Is a Whole note always 4 beats? Why is a dotted quarter 1.5 beats in common time but 1 beat in 6/8 time? how many beats is a quarter note in 6/8 time? I'm really confused.
The US names whole note, half note, quarter note, eighth note, etc have some obvious advantages over the UK names semibreve, minim, crotchet, quaver, etc. However, they introduce a different confusion since the whole note is only the whole bar in 4/4 or 2/2. In 3/4 a whole note is more than a bar and hence not used. In 5/4, it does not fill a bar.
6/8, and other time signatures with lower number 8, would suggest that the beat is an eighth note or quaver. However, these are used when it is wished to be able to divide a beat into 3 rather than 2. There will be 6 8th notes (or others of the same total duration) per bar they will not each be considered a beat. A beat will be 3 of these hence a dotted quarter note. These time signatures are called "compound time".
The system is not entirely logical, you just need to remember this quirk.
This Wikipedia article is quite good.
To further confuse - a 'whole note' is called a semibreve which in itself is half of a breve - a very old note, which ironically translates as 'short'!! That semibreve has become known as a whole note and takes up the full bar (measure) in the common 4/4 time signature.
Its actual played length or duration fills any bar of 4/4, no matter what the tempo (bpm) of the piece is. That tempo is usually the only deciding factor as to how long a beat is.
6/8 time comprises two beats, but each one is actually written as a dotted crotchet, so the whole bar is 3 crotchets long - or put another way, 6 quavers (eighth notes), and counts in two ways - a slow 1-- 2-- or a quicker 123456. This is where it all seems to lose credibility, as we're now in compound duple time. Compound = not standard/straightforward ! But - we're not in 4/4, or even 3/4, so the 'rule' that applied there doesn't have to here.
In 6/8 (and 9/8 and 12/8) one 'beat' is recognised by a dotted crotchet (dotted quarter note), as when writing the music out, it conveys more sense as to the feel of the piece. So, in one way, a 'beat' = a crotchet doesn't transfer happily to this compound time, as now a 'beat' = a dotted crotchet.
In this answer, I've referred to a 'beat' as the overriding pulse of a piece - the foot tapping part.
Yes, most seem to find this concept confusing, but it would appear that no better way of portraying it has revealed itself - yet !
I've had guitar students who struggle with the issue, so I sympathise with this question. Let me suggest a few thoughts that might help.
I would start with one note and get it absolutely clear in my mind. For instance consider the quarter note. This note is also called the crotchet, and it is one beat long. So it has three different labels. We need to take the time to understand how they are related.
"Quarter note" describes the note proportionally. The most common time signature is 4/4 time (four beats in each bar). In that music a whole note is a note that lasts for a whole bar. By correspondence a half note is two beats, a quarter note is one beat, and so on.
But this does not mean that the quarter note is variable. It is not proportional in all time signatures. What has happened is that 4/4 time, being the most common time signature, has been the foundation for defining the length of a quarter note, and that same label and length flows through to other time signatures.
A good way to get a feel for this is to set up a simple rhythm: listen to a metronome, follow the ticking of a clock, tap your foot. Then start counting 1-2-3-4-1-2-3-4-1-2-3-1-2-3. What you are doing is toggling between 4/4 time and 3/4 time. The length of the note hasn't changed. It's just that in 4/4 time there are four quarter notes, while in 3/4 time there are three quarter notes.
So we have three ways of describing the note. We can call it a quarter note, based on the proportion of a bar that it fills in the default 4/4 time. We can call it a crotchet, based on its historical origin. (I think it is French and has to do with the appearance of the symbol for this note.) Or we can name it a one beat note, based on its length.
In music lessons my habit and my suggestion to my students is that they call it a one beat note until they are confident with note lengths. This has the huge advantage of consistency and clarity. We see this in the OP:
Why is a dotted quarter 1.5 beats in common time but 1 beat in 6/8 time?
In fact this is not the case. The quarter note has the same length in both settings. In 6/8 time the 8 tells us that the bars have an eighth note or half beat rhythm, while the 6 tells us that there are 6 of those half beat notes in each bar. If there are two even notes played in a bar, then those notes will each be three half beats long. That equates to 1 and a half beats, hence the dotted quarter note. (If this is complicated, that illustrates the importance of starting simple and building one's understanding.)
The length of a note is always the same. Neither does it change with time signature, nor with the tempo.
Working in the 4/4 time signature I will explain how this works:
This is a whole note. A whole note is always 2 half notes, 4 quarter notes, 8 eighth notes and etc.
These are two half notes. They together form a whole note.
There are 4 quarter notes. They together form a whole note. In pairs they form two half notes.
This can go on and on. This is how the note duration tree looks:
Here is how dotted notes are formed. I guess you can get the pattern:
If we take the 6/8 time signature (6 notes which are eighth) then a whole note will be a quarter note more than the time signature allow (or two eighth more):
We can count 6/8 (8-2 eighth notes) as 3/4 and this way it's easier to understand that 3/4 is 4-1 quarter notes. Notice that you can always divide the notes using the tree mentioned earlier.
So to fill a 6/8 (3/4) bar we need something that is a quarter less. We can place a half note and a quarter note:
We can also use a different pattern and place a dotted half note which is a half note + a quarter note or we can place two dotted quarter notes:
Each quarter note is a quarter note + an eighth note so we get this:
And each quarter note as we know is formed of two eighth notes so we get this:
Now if we count the amount of eighth notes we get 6. so 6/8 is 6 notes which are eighth.