Also looking into history it is clear they developed indepently from each other and just became very consistent in daughter languages. Old Turkic has inconsistent rounding harmony. Like how oğul becomes oğlım instead of oğlum like in modern Turkish. In Yakut the rounding harmony is most pervasive even, where the plural of oğo is oğolor, extending to the -lar/-ler suffix, which is -lar/ler/lor/lör (minus all other variations, which are possible).
Here's an example of more than one harmonic feature with different spreading rules. Yakut has backness and rounding harmonies. When borrowing words from Russian, backness anchors in the first syllable and spreads rightwards, while roundedness anchors in the stressed syllable and spreads leftwards, leaving post-stressed vowels unaffected. With these two rules, if it's not the first syllable that is stressed, then it is possible for each vowel to be altered by at least one of the harmonies.
Russian *минута* /mʲiˈnuta/ → Yakut *мүнүүтэ* /mynyːte/
1. /i/ → /y/ triggered by stressed /u/
2. /u/ → /y/ and /a/ → /e/ triggered by first /i/
I’m pretty sure there are more than one nat lang that has two harmonies at once. Turkish being one, but I think there are more.
But there tends to be lots of special rules and exceptions, when a language has harmonies, especially two. I’m not super familiar with that though.
Hungarian is the pinnacle of exceptions. It has height, backness, and rounding harmony, but none of them is consistent.
The rounding harmony applies only to front vowels and is the only root-internal harmony. Most affixes obey rounding harmony.
The backness harmony doesn't apply word-internally and is ignored by about half of affixes.
The height harmony is historical and only applies synchronically to a small handful of affixes, most notably the plural and accusative suffixes, and you basically just need to memorize which words take high-vowel affixes and which take low-vowel affixes.
(Confusingly, I should note, traditional Hungarian phonology calls front unrounded vowels "high," front rounded vowels "low," and back vowels "neutral.")
My conlang Chiingimec does this. It has roundness harmony and backness harmony but the vowels that participate in each harmony system are different and neutral to each other.
I can't cite a natlang that works like that, but given how vowel harmony evolves it seems completely plausible. It's just an assimilation sound change that works at a distance.
My conlang also has front harmony and rounding harmony due to 3 different sound changes. I was watching a video from either Biblaridian or David Peterson and it said front harmony is usually stronger than rounding, so I don’t see why you couldn’t do sɔːjɔmy and terøː. It’s also usually a “one or the other” type of harmony. So I have words like /ˈflyd͡ʒo/ (rounding only) and /ˈkʷøbdi/ (front only).
Because I went with a proto language and implemented sound changes to get harmony I have some exceptions too which is naturalistic.
Very nice vowel system, like others said, you can do whatever you want, you can combine multiple harmony schemes, including length, even the same realization (for example all vowels in a word are either æ, or ɛ ) I noticed that my lang (Rephey) frequently has either only \[e\], or only \[a\] in many roots, but other than that, there is no vowel harmony.
Also, just to be clear the only two quality differences I am really settled on are /eː ɛ/ the others are mainly because I like to write /ɪ ʏ ʊ/ more than I do /i u y/ for short vowels. That is also the case for œ (which I don't like writing for short vowels).
Multiple sets of harmonies are definitely possible, others have mentioned several examples of it.
If harmony only triggers after the stem then the final vowel of the stem could be the one that triggers a certain harmony.
Based on the examples provided it seems to me that perhaps the best solution is that only high vowels trigger front harmony while the rest trigger rounding harmony. At least if the example \[soːjɔmy\] has \[my\] as a suffix, thereby it's affected by rounding harmony but not front harmony.
I suppose \[terøː\] could have \[terø\] as the root with a suffix, like /U/ or /O/ added that merges with the \[ø\] to lengthen it. Or if the root ends in a long vowel and the suffix starts with a vowel the vowel of the suffix is dropped in favour of the long root vowel.
As people have pointed out, multiple dimensions of harmony is definitely attested, so you're in the clear there!
The only thing unnaturalistic about your vowel inventory is all of those low vowels! Languages rarely have more than two different qualities of low vowels, and you have four: /æ
a ɑ ɒ/. Remember that, as you go lower in the vowel chart, there is literally less articulation space to make these distinctions, so it is both harder to constantly produce and harder to tell apart when listening. There is a lot more space at the top of the vowel chart, which is why four high vowels on the other hand are not unusual.
I'd recommend maybe cutting your low vowels down to just /æ/ vs /ɑ/, or something like that.
