i have a slightly modified way to remember this.
C,G,D,A,E,B,F#,C# - cars get dirty after every burnout, Fast Cars - Sharps 0 - 7
F.Bb.Eb.Ab,Db,Gb,Cb - Forget Beer Eddie And Drink Good Cola - Flats 1 - 7
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Another insight : The method is also fool proof for sharp major scales like B# Maj or A# Maj. In this case instead of sharp notes we obtain double sharp notes.
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I enjoyed this lesson back in Nov. 2022 and now that I have finished grade 3, I am reviewing again. I never used any of the mnemonics mentioned (although I thought @joselp post on April 2022 was cool). Instead, I used what I had posted back in 2022. I know the notes of the the open strings and the first position notes learned in grade 2. If you look at the notes of the open strings and imagine that the first two strings are fretted at the first fret on the F and C notes
I can easily see that in my head and go from right to left:
F C G D A E
Of course, this works because the guitar is tuned in intervals of 4ths, which is a 5th going high to low, except for G to B, which is a 3rd or 6th in reverse, thus the need to fret one semitone higher.
I know from blues studies that B is the 5th chord in E (or you can look at the open
E to B in the picture above) so I add a B at the end
F C G D A E B
Move the F from the beginning to the end. Since every 5th scale degree step up is 7 semitone steps on the note circle I can count in my fingers from B to F#
C G D A E B F#
Finally double this series (I like putting the double lines between the F and C)
C G D A E B F# // C G D A E B F#
And flat every note on the left (and add C# to end, which 7 semitones higher than F#)
<————- flats Sharps ————->
Cb Gb Db Ab Eb Bb F // C G D A E B F# C#
7 6 5 4 3 2 1 // 0 1 2 3 4 5 6 7
Anyway, this works the best for me, as far as thinking about the progression of keys and the circle of 5ths.
BTW - notice that Cb, Gb and Db on the left side are enharmonic equivalents to B F# and C# on the right side.
I did an interesting exercise by using the major scale chart filled out earlier and changing or swapping the enharmonic equivalent notes in the keys Cb, Gb and Db and proving to myself they resulted in the scale notes for the keys of B, F# and C#.
If you extended the series out by adding G# D# A# E# and B# to the right side, they would be enharmonic equivalents to AB Eb Bb and F of the flat keys on the left. So you really don’t need to add the extra sharp keys.
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Just one additional post, since this is such a fun topic to think about. Here is an illustration of a map of the keys using the first position fretboard diagram:
Start with the blue notes in the second fret, starting with the Fb and Cb (enharmonic equivalents to E and B).
Read to left and when you get to the 6th string, go back to the 1st string on the same fret and continue. You’ll end up with the virtual F# and C# notes that I added that are 1 semitone lower than the open notes G and D. If you tuned your guitar down 1 semitone to play SRV then the F#/Gb and C#/Db notes would be real and all the notes would slide up the fretboard 1 semitone 
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here is my system of classifying and understanding music theory concepts and their tight relations with guitar.
three copies of beadgcf fcgdaeb cover all keys. their notes. order of sharps and flats, chords in key etc. Any 6 adjacent notes from the stream of letters above is how notes go on guitar from 6th to 1st string if you go strictly in 4ths (i.e. adjust for B string). thickest three strings give you minor chords in key and thinnest three give you major chords if you take the note on 2nd string as root. works on any fret. if you want keys which have double sharps and double flats, just extend adding one more copy of beadgcf on each side. this is all 4ths. guitar is tuned in 4ths. guitar is THE theory instrument.
adding numbers (scale degrees) to the game.
modes are covered too.
so, darkness of modes is tightly related i.e. following the circle of 4ths.
moreover any 12 adjacent notes from the first picture i posted above give you the chromatic scale only in 4ths and not chromatically. Going across the strings winds the chromatic scale (if you stick to 4ths) until you hit an octave of the note you started from as seen in the diagram by @SteveL_G99 if you go from F to F. works for any starting note/fret.
What is guitar then? Chromatic scale along a string in halfsteps (fret by fret) and vertically in 4ths (string by string, corrected on B string to make it a 4th, and replicating the note on thinnest e to thickest E string).
why would you practice stuff on the guitar following circle of 4ths. because by going through it 12 times you covered all keys (i.e. you have covered all notes in chromatic scale which are all the notes).
3 copies of letters (note names), 3 copies of numbers (scale degrees) encapsulate a whole lot of theory. Really a lot. And the step of 4ths/5ths is the essence of guitar tuning.
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I found the same thing, but checked back before posting. I thought of it slightly differently wrt the next flat or sharp.
For sharps:
The 5th degree is the next key (no different to the quoted post)
Descend a semitone from the new key to get the next sharp, which is also the 7th degree of the new major scale,
So starting at C (no sharps), G is the 5th degree, descend a semitone to F sharp (the next sharp and 7th degree of G major). The 5th of G is D, so C sharp (a semitone drown from the new key) is the next sharp and 7th degree of Dmajor.
For the flats:
The 4th degree of the scale gives the next key (no different)
Descend a tone from the old key centre to get the next flat (which is also the flat 7th of the old key centre)
So again starting at C:
The 4th of C is F (the new key).Descend a tone from C to Bb (the new flat).
The 4th of F is Bb; descend a tone from F to Eb for the next flat.
The 4th of Bb is Eb, descend a tone from Bb to Ab (the next flatj
So the flats have revealed another patten: the next flat becomes the new key when you add one more flat. This is obvious going anti-clockwise on the circle of 5ths, but I hadn’t noticed it until writing this post (and I have previously used the circle of 5ths and read sections of @Richard_close2u ‘s post on the circle of 5ths). I guess it show how people’s brains work slightly differently, and it probably follows that what’s a lightbulb moment for one person won’t necessarily be for another.
That connection to the circle of 5ths might have revealed something else to me.
As you proceed anti-clockwise you are adding flats (I.e. going to the key with one flat more), then when you get to the Gb / B section at the bottom you start subtracting sharps .
Likewise proceeding in a clockwise direction you add sharps, then subtract flats.
Maybe that’s of academic interest…….but:
The circle of 5ths is a way of training fretboard knowledge. I initially didn’t find it very helpful as I was just trying to remember the next key. Now that I know the next note is the key with one more sharp or one more flat (I know these by heart from theory based on playing the piano for 50+ years,), I don’t need to think what comes next on the circle of 5ths and can just focus on the note names. One less thing to think about will hopefully speed up the fretboard learning process.
This might be a lightbulb moment (for me at least 
)
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Is this Mr. Cato by any chance Tom Cato Amundsen, the author of the free ear training application GNU SOLFEGE?