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but does the 4/4 mean that 1/4 notes are one beat? Otherwise, wouldn't it be 2/4?
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2/4 time would mean there are 2 beats per measure and that a quarter note gets the value of a beat.
In a more generic definition, the upper number is number of beats per measure (2) and the lower number identifies what type of note gets a beat (quarter note).
If one wrote out a song, you could write it in lots of ways. I could write it in 2/4 or 4/4 and the only difference would be how many beats there are in a measure. The 2/4 version would take twice as many measures, but would sound identical. The only difference is the person reading the music would be counting 1, 2, 1, 2, 1... in the 2/4 version and 1, 2, 3, 4, 1, 2... in the 4/4 version. If someone really liked counting higher numbers you could write it as 8/4 and use half as many measures as the 4/4 version but counting 1, 2, 3, 4, 5, 6, 7, 8, 1. This isn't usually done, but you can do it.
Back to the confusion in the original post: If I go to a jam and some girl next to me is playing the same song, in my mind it could be 200 BPM and in hers 100 BPM. The difference would be for every time I play an 1/8th note, she is playing a 1/16th note, when I am playing a 1/4, she is playing an eighth (and a similar relationship of 2 to 1 would exist for all note lengths). Neither one of us is wrong. The time value of a note gets tied to a particular type of note only when you are writing or reading music.