Depolarization and hyperpolarization in stereocilia of the inner ear
There are roughly two modes of pitch coding in the cochlea: place-coding and temporal coding. The place-theory is the most prevalent accepted model of how the cochlea realizes pitch coding (e.g., Zwislocki, 1991). Basically, it is based on a frequency-to-place Fourier transformation on the incoming sound, where each frequency is coded on a different place on the basilar membrane, as described accurately in the question.

However, there is another, much overlooked way of coding pitch, namely temporal coding. Up until about 1 kHz, spiral ganglion cells in the auditory nerve and acoustical brain stem regions (such as the inferior colliculus) have been found to respond in a phase-locked pattern (Du et al., 2011). Electrophysiology in auditory nerve fibers illustrates the phase-locked activity in response to low-frequency sounds (Fig. 1). This phase-locking behavior of neurons in the auditory system is called the frequency-following response (FFR).


fig. 1. FFR in auditory nerve fibers. Upper trace shows the stimulus, middle trace a single fiber, and the lower trace the compound activity of many fibers. The group-response faithfully codes the stimulus waveform. Source: New York University.   

However, as you rightfully state in your question - why would inhibition of auditory nerve fibers (ANFs) be helpful? Figure 1 illustrates nicely that the auditory nerve FFR is rectified, i.e. only the upper half is coded, while negative action potential counts obviously don't exist; ANFs fire, or they don't, they don't fire negative action potentials. However, one has to realize that a relatively large proportion of ANFs in healthy ears spontaneously fire, i.e., in the absence of sound the auditory nerve is still remarkably active. Spontaneous firing rates vary from 0 to more than 100 spikes/s (Jackson &amp; Carney, 2005). Hence, suppression of spontaneous activity codes pitch too. 

Moreover, as a theoretical and un-referenced side note - the auditory system processes acoustic information on the sub-millisecond level. Left/right sound localization is performed by the auditory system by resolving inter-aural time differences (ITDs). By analyzing the delay at which sound arrives in one ear with respect to the other, the localization in the horizontal plane can be estimated. ITDs do not exceed 0.8 ms, given the speed of sound and the size of the human head. This is well below the time it takes for any action potential to develop(!). Hence, by deploying both phases of the sound, one phase being excitatory, the other inhibitory at the ANF level, no time is lost when a sound wave happens to enter the cochlea with its inhibitory flanking phase first.   

References
- Du et al., Neurosci Biobehav Rev (2011)
- Jackson &amp; Carney, JARO (2005); 6: 148–59
- Zwislocki, Acta Otolaryngol (1991); 111(2): 256-62