I don’t think overtones are applicable here. If you think about how we generate sound, it is usually with a vibrating string or column of air, where multiples of the basic wavelength naturally fit. Most visible light on the other hand either comes from electrons jumping between different energy levels in molecules (including near ultraviolet) or molecules switching between different levels of vibration (including near infrared).
You do get something like overtones from that, but partly because it involves transitions and even more because the underlying system isn’t linear, they don’t form a harmonic series. The hydrogen spectral series is an example; at best it’s like overtones on a drum. So exact integers ratios aren’t common, and in the end we don’t pick up the specific frequencies like we do with sound anyway, just which of our visual pigments are capable of absorbing them. It’s not even a linear scale, it’s done by buckets.
As has been said, lots of animals respond to frequencies beyond what we see, but not by much. Many insects see ultraviolet, but only near ultraviolet within about 300 nm, and then they generally don’t see red. Likewise rattlesnakes famously see infrared, but that doesn’t mean a larger range – I couldn’t find much on them, but a little summary here lists them as “different” rather than “more”.
Birds do better, as many can see both red and ultraviolet; here is a paper exploring their color space that I found interesting. While many marine fish see only blue colors, there are some like damselfish that have this same kind of expanded range. Some freshwater fish apparently see near infrared to about 900 nm, but I’m not sure if any can see that and near ultraviolet. In any case, that means that some of the vertebrates make it a full octave range and maybe a little bit more, though not by too much.
Mostly I think advantages or disadvantages would apply to different frequencies more than the total range. For instance ultraviolet is potentially harmful not to screen out but lets you see markings on flowers, or stripes on other damselfish, hidden to animals like us. But I guess you could say it gives them more of opportunities for visual signals to one other, if they are able to produce the colors somehow, as studied in the bird paper.
For all their wonders, octopuses are said not to have color vision. It used to be considered that color wasn’t much use in the ocean in anyway, since past a few meters down all the ambient light is blue. However, it’s been discovered many fish are exceptions that see things like the fainter fluorescent colors down there. So maybe some cephalopods have developed the same, but it’s not really what you expect from them; they pick up on things like the polarization of light instead.
For the record most people don’t see ultraviolet as purple, they see it as nothing, because our eyes filter it out. So you could tell if someone can respond to it or not. Things like black lights will look purple only because they aren’t actually monochromatic, and you see the little bit that makes it into your visible range.