There are two distinct areas or domains to keep in mind when thinking about the idea of pitch, the acoustic signal (in your case a complex tone composed of a set of harmonics based on a fundamental frequency) and our perception of that acoustic signal.
Let's say you compose two complex tone signals, one signal includes the fundamental and a set of harmonics and the other signal has only a subset of the harmonics with no fundamental frequency. Using a 100 Hz fundamental frequency, an example set of frequencies for the first signal might be (100, 200, 300, 400, 500, 600; all values in Hz), while an example set of frequencies for the second signal might be (400, 500, 600; all values in Hz). I've generated spectrograms and links to WAV-files for each of these signals.
The spectrogram of the first signal is shown below. This signal has energy at six frequencies - the fundamental and five harmonics. A 3 second WAV-file of this signal.
The spectrogram of the second signal is shown below. This signal has energy at only four of the harmonic frequencies of the first signal. A 3 second WAV-file of this signal.
When each of these signals is individually presented to a listener, the only regions of the basilar membrane (BM) that are excited are the ones where there is acoustic energy present. Since the first signal includes the 100 Hz fundamental frequency and all of the harmonics, it will produce mechanical activity on the BM at these frequencies. The second signal will only produce activity at the frequencies that make up its composition.
If you listen carefully to these two signals they'll sound similar but they will have slightly different acoustic coloring, which is called timbre. Importantly, the 100 Hz fundamental is noticeable in both signals. Although I can detect the fundamental frequency in the second signal, the fundamental frequency is more pronounced in the first signal because the fundamental frequency is present in that signal. Pitch is influenced by the composition of a complex tone as well as other factors such as the intensity at which it listened to. These elements may play a part in how these two signals are perceived. For reference, here is a 3 second WAV-file of just the 100 Hz fundamental frequency.
Our perception of these two signals and our detection of the 100 Hz fundamental in the second signal, often called the residue pitch, is a function of our auditory system that extends beyond the BM of the cochlea.
There are two classes of models that attempt to explain the residue pitch, pattern recognition models and temporal models. For information about these models I suggest the textbook "An Introduction to the Psychology of Hearing", Fifth Edition, B. C. J. Moore, Emerald, 2008, specifically. Another very good textbook is "Psychoacoustics, Facts and Models", Third Edition, H. Fastl and E. Zwicker, Springer, 2007.
I hope this helps you.