Hope I don't screw things up by adding confusion to the topic.
A Time Signature functions by measure and may affect Tempo by how the piece is written (2/2 faster than 4/4) but that isn't the end-all. Most early composer's used words to describe Tempo in their pieces (Larghetto, Allegro, etc....) to express Tempo in the Time Signature, from Slow, Not Too Fast, A bit faster, Moderate, etc....... Metronomes, BPM and the such were introduced later and seemed almost mechanical by many Old School Conductors who only used a Baton to keep Tempo and instruct each beat exact for everyone in the group.
Nowadays, when Musicians refer to a "Beat" in terms of Tempo, it's normally referring to a quarter note, regardless of the Time Signature. If the Tempo is 100 BPM, it's equal to 100 steady quarter notes per minute.
To find out how many beats a note takes, you must know the Time Signature.
To find out how long a beat is, you need to know the Tempo.
4/4 Time Signature can be played in many ways; (Tempo - 60 bpm)
A) 4/4----4 quarter notes= 4 beats to the measure, quarter note gets one beat;
1........2........3........4
B) 4/4----8th notes= 4 beats to the measure, quarter note gets one beat;
1 and..2 and..3 and..4 and
C) 4/4--- mixture of quarter, 8th and 16th notes= (4 beats, same as above)
"1....2 and..3....4" "1....2 E and ah...3 and...4"
Although the notes played per beat increased, the Tempo did not increase.
Another important concept is Tempo verses Rhythm. You can increase or decrease the Tempo of a song without changing the Rhythm. Rhythm is the glue that connects the notes in the Time signature with placement.
Hope I didn't confuse anyone beyond repair..........
