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The 3-half-step harmony of E₅ at 659 Hz and G₅ at 784 Hz creates an illusory note at 534 Hz, as it is separated by the same pitch difference. That new pitch is near enough to C₅ at 523 Hz to form a C major C-E-G triad but misses by enough to sound blatantly out of tune.

This is a minor point, admittedly, but has an irresistible visual analogy. Do you sense faintly-dark blobs where they don’t exist in some corner gaps of this Hermann  illusion?

The 3-half-step harmony of E₅ at 659 Hz and G₅ at 784 Hz creates an illusory note at 534 Hz, which is separated by the same pitch difference. That new pitch is near enough to C₅ at 523 Hz to form a C major C-E-G triad but misses by enough to sound blatantly out of tune.

This is a minor point, admittedly, but has an irresistible visual analog. Do you sense faintly-dark blobs where they don’t exist in some corner gaps of this Hermann  illusion?

Some two-note harmonies indeed have “a scientific basis for why adding a third note suddenly creates a qualitatively different effect.”   Three effects that sometimes play are role can be understood after a mention of combination tones.
1.  Timbre   (a subtle factor)
2.  Intonation (a significant factor)
3.  Tangibility  (a less scientific factor)

Many two-note harmonies go beyond implying a third note to actually creating an audible illusion of three-note harmony. The result creates occasional problems and opportunities, as introduced nicely in the weblog A Mind for Madness.

Some two-note harmonies indeed have “a scientific basis for why adding a third note suddenly creates a qualitatively different effect.”   Three effects that occasionally come into play can be better understood after a mention of combination tones.
1.  Timbre   (a subtle factor)
2.  Intonation (a significant factor)
3.  Tangibility  (a less scientific factor)

Many two-note harmonies go beyond implying a third note to actually creating an audible illusion of three-note harmony. The result can create both problems and opportunities, as introduced nicely in the weblog A Mind for Madness.

For some two-note harmonies there is indeed “a scientific basis for why adding a third note suddenly creates a qualitatively different effect.”   Three effects that sometimes play are role can be understood after a mention of combination tones.
1.   Timbre    (a subtle factor)
2.   Intonation  (a significant factor)
3.   Tangibility   (a less scientific factor)

Many two-note harmonies go beyond implying a third note to actually creating an audible illusion of three-note harmony. The result occasionally creates problems that can be masked by a truly-played third note. Unproblematic results, though, are introduced nicely in the weblog A Mind for Madness.

When things go poorly, however, an actual played third note can rescue a well-intentioned two-note chord gone bad, mainly by overshadowing disruptive effects.

Moreover, these tones will appear with different strengths and blends at different intervals during a two-note harmonic passage, like a third player who keeps changing instruments and styles.

Some two-note harmonies indeed have “a scientific basis for why adding a third note suddenly creates a qualitatively different effect.”   Three effects that sometimes play are role can be understood after a mention of combination tones.  
1.   Timbre     (a subtle factor)
2.   Intonation   (a significant factor)
3.   Tangibility    (a less scientific factor)

Many two-note harmonies go beyond implying a third note to actually creating an audible illusion of three-note harmony. The result creates occasional problems and opportunities, as introduced nicely in the weblog A Mind for Madness.

When things go poorly, an actual played third note can rescue a well-intentioned two-note chord gone bad, mainly by overshadowing disruptive effects.

Moreover, these tones will appear with different strengths and blends at different intervals during a two-note harmonic passage, like a third player who keeps switching instruments and styles.