CoF is actually also much easier than it looks... especially when you use a keyboard (much weirder on guitar). I'll try to be more concise this time... lol.
One of the main things to remember is that in diatonic (seven note scale) theory we never skip or repeat a letter name. In ALL 12 keys we use ALL seven note names which of course are ABCDEFG. All we do is add sharps OR flats (never both) and change WHICH letter name we start on to create our scale pattern.
When constructing the Circle of Fifths we use the Major scale pattern of TTSTTTS. Of course after we are done we can then break each Major key down into its individual modes.
You start with the key of C Major which has no sharps or flats (so you only play the white keys from C to C).
C D E F G A B C
Now add the scale steps. We use Roman numerals for this (I have kept then all capitals so they do not indicate any Major minor chordal stuff... that is another lesson all together and would jsut confuse things). Our most important steps in all this will of course be I and V.
I II III IV V VI VII IC D E F G A B C Now write down the Tone/Semitone sequence underneath it (sorry if the formatting/display is weird... this forum mangles text based graphs and charts but obviously the T's and S's represent the gaps BETWEEN the note names... so C to D is a whole tone, D to E is a whole tone, and so on).
I II III IV V VI VII IC D E F G A B C
T T S T S T S
Play the scale on your keyboard. Pay attention to how it sounds and where the black keys are (which of course represent the Tones and Semitones... the spaces between the notes).
Now all you do is count up to the V, take the note that appears at the V and drag it backward to the I position. In this case the V is G so G becomes the new I. Also do not get confused by the two I notes. That last I is just the octave and is only there as a placeholder to show the final semitone jump. Of course once we move on to G Major that last I will become a G. A diatonic scale is only 7 distinct notes that resolves on it's octave/starting letter note (and hopefully I did not confus eyou more with that... lol, just drag the G backward to the I).
So take this...
I II III IV V VI VII I
C D E F G A B C
T T S T S T S
and do this
I II III IV V VI VII I
G A B C D E F G
T T S T S T S
Now play those notes on your keyboard (so G to G on the white keys only).
It sounds KIND of like the Major scale you played in C but slightly different at the end... right? That's because we are still in the key of C and playing Dominant mode which is Mixolydian.
Mixolydian is identical to the Major scale (Ionian mode) with one small but crucial difference. It has a minor 7th (meaning the 7th note has been flatted one semitone in comparison to the Major scale).
Off topic but useful info about this little difference is that flatted 7th is what makes the chord we build on the Mixolydian the Dominant chord. If we only play the triad that builds a Major chord (I, III, V) it creates a plain old Major chord just like the one on the Root (I). It is only when we add the 7th to the chord that the difference becomes noticeable. It creates a Dominant 7th chord whereas if we played the I chord with the 7th we get a Major 7th chord. They both create their own distinct sound. Also the Dom 7 chord only appears once in any given natural key whereas there are two Maj7 chords (I and IV) and three minor7 chords (ii, iii and vi). Just a bit of chord knowledge for you. The only other unique chord (built from triads and 7ths) is the diminished chord that appears on the vii and is really only used to move up the tonic for flavor (unless you are a jazz freak or metalhead... then it allows for the Devil's 5th sound made famous by Black Sabbath in their song Black Sabbath).
But I digress, so what do we do about that flatted 7th to turn this into a Major scale? Well we sharp it of course.
So this...
I II III IV V VI VII I
G A B C D E F G
T T S T S T S
turns into this...
I II III IV V VI VII I
G A B C D E F#G
T T S T S T S
The F is turned into an F#. Now play the new key with the F#. We now have introduced one black key to the physical scale and one sharp to the 7th step of the scale on paper.
Now pay attention to this as it is the entire trick to understanding the Circle of Fifths. As we move through the keys one at a time we will sharp whatever new note appears as the 7th step. We do this until all notes are sharps. Then we'll have to do the ONLY particularly difficult thing about the CoF which is switch to the flat keys but there is a trick to that too I will show you.
So let's move on to the next key.
The V of G Major is D so let's move that to the I...
So take this...
I II III IV V VI VII I
G A B C D E F#G
T T S T S T S
and do this...
I II III IV V VI VII I
D E F#G A B C D
T T S T S T S
Notice that sharp we created last time moved to the III. This is what will always happen as we work through the circle. So now again we have merely changed "modes" (to Mixolydian) instead of keys so again we will sharp that minor 7th which in this case is the C.
I II III IV V VI VII I
D E F#G A B C#D
T T S T S T S
Play the notes and you'll see again we have added one more black key.
So the action is this...
1) Drag the V to I
2) Add a sharp to the VII
3) Repeat
Starting from C you can do this a total of five times before weird things start happening. So including C Major you get 6 of the twelve keys using this method before you gotta put your thinking cap back on.
Go back to C and start the circle on your own. Go as far as you can and pay attention to what happens as you get further into the circle. Particularly starting at the 7th key where something odd will happen and then again on the 9th key where something really odd will happen. I will explain what is going on and what to do about it below but try this first.
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Okay... done? You should have been able to create the keys of C, G, D, A, E and B easily. Then you may have continued but wondered what in the heck was going on witht he next two keys F# and C#. Essentially in those keys to continue the Circle and preserve the letter order (never skipping a letter name or doubling a letter name) you would have to make some seemingly nonsensical sharps.
When you go to create the key of F# by dragging the V to I the note that lands on the VII is E. Of course if you sharp the note E you actually end up with an F because E and F are only a semitone apart. There is no physical E#.
So what the heck do we do? Well there are two ways to handle this. One is to switch to flats which I will explain later but for now I want to show you a little theoretical slight of hand that is acceptable to use in this situation.
