I bought a vocoder to use as a layer for backing vocals but now I realize it sounds terrible when only used with one note. So I need to use it with chords or harmonies. But I only know very basic music theory. I tried the perfect 5th, 4th, triads but they all impart an emotion on the sound which I don't want. I just want a neutral sounding harmony to blend with my vocal, like vocal doubling. Will negative harmony work? How do I create a perfect 5th and a negative harmony of the perfect 5th?
Unfortunately it's not as straightforward as that... :)
The question of whether a particular backing note sounds neutral varies case by case, depending on a variety of things such as whether the overall piece is in a major or minor key, what other harmonies are present before, during and after the backing note, and so forth. And we don't know your song...
Probably the most that can be said is that for your vocoder part, you should experiment with partial chords that don't contain thirds, so as to allow major/minor ambivalence for the required note duration. I think you have tried this to an extent, but to be specific I mean the root plus a fifth (do+so). When experimenting, I recommend you try the effect of including multiple octaves, which may fill out the sound by increasing the vocoder's ability to find something to do (do1+so1+do2+so2...), as well as bare octaves (do1+do2, or do1+do2+do3...). And try all this in different inversions and at different points on the keyboard, both above and below middle C.
You may even find more success in particular cases by adding 7ths, 9ths etc but that would be likely to impart emotion, which you say you want to avoid doing. Ultimately you will probably have to accept that a vocoder is not the tool for what you are trying to do. As if to underline this point, at the end you ask "How do I create a perfect 5th and a negative harmony of the perfect 5th?" and again this is not a job for a vocoder, which cannot generate specific notes but only sounds that result from processing a carrier signal (voice) by modulating it with another signal (chords, noise etc). There may be other signal processing tools that can do it, but to return to my opening point, I think the question of neutrality depends on many things and it would be hard to reduce it to an algorithm, hence writing backing harmony would be likely to remain an art, and one that may need to be learned over time.
Single notes at a fixed interval don't sound good. I'm afraid you're mistaken in thinking that triads will sound any better.
What you may be looking for is harmony in thirds. Unfortunately they will need to be diatonic thirds (i.e. not consistently major or minor thirds) and they'll need to be sometimes below, sometimes above the melody note. And sometimes a unison will be better. A reasonably simple and learnable composition/performance technique but not well-suited to mechanical generation.
There's another approach to creating vocal harmonies that 'fit' in a bland, non-challenging sort of way. It's simple but remarkably effective. Find the 5th note of the scale. Do, re, mi, fa, so - yes that one. That's your harmony note. Stick to it until your ear tells you that you MUST move one note up or down. But, as soon as possible, get back to 'so'. Again, pretty easy to DO, not all that simple to automate.
The 'vocal processors' that CAN achieve this sort of result need to be programmed with the chord sequence of the song.
Avoid triad-based harmony because, as you discovered, it sounds centred. Also avoid chords containing a tritone, e.g. C-F# as this creates a harmonic pull towards another chord.
Instead, play around with harmony based on descending fourths. This sounds much more ambiguous, so for any three-note or four-note chord you construct using fourths there are several bass notes that could work. Your listeners will hear the chord and it won't be immediately apparent what key you're actually in. Such chords are stable without pulling towards another chord. And because of their ambiguity you can sustain them longer; they harmonize with more chords.
But I only know very basic music theory.
It may help to try to improve that.
I tried the perfect 5th, 4th, triads but they all impart an emotion on the sound which I don't want.
Perfect fifths and fourths create very specific sound that you may like or not. Concerning chords – perhaps your plugin simply generates other chords than the ones you need? This might be the direction to explore.
I just want a neutral sounding harmony to blend with my vocal, like vocal doubling.
Then try exactly that: vocal doubling. Many artists simply record the doubled vocals, but you can also try generating them with the plugin: experiment with adding slight effect: short delay, chorus/detuning, whatever else is available. This might be the way to "thicken" the vocal track.
As your plugin allows MIDI input, you can use to generate the exact harmonies you want. A particular idea that comes to my mind are drone sounds: harmony melody consisting of a single note (or several notes, e.g. in fifths, but not changing). This might be the "static" harmony sound you seek.