I understand that 60 BPM means 60 Beats Per Minute. And if the "BPM beat note" is a quarter note, it means a quarter note has the duration of 1 second (each quarter note should be held 1 second when playing). But what if I don't want hold each beat 1 second but instead 0.5 seconds, yet still keep the frequency of them (still 60 BPM). Meaning, there will be a gap of 0.5 seconds between the actual beats. So I will actually play 1/8 notes as the beat.
Whats's the correct notation for this so it all fits together? If I haven't missed anything, I can't really use the BPM notation to address this, right? I assume I should notate the beat note as 1/8 notes. Should then the time signature address this? Example: 60 BPM - Quarter note, Time signature 4/8 (the "beat" note of the time signature = 1/8 note). Or maybe it doesn't really matter, and I could simply use 4/4 as the time signature?
I heard that BPM and the time signature are independent of each other, but as they both address the beat note type, I assume they in most cases should always be the same: if the BPM specifies 1/4 note as beat note, the time signature should logical also be x/4. If not, why?