Are there any modern theories on 5 (or more)-part harmony? Preferably something involving overtone series and chromatic chords
It's true that from some points of view, five and six part harmony is mostly the same as four. If you're working in the style of Bach chorale's, you're still going to (mostly) respect rules about resolution of sevenths, dominants tending to tonics, don't double this voice or that one, be wary of parallel motion, and so on.... The primary difference is that there will be more interactions between the voices so you'll have more to keep track of, and you'll need to be more conscious about maintaining independence of voices. If that's your concern, a big book about orchestration is your friend, maybe Samuel Adler's for example.
But I don't think that's what you're interested in. Personally I find that I need the option to employ at least 6 simultaneous tones to achieve the harmonic complexity I'm looking for, in part because, as we start to add more voices into the harmonic picture, our nondiatonic possibilities greatly expand. Four and less voices simply do not support counterpoint that heavily features equal divisions of the octave (augmented and diminished harmonies), because, in order to establish the harmonic character, we've already used up our voices and constrained ourselves to highly parallel motion. (Additionally, in closed progressions, with the notes close together, none of the fundamental or lower overtones coincide !)
To respond to your comment above, I don't think it's Neo-Riemannian theory that you're looking for in this question. Neo-Riemannian theory works by distilling music to a skeleton of fundamental chordal objects (either major/minor triads or dominant/half diminished sevenths) and then abstracts to the transformations between these objects, and the focus of the theory is on the latter: more voices are essentially obstacles to the theory, or auxiliary coloration, depending on how you look at it. There's a lot of insight to be gleaned from the Neo-Riemannian approach when it's sagely applied (to the right music), but the theory only possesses the capacity to address triadic music. That said, sometimes it's very interesting to analyze music that is ostensibly nontriadic by trying to find a triadic reduction that seems to "accomplish" the same harmonic function. I'd highly recommend Richard Cohn's book "Audacious Euphony" for an extensive introduction, or Julian Hook's work if you enjoy the algebraic aspect of the theory. I should also say that, in theory Neo-Riemannian methods could hope to account for some richer harmonies by considering minimal perturbations of the six-part equal division of the octave (whole tone scale) as its dual chordal objects — analogous to augmented triads and fully diminished sevenths — but in practice this is totally uninteresting because there are only two transpositions of the whole tone scale, and so the voice-leading/transformational aspect is incredibly shallow.
When we allow ourselves to add more voices, and not just for the sake of doubling, a lot opens up. Shoenberg writes about this over a hundred years ago in the last sections of his classic instructional text, Harmonielehre. With a couple additional voices, you can organize your harmony to simultaneously convey qualities of conventional consonance and that ambiguity characteristic of the whole tone or diminished scales. These "vagrant" harmonies possess a degree of stability from their consonant internal interactions but also strongly suggest resolutions by proximal voice leading. You might think of the vocabulary of five and six note chords as being "derived" from the triadic vocabulary by minimally perturbing a couple voices. Two early 20th century movements treat this vocabulary quite differently: the impressionists considering the objects stable unto themselves and the late romantics as ways to fluidly transition between tonal regions, resulting in heavy chromaticism. You could also do as Schoenberg eventually did and start thinking about a vocabulary which comes largely from tone clusters and is not designed for bringing out independence of voices — rather the opposite — in which you take a pointilistic approach such that tones lose their individual identities and lend themselves to composite "colors".
Hindemith might also be of interest to you as a proponent of a "pandiatonic" approach. The term is terribly abused, but here I mean it as someone whose relevant notion of consonance was to be confined to a diatonic scale, but still drawing harmony freely from it, often full of suspensions and dissonant sevenths and without resolution. In this way of thinking, you might be also interested in reading some of Dmitri Tymoczko's work about scalar networks, which is very useful in studying "nonfunctional harmony." The fundamental scalar objects in this theory are the "locally" diatonic scales, nameley: the diatonic, the octatonic (diminished), the acoustic (e.g. G A B C# D E F), and the whole tone scales. "Averaging" any whole tone step in an octatonic scale yields an acoustic scale, as does splitting any tone in a whole tone scale; meanwhile raising the fourth scale degree of an acoustic or diatonic scale yields a diatonic scale.
In essence, you can start thinking about harmony as the compact presentation of a scale, and focus your attention on modulating between scales. An always-good exercise if you play a polyphonic instrument is to lay out a path in the scalar network and improvise over it with a walking bass and large random-esque harmonies above.
I apologize that this post is all over the place and by no means thorough in any place, but I hope it gives you a few ideas about places to look.
Two through thirty-six or more part harmony is essentially the same. The only real change as the number of parts goes up is in the relaxation of the doubling rules. By happenstance, four-part harmony has the toughest rules on voice leading. Applying the usual ideas of avoiding parallel fifths and octaves still apply but successive octaves or fifths by contrary motion is permitted in five or more parts. It's easier to construct a full V-I cadence in root position in five part than in four part harmony (also easier in three part.)
Some of the older books like Ebenezer Prout's have a bit of discussion on the subject. http://petrucci.mus.auth.gr/imglnks/usimg/4/4a/IMSLP240170-SIBLEY1802.15874.d6d5-39087009937881text.pdf page 250ff.