We simply proceed as normal and pretend that F is actually E sharp. You write it in the key signature normally by putting the sharp on the top space of the Staff which denotes E. Now have preserved the letter order without using the F twice.
Example:
The following, if played, would create the tones of the F# Major scale but is theoretically improper because it has repeated the note F even though they are not the same pitch.
F# G# A# B C# D# F F#
This is correct if not a little odd conceptually and confusing to look at on a keyboard.
F# G# A# B C# D# E# F#
The same thing happens in the next key C# Major. This time it happens with the B which of course a semitone below C. This means by sharping the B we are actually playing an F but to avoid using B twice we just pretend as if B# exists. Also because we are using all the same notes as we did in F# Major as well we also still have that phoney E# as well.
C# D# E# F# G# A# B# C#
Weird, huh? Without going into the flat keys yet I will provide one solution for one key and point out a glitch in the Matrix that I just now noticed (Whoa!).
For C# Major you can instead use Db Major which avoids this quirk altogether.
For F# Major you can instead use Gb Major but unfortunately that ends up causing the same issue except you end up using a Cb which of course is a B. There is no solution I know of that avoids this problem for this key so this is the ONLY key out of the 12 where you are absolutely FORCED to use enharmonic trickery. I actually just realized that now while typing this all up so hey, here I am learning new stuff too... or I made a mistake but I don't think so. Freaky.
Just so you know I use a lower case "b" to denote flats when typing up note names. That has caused confusion in the past. So "Db" means "D flat". M'kay?
Moving on...
Now we have reached full saturation of sharps. If you try to take another step in the circle you would be forced to double sharp the note that appears at the 7th step.
The 5th step of C# Major is G# which results in this...
G# A# B# C# D# E# F## G#
Although double sharps (or flats) are used occasionally they are best to be avoided because they pretty much NEVER serve a useful purpose and will almost always lead to confusion.
Enter the Flat Keys and what is sometimes referred to as the Circle of Fourths.
There are two ways to continue. We could simply translate the key of G# to use flats but that is kind of a pain in the butt... especially when you are trying to work fast or are just learning the CoF concept.
The other, much simpler way is to start back at C and work backwards through the circle and add flats instead of sharps.
In the Circle of Fourths what we do is take the IV step and drag it back to the I like this...
Take this...
I II III IV V VI VII I
C D E F G A B C
T T S T S T S
and do this...
I II III IV V VI VII I
F G A B C D E F
T T S T S T S
Then instead of adding a sharp to the VII we add a flat to the new IV like this...
I II III IV V VI VII I
F G A Bb C D E F
T T S T S T S
Just an interesting thing to note is that F Major is the ONLY key with a non flat or sharp Root note that contains a flat. All the other keys with natural notes as their root will ALWAYS use sharps instead of flats. This can make it a little easier to remember/solve keys.
One more step and then you can solve the rest yourself...
Take this...
I II III IV V VI VII I
F G A Bb C D E F
T T S T S T S
move the IV to the I (notice that the flat you created in the last step is now the Root)...
I II III IV V VI VII I
Bb C D E F G A Bb
T T S T S T S
and add a flat to the IV...
I II III IV V VI VII I
Bb C D Eb F G A Bb
T T S T S T S
Now work through the keys this way until you run out of flats then continue reading...
This time you will have had to use seemingly incorrect flats once you got to Gb Major and Cb Major.
You will have run out of flats after Cb Major which means you would switch to sharps again to create E Major.
Now I could type up how to do the linear transition from Sharps to Flats and vice versa but this is, again, getting very long so I'll just close up with some important tidbits you'll want to be aware of.
The enharmonic keys (as in the keys that are tonally the same but can be constructed using either sharps or flats) are...
C# Major = Db Major
F# Major = Gb Major
B Major = Cb Major (Probably best not to use the flat version of this unless there is some odd application I am not aware of)
As noted above the ONLY key(s) where you are FORCED to sharp or flat a note "unnaturally" to preserve the letter order is F#/Gb Major where you have to either sharp the E (F# Major) or flat the C (Gb Major). Remember F# Major and and Gb Major are the same key tonally so this is just one key where you are forced to do this enharmonic trick in either version. For the other keys where this occurs you can use their enharmonic equivelents to avoid the issue.
Of course what I have shown above is the manual, step by step way of doing things. This was to illustrate the math. You can easily solve sharp key signatures by simply counting the number of sharps in the key sig then, starting at C Major, just keep finding the 5th until you have moved through the same number of steps as there are sharps. For flat key sigs you would do the same thing except finding the 4th instead. Keep in mind that C does NOT have a key sig so you would factor that into your counting.
Learning the CoF is not just about writing and solving key sigs. It is extremely helpful for knowing EXACTLY what notes appear in what keys in a quick, logical way. This is more useful on guitar than on piano (because we don't have handy dandy sharp/flat markers like those black keys) but if you know the key but are sketchy on the notes you can just move take a pencil and paper (or do it mentally) and quickly work through the circle until you reach the correct key. You then just add the appropriate sharps or flats. No guesswork or fumbling for notes.
By getting familiar with all this it also makes identifying relative minors and modes much easier. You will start being able to identify what key these relative scales belong to which helps navigating keys, writing and improvising easier. It will also make it easier to identify "accidentals" that don't fit into the key which could denote any number of things like a shift to Harmonic/Melodic minor (which use the same key signatures as their natural minor versions and the accidental notes are written within the music with sharps/flats/naturals) or non-diatonic scales like blues scales.
I guess, to make a reaching analogy, learning the CoF is to music what learning the multiplication tables is to basic math. Not really but kind of.
Hope that helps... and wasn't too rambling/confusing. Pardon any typos or brainfarts. Again I did not edit.
Cheers.