I'm trying to make sense of the Circle of Fifths/Fourths and hoping that someone can help me with this. I was playing around with it on paper, counting semitones to predict the next Perfect 5th and the associated accidentals. But, I saw no real reason for it. Even though it has a pattern I see no reason why these patterns are so special.

For instance, why does the CoF start on C? Why not start on A? What would be the difference? Why flats when going clockwise? Why not sharps? Etc.

Also, it fell apart after C#, producing nothing but more naturals, not the flats you would get from the circle of fourths. So, the two circles did not meet up, so to speak.

The whole thing seems arbitrary to me or worse. The name comes from interval theory, but, the actual content is notational. I don't think it makes sense. Let's say we were to construct another CoF starting on A using the same pattern as the CoF on C.

A: no accidentals
E: A#/Bb
B: A#/Bb, F/F
F#: A#/Bb, F/F, C/C
C#: A#/Bb, F/F, C/C, G/G
.....
I notice that constructing other CoF based on other notes besides C produces even more aribtrary CoFs. But, that doesn't explain why it's worth using a COF anyway.

How much do you want to know?
The Circle of Fifths helps musicians remember what notes sound good together, as in chords, and what notes or chords sound good in sequence, as in scales or chord progressions.

There is a theory of harmony attributed to Plato that says "harmonious" or "good-sounding" sounds have frequencies, or pitches, in the ratio of small whole numbers.
The most harmonious ratio is 2:1. Example: orchestral "A" has frequency of 440 Hz. 2 times that is 880 Hz, which is the next A up.
We call this an "octave" because it is 8 notes apart in a scale.

The next most harmonious is the ratio 3:2, which turns out to be 5 notes apart in the scale (except starting on ti),
Example: 3/2 times 440 Hz is 660 Hz, which is the theoretical frequency of the E above orchestral A, which is the fifth note of the A scale, so the interval, or distance between the notes, is called a "fifth." (Technically, it's a "perfect fifth". A "perfect fifth" is ALWAYS seven half-steps apart, no matter which note you start on.)

The next "fifth" would be 3/2 times 3/2 or 9/4, then 3/2 times that, or 27/8, etc.
That's the theory. After 12 fifths you come back to a note that is 531441 / 4096 = 129.746337,
which is very close to 128 or 8 octaves up from the starting note.

Around the time of Bach, musicians decided to start tuning instruments to make it a "perfect" circle,
which is called "Equal Temperament Tuning."
This makes it easier to play in any key
(that is, play a scale starting on any of the 12 notes) without retuning the instrument.
That gives you a 12-note "circle" which we call the "Circle of Fifths".
But theoretically, it is not really a circle, because it doesn't match up exactly with octaves.

Starting on F, seven successive notes from the "circle" give you the notes of the C scale: F C G D A E B. Of course the order is not the same, but if you take octaves of all the notes up or down until you get 7 notes "close" to each other, you will have the scale in the usual order.

Going up form B, the next five notes are called "sharps": F# C# G# D# A#. Going down from F, the next five notes are called "flats": Bb Eb Ab Db Gb. This gives two names to each of the five black keys on the piano: F# = Gb, etc. Start on any note, go in the direction of F C G... and you will get notes of the scale for the note you started on.

For example, Ab Eb Bb F C G D are the notes of the Ab scale, E B F# C# G# D# A# are the notes of the E scale, etc.

So what happens after A# or Gb? IF you keep going after A#, you get
the note one half-step above E, which on instruments is the same note as F, but in theory we call it E#.
Next comes the note a half-step above B, which on instruments is the same note as C, which in theory we call it B#. After B# (C) would come the note G again, but in theory it would be the "sharp" of F#,
which we call F-double-sharp and write with a symbol that looks like the letter "x" Fx.
In theory you could keep going around to triple sharps, etc. but nobody does that in practice.

Going the other way, after Gb (which on instruments is also the note F#) comes a half-step down from C, which we call C-flat, but it's also the note B. Then comes a half-step down from F, called Fb which is also E. Then a half-step down from Bb, which is A again, but in theory it's called B-double-flat, written with two flats close together, or Bbb. And so forth for Ebb, Abb, Dbb, Gbb, etc.

So why all the fuss?
It's the harmony between notes a "fifth" apart that makes the circle so important.
From the circle, we can pick the notes of a scale, and identify the important chords of that scale. (The most important chords are next to the keynote on the circle, e.g. G and F are the most important chords in C). We can also identify "closely related" scales or keys--scales that have 6 of the 7 notes in common,
like "C" and "G" or "C" and "F".

That's probably more than you wanted to know about the Circle of Fifths.

BTW, Plato's theory says the next most harmonious ratio are 4:3, 5:4, and 6:5.
These ratios corresponds to "fourths", "major thirds", and "minor thirds",
which in half-steps are 5, 4, and 3 respectively.
These intervals along with fifths are the basis for forming chords and harmonies.

If you put another circle-of-fifths inside the first with the inner A between the outer F and C,
the zig-zag diagonals between the two circles give you the major and minor thirds.

<Added>

CORRECTION: The theory of harmony is attributed to Pythagoras, not Plato.

Oops! Pythagoras, not Plato, was the dude responsible for "Pythagorean" harmony theory, per my recent post.

