I've seemed to stumble upon a possible variant of the Blancmange Curve, using recursive Sine waves using the following function:
$$
f\left(x\right)=\left(\sum_{n=0}^{a}\frac{\left|\sin\left(2^{n}x\right)\right|}{\frac{1}{2^{n}}+2^{n}}\right)
$$
Where '$a$' is the number of iterations. The result is this: 
I've come to notice that this curve bears a moderate resemblance to the Blancmange Curve (also referred to as the Tagaki Curve):
$$
s\left(x\right)=\left|\lfloor x+\frac{1}{2}\rfloor-x\right|
$$
$$
\sum_{n=0}^{k}\frac{s\left(2^{n}x\right)}{2^{n}}
$$
But I do note, there are many differences, both in the formula, and in the resulting curve, so if anyone could clarify what this curve is, if it is indeed a different curve, this would help.