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Let a, b, c denote three different musical lines. My question is, would it be true to claim that,

If a and b sounds good together and if b and c sounds good together, then a and c sounds good together. In other words, is there a kind of transitive property in music?

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Interesting way of framing this question!

The answer, alas, is a resounding no.

Imagine you have a C in Line A. Cool.

Now imagine you have an E in Line B. This makes a great major third (a consonance) between Lines A and B. Perfect.

Now forget about Line A and imagine you have a B in Line C. Between Lines B and C, you have a perfect fifth. Perfect again (and literally this time!).

So Lines A and B sound great together, and Lines B and C sound great together.

But Lines A and C (pitches C and B) create the harsh dissonance of a major seventh. Womp womp.

Transitive property of music fails. QED.

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  • Although I would say that a major 7th isn't that dissonant. Better to create a P4 between the B that is a minor 2nd below the C and the E above the C. Then you have M3 from C up to E, P4 from B up to E, and then the "bad" m2 from B to C.
    – Todd Wilcox
    Commented Jul 13, 2016 at 17:43
  • Without getting into a debate about the goodness or otherwise of an isolated 7th (after all, this is the 21st century, not the 16th) try this for an example. Line a: G A B C D E F G. Line b: E F G A B C D E (a third lower.) Line c: C D E F G A B C (a 6th higher than Line b). We can probably all agree that a+b and b+c sound fine but a+c doesn't quite make the grade.
    – user19146
    Commented Jul 13, 2016 at 22:07
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No for two very important reasons. The first and most important being harmony is a subjective subject. There's no guarantee if you like how lines A and B sound that anyone else will. How good something sounds is in the ears of the beholder.

The second is from a counterpoint perspective of harmony. How lines A, B, and C interact is completely independent of each other. Just because A and B are good counterpoint and B and C are does not mean A and C will be. They could be sound from a counterpoint perspective or the could be terrible like having parallel 2nds. If you listen to fugues you will start to understand how adding additional lines adds complexity and may not "sound good".

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Richard and Dom give a good explanation of why not, so I'll just give a proof by demonstrating a specific counter example, which you may find interesting: Telemann's Canonic Sonatas.

These are strictly imitative canons, so the second voice plays a time-delayed repetition of the first. If you consider a melody that contains the phrases A-B-C, then when the first voice begins playing B, the second voice is playing, A (so A and B sound good together). Then, when the second voice reaches B, the first voice has moved on to C (so B and C go well together).

What is astonishing about these sonatas is that they modulate (change keys). This means that phrases A and C might be in 2 different keys, with phrase B cleverly designed to be tonally-ambiguous so that it can fit into either key. Obviously, A and C, being in different keys, can be shown to not work well together. Yet B blends seamlessly with both of them.

Telemann wrote a number of these, but here's one example:

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