or in a global way (what were the higher-order regularities of these sounds). They then displayed these sounds in pairs for variable amounts of time. 

They found that people were better at discriminating sounds from each other when the sounds lasted longer - which is logical - but this was only true if the complex sound textures were different from each other. When the textures were mutually similar, people actually got worse if they listened to the (potentially different) sounds for a longer time. 

The interpretation the authors gave is that the auditory system first encodes simple features of sounds, but, once confronted with too much stimulation to keep track of, it instead extracts higher order regularities and relies on these for perception. 

I do not have a technical background so I can't go into detail about what these higher order regularities represent, mathematically speaking. But the links come with some images with explanations, and you might be able to glean more information from them than I.

Here is the abstract:


  Sensory signals are transduced at high resolution, but their structure
  must be stored in a more compact format. Here we provide evidence that
  the auditory system summarizes the temporal details of sounds using
  time-averaged statistics. We measured discrimination of 'sound
  textures' that were characterized by particular statistical
  properties, as normally result from the superposition of many acoustic
  features in auditory scenes. When listeners discriminated examples of
  different textures, performance improved with excerpt duration. In
  contrast, when listeners discriminated different examples of the same
  texture, performance declined with duration, a paradoxical result
  given that the information available for discrimination grows with
  duration. These results indicate that once these sounds are of
  moderate length, the brain's representation is limited to
  time-averaged statistics, which, for different examples of the same
  texture, converge to the same values with increasing duration. Such
  statistical representations produce good categorical discrimination,
  but limit the ability to discern temporal detail.


edit: I found a link to the pdf of the second paper I mention.