# Why does the circle of fifths only show some notes that are enharmonically equivalent?

I was wondering why the circle of fifths only has three keys that are enharmonically equivalent at the bottom of it namely, C#/Db, Gb/F#, B/Cb. Why only those three? Can’t the other notes in the circle also be called different names such as A#/Bb, Eb/D#, C/B#, etc. One thing I noticed is the flatted keys are on the left, and sharp keys are on the right of the circle. Why then did they show C# on the left.

• There are too many enharmonic equivalents to show them all without making the whole thing a mess. Apr 11, 2019 at 14:54
• I think if you really want a chart for the key signatures it should have 15 positions. Apr 11, 2019 at 15:16
• It's already a mess as it is! What value is an intersection to connect G# minor with Cb major? A practical reason for a reference chart? Who needs a quick reference to know that G# minor's relative major's enharmonic equivalent is Cb major? Apr 11, 2019 at 17:45
• Because an infinitely large circle of 5ths would be completely useless to everyone. Apr 11, 2019 at 18:10
• The graphic doesn't show enharmonic notes, it illustrates keys. And while e.g. g flat minor is theoretically a perfectly valid construction, in practice no one ever uses it because f sharp minor is more convenient - so there is no point in illustrating it either. Apr 12, 2019 at 6:42

The seven major scales that only include single-sharped notes (G, D, A, E, B, F♯, C♯) + the seven major scales that only include single-flatted notes (F, B♭, E♭, A♭, D♭, G♭, C♭) + C major are the only major scales that can be constructed using no "double-and-above accidentals" (double-flat, double-sharp, etc.). Of those 15 scales, there are three groups of two where both scales are tonally the same. Hence there are three tones on the circle of fifths with enharmonic equivalents printed, thus the circle of fifths represents all 12 tones of Western tonality (15 - 3 = 12).

• The first sentence is a bit of a circular statement, but this is the only answer with a plausible and logical reason to explain the chart's inclusion of C-sharp major and C-flat major. Apr 13, 2019 at 4:57
• @phoog I edited it to hopefully make it less circular. Does this make more sense? Apr 15, 2019 at 18:48
• Hm, yes, it's less circular, but more verbose and I suspect less clear. Perhaps something like "ignoring key signatures that contain double sharps or double flats, there are only fifteen possible key signatures: seven with sharps, seven with flats, and one with no signs at all." Apr 15, 2019 at 19:31
• I figured with the addition of the parenthetical explanations, the meaning becomes clear. Apr 15, 2019 at 20:37
• Probably so. I've been thinking about it for too long to have a good idea about what would or wouldn't be clear to someone reading it for the first time. Apr 15, 2019 at 21:01

It's a completely arbitrary limitation.

Professional quality music notation software will let you write in any key you like, between 7 triple-sharps and 7 triple-flats. (Of course you don't see those key signatures very often, but in case your don't believe they exist, here they are...)

You won't find those keys on a circle of fifths chart because the only people who need the chart are beginners.

I assume the OP doesn't keep a chart of the letters in the English alphabet next to his/her PC to help write questions on this forum, but you will find them on the wall in most elementary school classrooms. Charts of the circle of fifths are at about the same level in musical theory education as those alphabet charts. After a very short while, you simply don't need them any more.

• "Professional quality music notation software will let you write in any key you like, between 7 triple-sharps and 7 triple-flats." No it won't. Unless you have a very narrow definition of 'professional quality'. Apr 12, 2019 at 14:34
• Which software did you write the triple-sharp/flat key signatures in? Apr 13, 2019 at 7:09
• The circle of fifths is not just for beginners. See my comment about modes on Laurence Payne's answer. Apr 15, 2019 at 18:58
• @Laurence Even alternative tunings (assuming equal divisions of the octave), where triple (and higher) sharps and flats are distinct notes, still have enharmonic equivalents, even if they involve microtonal accidentals. In most such tunings, anything more than double sharps and flats are not needed.
– user59346
Dec 25, 2023 at 18:40

Why? Because the designer of that particular chart decided to. I think it would have been more logical to just show F#/G♭, the only one where there is no default spelling.

