It has been defined that a Majority gate follows this formula concerning its behavior about outputs: $$\operatorname{Maj} \left ( p_1,\dots,p_n \right ) = \left \lfloor \frac{1}{2} + \frac{\left(\sum_{i=1}^n p_i\right) - 1/2}{n} \right \rfloor\>\>\text{(Wikipedia)} $$
It is true when more than 50% of its inputs are true. I have found that an exclusive one can be made by omitting the $-1/2$ by the summation.
My question is: is there a formula for finding if $x\%$ of the inputs are true? For example, $100\%$? $0\%$? $23.4\%$?
I am aware such constructions can be more-or-less obtained by using multiple logic gates (e.g. $100\%\equiv p_1\land\cdots\land p_n,0\%\equiv\top$, etc.). This is not what I am looking for.
$$k=\text{decimal equivalent to percent necessary}$$
$$\operatorname{Maj}_k \left ( p_1,\dots,p_n \right ) = \left \lfloor 1-k+\frac{\left(\sum_{i=1}^n p_i\right) - 1/2}{n} \right \rfloor$$
$$\text{OR: }k=\text{percent}$$
$$\operatorname{Maj}_{k\%} \left ( p_1,\dots,p_n \right ) = \left \lfloor 1-\frac{k}{100}+\frac{\left(\sum_{i=1}^n p_i\right) - 1/2}{n} \right \rfloor$$