If I open 2 tabs of https://www.szynalski.com/tone-generator/ and then listen to 440+880, and then change 880 to 850 it is a world of difference. I would definitely describe that difference as dissonance and consonance.
Now the overtone series IS important and is not always 'simple ratios', a good example in a real instrument is the strong minor third overtone of a carillon, and as expected writing in major for that instrument is hard.
Thank you for that link. I never thought of my browser as a test bench before. (Of course, now I want a DVM, function generator, scope, logic analyser, spectrum analyzer and all the other goodies ;) ).
I'm not sure if this is the same thing as consonance/dissonance, but the graph of sin(x) + sin(2x), an octave, is regular and pretty and the graph of sin(x) + sin(sqrt(2)x), a tritone, is much less so.
Except that if you use a frequency ratio of sin(x)+sin(2.01 x), which is really very close to an octave and really sounds just as consonant as an octave to almost all people, you almost the same "dissonant" picture:
The strange thing is, none of these "simple ratio" theories account for the fact that our brains allow a lot of "fuzziness" around these simple ratios, so much that you can't really call them simple ratios as they encompass a whole bunch of not-simple ratios as well.
That sine wave sounds a bit "fuzzy" to me, maybe the generator adds a small amount of overtones or aliasing. I tried another generator (http://onlinetonegenerator.com/) and the consonance feels weaker.
Interesting, I still hear it the same, dissonant and consonant, perhaps western music ruined me. Thanks for sharing.
Edit: Didn't see the url, makes my old reply obsolete:
Interesting. I tried to avoid clipping/aliasing by using audacity with as high quality audio as my system allows and I can still reproduce pretty much exactly what you hear on those websites. https://vocaroo.com/i/s0Be5CexLgVs is 440hz, then 440hz+880hz, then 440hz+850hz. But I would be interested in any repeatable signal that does not harmonize at all so do share!
Now the overtone series IS important and is not always 'simple ratios', a good example in a real instrument is the strong minor third overtone of a carillon, and as expected writing in major for that instrument is hard.