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how can we theoretically understand that chords are compatible? let's imagine that there is a chord X1 and X2, what rules/conditions must these chords follow in order to be compatible? I have a similar question about dissonances, because their good sound depends on the context, but how is this explained theoretically - is there any algorithm to check the compatibility of chords?

upd: Let's just imagine that we are standing at the piano and I am playing two chords at the same time, the question is: "why do they sound harmonious?", or vice versa: "why don't they sound harmonious? ", or even like this: "why does this dissonance sound normal with such chords? ". Everyone had a moment when you made a mistake in the note when playing and it sounded terrible, but where is this line between bad and normal? Where can I find a theoretical answer/musical algorithm like: "with such a chord, such notes/chords will sound bad"...or..."being in such a scale or key, such notes will sound bad" (please, please don't talk about the quarto-screw circle - there are too few variations)

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A partial answer is to consider the three Common Practice Period interval classes.

Perfect consonance: unison, octave, fourth*, fifth. Imperfect consonance: major and minor sixths and thirds. Dissonance: all seconds and sevenths, all augmented or diminished intervals. *fourth: dissonance against the bass note, else consonant

The number of dissonant intervals approximates the overall dissonance as each dissonant interval (according to counterpoint rules) has to be resolved to a consonant interval.

Of course, the music may not call for the resolution of dissonance at all.

Ludmila Ulehla's book, "Contemporary Harmony" has several lists ranking chord types by dissonance. This may be useful.

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  • Upvote for assuming the frame of reference that the question assumes, even if there's no universal answer.
    – Andy Bonner
    Commented Dec 5, 2023 at 17:31
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    @AsdFgh Note, even though this is the closest you can get to an answer, there will still be problems. For instance, you might find a C and an F# played simultaneously to be dissonant (a tritone). But add D# and A, another tritone, and you have a diminished 7th chord, which is used often enough without objection.
    – Andy Bonner
    Commented Dec 5, 2023 at 17:35
  • Or add an Ab and Eb to the C and F# to get a dominant 7th chord. Or a D and an A to get a different dominant 7th chord.
    – Todd Wilcox
    Commented Dec 5, 2023 at 19:05

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