Andrzej K. answered 5d
Piano Lessons with a Professional Concert Pianist & Composer
A time signature has two parts to it:
1) The top number giving you the number of beats/counts per measure.
2) The bottom number giving you which note value receives that beat.
A time signature can be looked at as a fraction. For example, when you have a time signature of 4/4, it's telling you that a measure has FOUR QUARTERS in it, or anything that adds up to four quarter notes. Another example is 6/8, which is telling you that each measure gets SIX EIGHTHS in it, or anything that adds up to six eighth notes.
So when you look at a time signature, the top number is straight forward; if its a 3, it means there are three beats per measure, and that same rule applies for any top number in a time signature. This number can literally be anything the composer wants it to be.
The bottom number you can look at as the denominator (bottom number of a fraction), and if you place a 1 over it, that will give you the note value that received one beat. Ex. if the bottom number is 4, if you put a 1 over it, giving you a fraction of 1/4 (quarter), that is the note value that receives one beat. If you bottom number is 16, you place a 1 over it, giving you a fraction of 1/16 (sixteenth), that is the note that receives one beat. The bottom number can only be 1, 2, 4, 8, 16, 32, 64, 128, as those are the note values that really exist. Any other than these numbers, you're getting in to 20th century or newer music, and these are called irrational/dyadic time signatures.
That is the basic part of reading a time signature what the numbers mean. Though there are a few other rules to consider when looking at a time signature, like whether the signature is a simple or compound. One example is the time signature of 6/8, which is commonly felt in 2 beats per measure, with a subdivision of three for each beat and not actually felt in 6 beats per measure.. For that, we'd have to chat a little longer beyond just this message thread to understand all that goes into this part.