Thanks Doug - I'm glad you're finding the lessons useful.
As for the bottom number of the time signature, it's a good question because it confuses a lot of people.
To expand a bit on Nutty's good answer, the top number is the important one. In fact it's the only REAL one. When you hear a piece of music, you can usually hear the music naturally divides into an equal 'measure' of beats. On hearing it we can usually say it has three beats per measure, or four, or whatever. But's that's ALL.
The bottom number doesn't exist until we come to notating the music. So let's say you compose a simple and very short song with just 10 equal length notes:
C E E |C E E |G A B |C--||
If you play those notes with a little emphasis on the bold notes, the "three beat per measure" time is unmistakeable. And we can hear that every note coincides with the beat
So when it comes to notating it, you know that the top number of the time signature MUST be THREE.
But how do you write the notes of your song? Do you choose 10 quarter notes? or 10 eighth notes? or even 10 half notes?
The good news is that it's up to you because whichever note length you choose they'll all sound the same. The tempo of the song is unaffected because it's measured in beats per minute, regardless of what note length you choose to equal one beat.
If you decide to write it as 10 quarter notes - then, as Nutty says, the quarter note will be the beat - and your bottom number will be four.
If you decide to write it as 10 eighth notes, then the bottom number will be eight.
Either way, they'll sound exactly the same, They'll still have exactly the same tempo in beats per minute. Whether it's 60 quarter note beats per minute or 60 eighth note beats per minute, they'll sound the same because 60 beats per minute is 60 beats per minute, whatever the beat happens to look like.
Here's an analogy - If I recite a poem to you from a sheet of paper. You can hear the words but you've no idea whether the words on the sheet are written using all capital letters, or lower case or even shorthand. And if you like the poem and decide to copy it yourself by ear, it doesn't matter if you write it the same way as on my sheet - The words will come out exactly the same when you recite it.
A common cause of confusion that you hear is when someone says something like "that song is in three-four". Many of us do it because it's a very common time signature, but strictly speaking, there's no way of knowing the bottom number without seeing the notation. Strictly, we should say it's in 'triple time'.
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