What is the frequency of this note?
Japt, 41 37 35 34 bytes

I finally have a chance to put ¾ to good use! :-)

55*2pU¬®-1¾ª"C#D EF G A B"bZ /C} x


Try it online!

How it works

          // Implicit: U = input string, C = 12
U¬®    }  // Take U, split into chars, and map each item Z by this function:
-1¾       //  Subtract 1.75 from Z. This produces NaN for non-digits.
ª"..."bZ  //  If the result is falsy (NaN), instead return the index of Z in this string.
          //  C produces 0, D -&gt; 2, E -&gt; 4, F -&gt; 5, G -&gt; 7, A -&gt; 9, B -&gt; 11.
          //  # -&gt; 1, and b -&gt; -1, so we don't need to calculate them separately.
/C        //  Divide the index by 12.
x         // Sum.
2p        // Take 2 to the power of the result.
55*       // Multiply by 55.


Test cases

All the valid test cases come through fine. It's the invalid ones where it gets weird...

input --&gt; output       (program's reasoning)
A0  --&gt; 27.5           (Yep, I can do that for you!)
C4  --&gt; 261.625565...  (Yep, I can do that for you!)
F#3 --&gt; 184.997211...  (Yep, I can do that for you!)
Bb6 --&gt; 1864.6550...   (Yep, I can do that for you!)
A#6 --&gt; 1864.6550...   (Yep, I can do that for you!)
A4  --&gt; 440            (Yep, I can do that for you!)
D9  --&gt; 9397.27257...  (Who says that's too high?)
G0  --&gt; 24.49971...    (I've heard that note before.)
Fb5 --&gt; 659.25511...   (Wait, Fb isn't supposed to be a note?)
E   --&gt; 69.295657...   (I'm gonna guess that the missing octave is 1¾.)
b2  --&gt; 61.735412...   (I assume that b means Cb...)
H#4 --&gt; 261.625565...  (H# is C!)