If an eighth note is half the duration of a quarter note, would the duration of an eighth note played at 60bpm be equal to the duration of a quarter note at 120bpm?
You are correct, OP.
An eighth note at 60bpm is in principal the same time duration as a quarter note at 120bpm.
We can set up a simple equation to prove this:
eighth note * 60 beats per minute = quarter note * 120 beats per minute
[We put beats, not minutes, in the denominator because it correlates to the fraction of a beat.]
1/8 * 60/60 = 1/4 * 60/120
60/480 = 60/480
If yes, why was the concept of different notes, like half, quarter, eighth, formed?
Cutting up a measure into equal parts made it easier for musicians to communicate.
And, I think the answer to this question is just that simple. Cool!
Wouldn't it be easier to have just one base/reference note and vary the tempo of the song whenever necessary?
This is actually pretty much how it works. Most songs have their quarter note set to a certain number of bpm.
We can consider the quarter note the 'default base/reference note'.
Composers and arrangers that choose otherwise, such as in marching bands that use 'cut time', do so in order to make more of the music written in quarter notes. The idea is that this makes the music easier to read.
Some pieces of music use a half note as a 'default base/reference note'.
Of course, there will always be composers and arrangers that do not follow 'good musical grammar' rules. They will not make the beats visible in a measure, or they will choose a *crazy time signature like 10/8 for seemingly arbitrary reasons. Music is an art, I am aware, but I am sure we all prefer to read legible music when I can.
Best practice in musical notation should be to make your music as easy to read as possible, so do whatever you can to that end.
And, is it easier to set your bpm to a quarter note or a half note? That depends on how fast you want to go. A quarter note set to 120 is normal, and a half note set to 120 is twice as fast!
I enjoyed your questions very much. Thank you for your post.
Robinson Tyndale G.
*Please excuse my snark. Some crazy time signatures are integral to certain grooves.
I would also like to say, though, that if you are a composer and you elect to use an abnormal time signature, you should have a really good reason.