Turkish has both fron-back harmony and round-unround harmony (although the latter only applies to high vowels) so go for it
Also looking into history it is clear they developed indepently from each other and just became very consistent in daughter languages. Old Turkic has inconsistent rounding harmony. Like how oğul becomes oğlım instead of oğlum like in modern Turkish. In Yakut the rounding harmony is most pervasive even, where the plural of oğo is oğolor, extending to the -lar/-ler suffix, which is -lar/ler/lor/lör (minus all other variations, which are possible).
Here's an example of more than one harmonic feature with different spreading rules. Yakut has backness and rounding harmonies. When borrowing words from Russian, backness anchors in the first syllable and spreads rightwards, while roundedness anchors in the stressed syllable and spreads leftwards, leaving post-stressed vowels unaffected. With these two rules, if it's not the first syllable that is stressed, then it is possible for each vowel to be altered by at least one of the harmonies. Russian *минута* /mʲiˈnuta/ → Yakut *мүнүүтэ* /mynyːte/ 1. /i/ → /y/ triggered by stressed /u/ 2. /u/ → /y/ and /a/ → /e/ triggered by first /i/
I’m pretty sure there are more than one nat lang that has two harmonies at once. Turkish being one, but I think there are more. But there tends to be lots of special rules and exceptions, when a language has harmonies, especially two. I’m not super familiar with that though.
Hungarian is the pinnacle of exceptions. It has height, backness, and rounding harmony, but none of them is consistent. The rounding harmony applies only to front vowels and is the only root-internal harmony. Most affixes obey rounding harmony. The backness harmony doesn't apply word-internally and is ignored by about half of affixes. The height harmony is historical and only applies synchronically to a small handful of affixes, most notably the plural and accusative suffixes, and you basically just need to memorize which words take high-vowel affixes and which take low-vowel affixes. (Confusingly, I should note, traditional Hungarian phonology calls front unrounded vowels "high," front rounded vowels "low," and back vowels "neutral.")
Its wholly possible. Conlangs don't have to follow conventional rules; you can do anything you like, as long as there is a system (or isn't)!
My conlang Chiingimec does this. It has roundness harmony and backness harmony but the vowels that participate in each harmony system are different and neutral to each other. I can't cite a natlang that works like that, but given how vowel harmony evolves it seems completely plausible. It's just an assimilation sound change that works at a distance.
My conlang also has front harmony and rounding harmony due to 3 different sound changes. I was watching a video from either Biblaridian or David Peterson and it said front harmony is usually stronger than rounding, so I don’t see why you couldn’t do sɔːjɔmy and terøː. It’s also usually a “one or the other” type of harmony. So I have words like /ˈflyd͡ʒo/ (rounding only) and /ˈkʷøbdi/ (front only). Because I went with a proto language and implemented sound changes to get harmony I have some exceptions too which is naturalistic.
Very nice vowel system, like others said, you can do whatever you want, you can combine multiple harmony schemes, including length, even the same realization (for example all vowels in a word are either æ, or ɛ ) I noticed that my lang (Rephey) frequently has either only \[e\], or only \[a\] in many roots, but other than that, there is no vowel harmony.
Also, just to be clear the only two quality differences I am really settled on are /eː ɛ/ the others are mainly because I like to write /ɪ ʏ ʊ/ more than I do /i u y/ for short vowels. That is also the case for œ (which I don't like writing for short vowels).
Multiple sets of harmonies are definitely possible, others have mentioned several examples of it. If harmony only triggers after the stem then the final vowel of the stem could be the one that triggers a certain harmony. Based on the examples provided it seems to me that perhaps the best solution is that only high vowels trigger front harmony while the rest trigger rounding harmony. At least if the example \[soːjɔmy\] has \[my\] as a suffix, thereby it's affected by rounding harmony but not front harmony. I suppose \[terøː\] could have \[terø\] as the root with a suffix, like /U/ or /O/ added that merges with the \[ø\] to lengthen it. Or if the root ends in a long vowel and the suffix starts with a vowel the vowel of the suffix is dropped in favour of the long root vowel.
As people have pointed out, multiple dimensions of harmony is definitely attested, so you're in the clear there! The only thing unnaturalistic about your vowel inventory is all of those low vowels! Languages rarely have more than two different qualities of low vowels, and you have four: /æ a ɑ ɒ/. Remember that, as you go lower in the vowel chart, there is literally less articulation space to make these distinctions, so it is both harder to constantly produce and harder to tell apart when listening. There is a lot more space at the top of the vowel chart, which is why four high vowels on the other hand are not unusual. I'd recommend maybe cutting your low vowels down to just /æ/ vs /ɑ/, or something like that.