When you play multiple notes on a bass, you get the notes you're playing, plus some difference frequencies. The difference stuff is due to non-linearities in our pickups, amps, speakers, and ears. I'll skip the Physics/Psychoacoustics lecture here, and assume that's a good enough explanation for this thread) If you play a Low E (41 Hz) and an octave higher (82), the difference note (82-41) is 41 Hz - that same as your low E. That works. But a Low E and a third up, the difference note is REALLY low in frequency - that's the source of your mud.
As you go up in pitch, you can play closer intervals and get away with it. Octaves work anywhere. A 5th (which is a note at one and a half times the frequency, so the difference note is an octave below the lower note) works OK somewhere a few frets up on the A string and up (exactly where it starts to sound OK depends somewhat on your gear) - I sometimes use the C. C#, or D on that string plus a fifth up at times to make it sound thicker (since I can't play it lower - assuming I'm on a 4 string). You can play thirds, but as they're closer n pitch than 5ths, you need to be even further up in pitch for that to sound OK.