The Actual Purpose of the Bottom Number in Time Signatures [duplicate]

Question

I understand that 4/4 time signatures mean that there are 4 beats per measure, and that those beats are quarter notes. However this is the point where I get confused as I’ve listened to and have seen several compositions in 4/4 time signature where not a single note was actually a quarter note; however, the notes or combinations thereof within the measure doesn’t exceed a quarter note either.

Let’s say, I compose a piece with 4 quarter notes in the first measure, a whole note in the second measure, and 2 half notes in the third measure, how is this still able to be considered 4/4 time signature even though none of the measures actually have “4 quarter notes”?

I’m confused because I figured that 4/4 time signature meant that every measure had to have four quarters notes exactly. However this doesn’t seem to be the case.

Considering this, I concluded that the actual purpose of the bottom number in time signatures is to represent the note value of each beat OR “any combination of notes as long as they don’t exceed note value which represents a beat. For example in 4/4 time signatures, you can have any combination of notes as long as the notes in each measure doesn’t exceed 4 quarter notes. But looking at it from this perspective makes the top number questionable as I thought it had to be exactly 4 beats in each measure as represented by the top number. Someone please provide clarification.

I’ve read in a few articles that says time signatures dictates the pulse of a musical piece, are there any references on which beats are accented and weak in each time signature? This probably would help a lot.

I’ve been teaching myself music for the past 3 years and I’ve pretty much grasped every concept thoroughly until the point where I’m doing advance ear training now. However this particular subject has plagued for the past 2 years.

Answer 1

Time signatures, X/Y, mean "the total allowed duration of a measure is X beats. Any combination of notes whose total duration is X beats is okay. One beat will be represented by a Y note."

In X/4 time, then, we set "quarter note = 1 beat". It follows that an eighth note is 1/2-beat, a half note is 2 beats, etc. A measure can contain any combination of notes as long as their total duration is X beats (or, the time equivalent of X quarter notes).

In X/8 time, we set "eighth note = 1 beat". it follows that a sixteenth note is 1/2-beat, a quarter-note is 2 beats, etc. A measure can contain any combination of notes as long as their total duration is X beats (or, the time equivalent of X eighth notes).

The below three pieces of music would sound identical when performed; they differ only in the notes used to represent time durations.

For more on the "feel" of a time signature, see How should I understand time signature? and Is there a rule for the different sounds of time signatures X/2, X/4, X/8 etc not just defined on a case-by-case basis?.