It isn't really true that the human eye sees "exactly one octave of light".
The visible spectrum does not have precise boundaries, so the choice of "350-700 nm" is a bit arbitrary and might even just reflect a human bias towards integer ratios. Wikipedia gives "380-750 nm" as the typical range for the human eye, and if you look around online you'll see lower limits that range from 350 to 400 nm, and upper limits from 700 to 830 nm.
For sufficiently bright sources, human perception may actually extend down to 310 nm and up to 1100 nm.
Humans can see the spectrum they do because of a combination of eyes being optimized for seeing in sunlight and avoiding the biochemical and biophysical challenges of detecting longer and shorter wavelengths.
Visible and near-visible light is the "sweet spot" for biological vision on Earth. The human eye could probably not operate over a much larger wavelength range, and a range much less would inefficiently waste available light. There is no reason for the visible range to be precisely a factor of 2, but it would be surprising if the range was very much larger.
As noted in the comments, the range of human vision almost exactly matches the wavelength region of peak solar irradiance on the Earth's surface. The full width at half maximum range for sunlight is about 300-900 nm.
As noted in the answers to "What is the biological potential for vision of wavelengths outside the human visual range?", shorter wavelengths tend to destroy biological structures and longer wavelength photons don't have enough energy to be easily detected individually.
The human eye doesn't "respond to harmonic frequencies" because it it detects light via quantum processes, which are not the same as the classical resonances such as a vibrating string. Just because a molecule absorbs photons of a given frequency, doesn't mean it will absorb photons of twice that frequency.
Another issue is that if you double the frequency of light, you double its photon energy, which for visible light takes into dangerous wavelengths.
Even for creatures whose eyes work into the ultraviolet, their range doesn't go much below 300 nm, which is not surprising since 300 nm (4 eV) photons can break carbon bonds and 250 nm (4.5 eV) photons can break C-H bonds.
Interestingly, however, the eye has been observed to respond to sub-harmonic frequencies. Intense infra-red light can be seen as green, because two infra-red photons can activate the rhodopsin receptor that normally is activated by a single green photon. This requires very bright infra-red light since such two-photon processes happen rarely.