At the risk of a derail into what should be a whole 'nother thread... "blue" notes are another matter entirely.
"blue" notes aren't "in-between" or 'minor-over major"... they're tunable notes, they just require an entirely different conception of harmony that extends to the seventh partial, whereas Western music only extends to the fifth.
In our Western harmonic system, all twelve pitches of the equal-tempered scale "stand for" just, resonance-based pitches. And all twelve can be arrived at by reckoning only third and fifth partials in various directions. In Indian solfege, "mi", or the third (fifth partial) is known as "ga", and "sol" or the third partial is known as "pa." That will be helpful as we go on...
If I start with "C" as my tonic, the third partial ("pa") is G and the fifth partial ("ga") is E. (that's three of twelve)
The third partial above the third partial (G) is D. (four)
The fifth partial above the third partial (G) is B. (five)
The third partial above the fifth partial is G# (six)
The fifth partial above the fifth partial is A. (seven)
The tonic is the third partial of F --Mathieu would call this "pa below tonic" (eight)
The tonic is the fifth partial of Ab, or "ga below" (which duplicates the G# in equal temperament, but is a slightly different pitch in just tuning; this is the more commonly used of the two. Still eight)
"Pa below" (F) is the fifth partial of Bb, or "pa below pa below" (nine)
"Ga Below" (or Ab) is the fifth partial of Db, this is "pa below ga below" (ten)
"Ga below Pa below" is Db again (but a different Db!), as this is the fifth partial below F. (still ten)
"Ga below Ga below" is Fb (or E natural) again, but a different E natural by a few cents in just tuning (still ten)
"Pa below Pa" is redundant... that's just our "C" tonic (still ten).
But "Ga below Pa" is Eb, our minor third (eleven).
The next few in the sequence are redundant in equal temperament (but are actually distinct pitches in just tuning), so let's skip ahead to the first place we get the twelfth equal-tempered pitch class--
Ga above Pa above Pa, or the fifth partial above the third partial above the third partial above tonic (F# above D above G above C).
Now we've got all of our twelve equal-tempered pitch classes, using only third partials and fifth partials. Western harmony is thusly referred to as a "five limit" system, as we can limit the harmonic series to five partials and still find all the pitches we need.
Now for the blues...
The blues is descended in part from some West African musical traditions that utilize the harmonic series up through the seventh partial, which in C is that slightly out-of-tune (to the Western ear) Bb harmonic.
And these origins are what inform the "blue notes". The blue notes are tunable pitches that are by degrees either in-tune or out-of-tune, sure as a perfect fifth or major third. They're not just random places in-between major and minor. And it's also not random that the I, IV, and V chords are commonly played as dominant seventh chords, and that this isn't well-explained by Western music theory. The lowered seventh is nothing more arcane than an imperfect approximation of the seventh partial which is so integral to the music's tuning and harmonic conception.
The "blue" minor third is the seventh partial below the fifth partial below the tonic. In C, "pa" below is F. And F is the seventh partial of a pitch lying somewhere between Eb and E. The blue minor seventh is, of course, just the seventh partial itself (which Mathieu extrapolates Indian solfege to call, appropriately "blu").
So... not really "bitonal"... a whole other system of harmony.
Now all the reckonings we did above in the five-limit system, imagine doing all of those in the seven-limit system. That is a massive amount of tunable, singable pitches. To try and explain those via our Western system of harmony and solfege is reductive, and doesn't really "work" so well to explain the blues.
Of course, Western hegemony being what it is, the accepted 20th century explanation was that blues was "primitive" music and was somehow naive to our "sophisticated" Western concepts of harmony.
Anything but, really.