My understanding of the bottom number in a time signature is that this is how many beats there are in a whole note. So if the bottom number is 4, a whole note gets 4 beats. If the bottom number is 2 a whole note gets 2 beats and if the tempo is 60bpm then it lasts 2 seconds instead of 4.
Is this right?
"the bottom number in a time signature is ... how many beats there are in a whole note"
I wouldn't think of it as beats, necessarily; but certainly the denominator represents the note duration that fits 'that many' in a whole note.
In the US, of course, the maths is done for you: a quaver is an eighth note, so anything over 8 is in eighth notes; a minim is a half note, so anything over 2 is in half notes.
The problem with thinking in terms of 'beats in a whole note', is that this doesn't help you much with 3/4, or 5/4, or 9/8; unless your maths is very good.
(And 3/8 is often beated in one; 6/8 is usually beated in 2; 4/4 might be beated in 8 or 2, etc, etc.)
Also: metronome speeds are usually marked in terms of a note duration e.g. 'minim = 60', rather than just '60bpm'. So the 'calculation' you're making of the denominator in terms of a whole note doesn't really apply.
A time signature is similar to a fraction. In that the bottom number tells what the divisions are, while the top tells how many of them there are.
So, in 4/4 the bottom 4 tells each 'beat' is a crotchet (quarter note), while the top 4 tells there are 4 of them in each bar. Similarly with 3/4. 3 crotchets per bar. 5/4 tells 5 crotchets per bar, unsurprisingly.
6/8 tells there are 6 quavers in each bar, which is the same as 3/4 mathematically, but actually doesn't reveal that the rhythm of the bar is quite different, though. That's often where the confusion between 3/4 and 6/8 is found. 7/8 unsurprisingly tells 7 quavers (eighth notes) in each bar - but again, a little like 6/8, doesn't give a clue in itself how those quavers are grouped, rhythm wise.
The concept of a 'whole note' :- a whole note is called a semibreve, which is made up of two minims,(1/2 notes) or 4 crotchets,(1/4 notes). It is the hollow egg-shaped note, with no stem. It doesn't have too much bearing on things - except that it is the 'whole' that is divided, as explained in this paragraph. A whole note will fit into one bar of 4/4, but not into a whole bar of 3/4. Hence 3/4 gets called 'three-quarter time' quite often. That, and after all, it looks exactly like the fraction - 3/4!
Yes, that's correct. Just keep in mind that a whole note need not fit in a single measure. For example, in 6/8 time, a whole note gets eight beats, but each measure only has 6. Similarly, for 3/4 time, a whole note gets four beats, but each measure contains only three.
See also I don't understand the bottom number in a time signature, which explains the same concept in a different way.
There is also a concept of "irrational" time signatures, such as 4/3 or 6/10. The rule holds here as well. In 4/3, a whole tone would receive 3 beats, and a measure would contain four 1/3-beats. ("Irrational" in this sense is distinct from the mathematical idea of irrational numbers.)
