Wikipedia gives multiple candidate definitions for the blues scale.
A musician showed me how to play blues e.g. in C by smudging say Eb and E.
The actual note lies somewhere between. But where? Is there any rationale for placing this note? Is it some ratio with regard the tonic?
Is there a mathematical formula for placing the notes in the blues scale with respect to the tonic?
A look over the Blue Note article on Wikipedia that Shevliaskovic linked talks a bit about the tuning theory behind Blue Notes, so I'd like to expand on that, as you mentioned wanting a "Mathematical" definition of these notes.
The article states that in order to overcome tuning hardships in keyboard creation in the 18th century, Equal Temperament was instituted and is the standard today. Rather than following the Harmonic Series (harmony in nature) or Pythagorean Temperament (the circle of fifths), it divided the octave into equal parts. That means that each note we hear in Equal Temperament is slightly out of tuning for "natural" harmony, which is what the African immigrants would have been accustomed to.
This brings us to a bit of Just Intonation, which is the practice of tuning notes to whole number ratios. Intervals between notes in equal temperament are expressed with exponents. For example, a minor third in ET would be 23/12 and a major third would be 24/12. There are many intervals in Just Intonation that are considered major thirds and minor thirds, but we'll look at two of the most basic ones. The major third is 5/4 and the minor third is 6/5. Without too much discussion into specifics, let's consider these the "natural" harmony that is ideal to our ears. You can see how they are tuned differently (in cents):
JI ET
Major 3rd 386.31¢ 400¢
Minor 3rd 315.64¢ 300¢
You'll see that the M3 is a little flat from Equal Temperament and the m3 is a little sharp. This is one way that a "blue" third may have pulled inward toward the half-way note that it is. It is also worth mentioning "neutral" intervals, which fall right in between major and minor intervals. 11/9, 60/49, 49/40, 27/22, and 16/13 are all examples of neutral third and they clock in at 347.4¢, 350.61¢, 351.33¢, 354.54¢, and 359.47¢, respectively.
This isn't all to say that these intervals are exactly what a blue note is, just how it may be derived. I can't seem to find an scholarly articles examining blues singers or players, so I guess the specifics of this mystery will still go unsolved.
That blues note is nebulous. It can be, and is, anywhere between a minor 3 and a major 3. Listen to blues players, and you'll hear it bent fully from min. to maj., or just hinted at with a tiny flick from minor upwards. The listener probably completes the bend in his mind's ear. It sometimes gets played as a straight major that gets wobbled down to minor and back, in a wide vibrato. As Shev says, it works better on some instruments than others - guitar and sax being a couple that use it a lot.
Each player will use a blue note in a different way in different circumstances, so there isn't an actual 'proper pitch' for it.
The formula you ask for is generally regarded as - minor blues, 1,b3,4,b5,5,b7 , and major blues,1,2,b3,3,5,6 - which actually translates to the relative major and minor notes, rather like the pentatonics ( and full major/natural minor).
The so called 'blue note' has its roots in the African immigrants in the States.
Back in Africa, they didn't have the piano to tune their voices to, so they sung what they liked best. When they came to the Western World, they found out the piano (and other instruments of course) and they learned to play it.
When they begun to sing the blues, songs based on what they used to sing back in their country, they found out that a note they liked wasn't on the notes on the piano. So, they played E natural and E flat at the same time and found that the note they wanted was in the combined overtones of those two notes. I don't think there is an exact 'position' of the note. Just that it is between these two.
It is famous in Blues, to play minor scales over major chords. That is why the musician you mentioned said to play Eb (from C minor) when you are over a C chord.
You cannot play this note in a piano. You can play it on other instruments, like a trumpet, a double bass or you can sing it.
You can read more on the Blue Note on Wikipedia.
The archetypal bluesy sound comes from bending and inflecting the notes within certain ranges. When soloing, I personally play the blues scale on guitar as a pseudo-pentatonic something like this (C tonic):
C
a 'window' around Eb, covering the range down to D and up to E.
F, bending up a little (maybe not as far as Gb)
G
Bb, with scope to bend up a little (but maybe more like a quarter tone - not as far as B)
(Adapted from my answer to Blues Scale Notes - major 6th?)