The CoF is part of music theory based on scales. To understand the CoF, make sure you understand clearly how the major scale works. It is the most common musical pattern in Western music. It is an 8 note scale from the key note to the same note an octave higher (or lower).

The CoF starts with C because C is the default key in music. In the key of C the scale goes C D E F G A B C. Why start with C? That's like asking why train tracks are a certain gauge. It would require a historical answer. For now, you and I need to learn scales and the CoF as they are, not what we think they should have been. (You could indeed start with A, but you would still need sharps and flats to describe other keys, and what turned out to be better for you on your instrument might be worse for other kinds of instrument.)

My advice is for you to stick to the small number of keys you know well, and learn the scale patterns intimately in those specific keys. Then gradually apply your knowledge of one key to a new key. You will build your knowledge, and the time will come when you get the insight to see the underlying principles (which is really what the CoF is trying to give you).

For instance, study the key of C and ask yourself why the relationship between G and F, or C and G are important. Do this not just in theory, but with actual music. Listen to the bass line in country music travel up and down the scale and find its natural resting place at those particular notes. Hear the melody of "Lean on me" by Bill Withers start on C, travel up the scale to F, then naturally reverse back down to C. Listen to "Piano man" by Billy Joel. Hear the bass line of the chord progression start on C and descend through the scale, one chord per bar: C, B, A, G, F, E, D - and then notice something interesting. He doesn't go from D to C. He goes from D to G, and then G A B C. This is the same interval in two different keys: D to G (in the key of G) and G to C (in the key of C).

hth.

WOW Amazing info here guys, thanks for sharing. I'm trying to make sense of the whole circle of fifths thing too. I just don't get it, but I'll read this stuff a couple times to see if I can make some more sense of it.

I have the same question

I also have same query, Please help!!

I'll try again!
Co5 is a way of organizing the musical notes that makes it easier to:
* figure out which notes are in a given scale or key
* figure out which notes are sharp or flat in a given key
* figure out which chords are most common in a given key
* figure out which chords "most naturally" follow one another in a harmonious progression

The order of the notes on a piano keyboard or guitar string have names in a weird sequence:
A A# B C C# D D# E F F# G G#, then starting over with A ...
(notice there is no name between E F or B C).

Or going the other way:
G Gb F E Eb D Db C B Bb A Ab, then starting over with G ....

You may have learned how to make scales and chords counting half-steps along this sequence:
whole whole half whole whole whole half.

But lets rearrange those names into the order of Co5:
Gb Db Ab Eb Bb F C G D A E B F#

(BTW, note that Gb and F# are the "same" note on standard instruments, so they close the circle.
But the letter pattern extends in both directions and goes on and on, round and round:
... Bbb Fb Cb [ Gb Db Ab Eb Bb F C G D A E B F# ] C# G# D# A# E# B# Fx ...
where Cb = B, Fb = E; E# = F, B# = C, etc.)

Now lets get our scales:

We can get a major (or relative minor) scale by picking the starting note, move left one, and then take seven notes moving right.
Examples:
key of C: F C G D A E B
key of F: Bb F C G D A E
key of Bb: Eb Bb F C G D A
key of A: D A E B F# C# G#
key of E: A E B F# C# G# D#

Try these with the "whole whole half..." method in the alphabetical order and verify that you get the same notes in each scale.

Now for the order of sharps and flats:

We can also notice, moving right from C, we add one sharp with each successive scale.
And moving left from C, we add one flat with each successive scale. And the order of adding sharps follows the same letter sequence F C G D A E B;
and the order of adding flats follows the same sequence in reverse: B E A D G C F.

Now lets find the chords of a scale.

Pick any note; it's the "tonic" or most important chord in that key. The letter to it's right is the next-most important chord, the dominant, often a 7th.
The letter to the left is the third most important chord, the sub-dominant.
Examples:
key of C: F, C, G7
key of Bb: Eb, Bb, F7
key of A: D, A, E7

Moving right from that group of three, the next group of three are the chords of the relative minor:
C major (F C G7) A minor (Dm Am Em)
Bb major (Eb Bb F7) G minor (Cm Gm Dm)
A major (D A E7) F# minor (Bm F#m C#m)

In some versions of the minor scale,
the third chord in the set becomes a 7th,
example: (Dm, Am, E7).

If you put two Co5 together, one inside the other,
offset in sequence d F a C e G b D f#,
where in this example one Co5 is capital letters and the other small letters,
you can actually get the notes of the chords.
Example: Dm = d F a, F chord = F a C; Am = a C e,
C chord = C e G, etc.

When you play a song with chords, you may notice that a chord is often followed by a chord with the letter to the left (or right) along the circle, e.g.
C G7 C or C F C or Em Am Dm(7) G7 C.
Those chords are also given Roman numeral names based on the positions of their notes in the scale:
examples: I V7 I or i IV I or iii vi ii V I (in decimal, 1 5 1, or 1 4 1 or 3 6 2 5 1) progressions.

So, if you learn to play your chords in those patterns, you'll be ready for the most commonly occurring chords and chord progressions. Or if you play by ear, knowing the Co5 helps you know which chords are likely to come next. In classical music, you'll find that a piece often changes key from section to section; the key changes also often follow the Co5 pattern, or the relative major/minor pattern, which is the Co5 pattern of triples.

So those are some examples of how Co5 useful.

Where did it come from? Why does it work that way? I tried to answer those questions in my first post.