I've written a melody in 4/4 time signature. The problem is that at 118 BPM the melody is ridiculously slow. I don't want to increase the BPM so I ignored the current time signature and simply shortened the gap between each note; making it faster at the same BPM. The problem now is that I can't seem to find the proper time signature for the faster version of the melody..
You could change it into 2/2 time; keeping quarter notes as is will make it twice as fast. Since the beat is now 2 quarter notes rather than one, it will be twice as fast.
If you want to have the melody change its speed without affecting the rhythm section, you're going to have a metric modulation - a modulation in tempo, in this case - only the melody will do a metric modulation. You could either do it continuous (where you increase the tempo over time) or disjointed (where you shift to the new tempo instantly). We call these modulations by their ratios - for example: double time, which is when the new tempo is twice faster than the old tempo, is called 2:1 (new tempo:old tempo). If only the melody were to go in double time, you could simply notate everything with half the note length (quarter->eighth and so on). Continuous, however, has a part when you're not in the old tempo and not in the new tempo, and that makes you sound "out of tune" rhythmically - analogous to bending a note. If you're gonna go for a continuous metric modulation, you might want to start it on the 1 beat and end it on the 1 beat (doesn't matter how many measures - if you're gonna go to a much higher tempo, you might want to make it 4 measures).
Good luck with your new polytempic music!