Wondering what the most complex examples are of tonal languages, and what its features are. In Chinese there are 4 or 5 tones, but they are relatively simple (contour changes, move up, down, down then up, etc.). Wondering if there are any that are fully musical like having absolute pitch, or if any have relative musical scales (like you would start at any random pitch, but then do a major third or whatnot). Wondering what an example of a tone is in the African register tonal languages.
Unfortunately, we don't have a musically-informed theory of tonal phonetics, and there is very little literature which applies notions like "major third" etc. to pitch relations. Indeed, the literature on the acoustic phonetics of tone is rather restricted, given how many tone languages there are. We do have a reasonable understanding of the basic contrastive types of tones.
First, a tone can be defined in terms of discrete levels. It's a bit problematic that there are as many as 6 levels employed in the language Chori, and 5 levels in a number of other languages like Bench Non and Trique (we theoretically expect there to be a power of 2 levels). Second, tone levels can be combined on a single vowel, so a Falling tone can be decomposed into the combination of High and Low. Theoretically that allows very rich inventories where a vowel could have the profiles 1, 2, 3, 4, 12, 13, 14, 24, 43, 42, 41 etc (where 1 is the lowest pitch and 4 is the highest pitch). Some languages have three-element contours (though always involving a 'change in direction') such as 132, 423.
There are, additionally, two global operators, upstep and downstep. The aforementioned levels are defined within a given pitch range (determined non-grammatically based on general social values e.g. "how high should your voice be", and individual factors ("don't expect me to say that at 400 Hz, I can't go that high"). Normally, people use a certain subrange of comfortable talking pitch, so a H tone would be at the top of the range, a L at the bottom, and M in the middle. However, that subrange can shift upwards and downward: if you have a "downstep", the pitch range is lowered (all tones get lowered), and "upstep" involves raising the pitch range.
The question of what the correct featural analysis of tone is remains an unresolved matter. The main problem is that most tone languages have 2 levels, few have 4 levels, very few have 5, and only 1 has six. Without a more robust sample of multi-level languages, it's really not possible to make proper generalizations qua claims about how human language works.
The complexity of tone systems really is not about the inventory of elements, it is about how tone plays a role in grammar, e.g. as a marker of tense or syntactic relations, and how underlying tones change radically in surface representations (for example, the tone of this word shifts into the next word).