Don't worry too much about the Circle of 5ths anyway. It over-illustrates a very simple concept. I think it's only included in textbooks because it's a pretty way of filling a page!

It illustrates (but doesn't explain) dominant-tonic relationships. It makes beginners worry over why ♭III, ♭VII etc. are 'allowed' - after all, they're SO far away in the Circle! It shows you 'close' keys that aren't really close at all - moving a song's melody up a 5th or down a 4th will just sabotage the vocalist!

• I thought the circle of fifths is the most important diagram in all of music.
– user34288
Apr 11, 2019 at 15:11
– user34288
Apr 11, 2019 at 16:17
• The circle of fifths is an incredibly valuable tool, especially for working with modes. If you remember the abbr. LIMDAPL, then constructing modes becomes SIGNIFICANTLY easier: associate "I" (Ionian) with the major scale of the tonal center you want to construct, "L" (Lydian) goes to the right ("one sharp greater") and the rest of the abbrv. goes around the other way, one letter per key signature at a time. So A Mixolydian is an A scale with the key signature of D maj; E Locrian is E with the key signature of F major, and so forth. Apr 15, 2019 at 18:55

The scale of B# major would be: B# Cx (double sharp) Dx E# Fx Gx Ax B#

As you can see every single note is enharmonic to the note next to it. It's just not practical or easy to read or play.

It is much simpler to just use C.

• well C was just one example, I meant why just two on the circle have enharmonic equivalents
– user34288
Apr 11, 2019 at 14:31
• Pick a key that's not there and try to spell it out. You will find the ones listed are the only ones that really make sense.
– b3ko
Apr 11, 2019 at 14:32
• @foreyez Because that is a bad circle of fifths chart. I think there should be more enharmonics listed, especially becasue the enharmonic 'flipping' will depend on whether major or minor. I.e. `G#` minor verus `Ab` major. Notice how what is missing on the chart you found? Apr 11, 2019 at 14:33
• Pick D# for example. D already has two sharps. All the natural notes will become sharps. F# and C# become double sharps. Much easier to use E flat with the three flats.
– b3ko
Apr 11, 2019 at 14:34
• @MichaelCurtis fair enough, I traded it for a better circle
– user34288
Apr 11, 2019 at 14:38

Obviously the two enharmonics are listed to show they are equivalent.

But I think this is a bad circle of fifths chart.

A good one should list the enharmonics for all of the common key signatures. Notice that `G#/Ab` is missing. A good chart should somehow visually explain the common choices are `G#` minor or `Ab` major.

...you just changed the chart!

This chart seems to split the difference.

It doesn't list all the standard enharmonics with each position because the inner circle for relative minor key makes you figure that out. In other words you need to mentally say "Ab is enharmonic to G#, let me find a G# somewhere else... there it is G# minor"

The way you make such a chart depends on what you really want it to explain.

This one combine at each position a combination of enharmonic equivalents and relative keys. It sort of wants to arrange things so that each position has the same count of sharps and flats which of course works for the relative keys, but it all breaks down in the enharmonic area. So...

Why only those three?

Because the designer doesn't seem to have a clear sense of what they want to explain. Is it about knowing the count of accidentals in a key signature or knowing the common keys for enharmonic tonics? It isn't clear what they want to show.

I think if you really want a chart for the key signatures it should have 15 positions. It would not pair up enharmonic equivalents like Ab/g#, but instead consistently list by accidental count like Ab/Fm. But the point of that would be to recognize key signatures rather than enharmonic tonics. I may draw one of my own - just to get the nagging out of my head - I'll come an update my post.

If a chart for explaining key signatures is desire, this seems to be a straight-foward design...

...emphasis on straight, I don't see any benefit to learning key signature by forming a circle. A circle is good for counting in fifths, but that is different than explaining key signatures.

If you warped that into a circle, it would be something like this, but I wouldn't wrap it around to try overlapping enharmonic equivalents, it will just become a mess...

You might ask "why does the typical circle of fifth switch at the F#/Gb equivalency point in the first place?" I think the obvious answer is so the accidental count in the key signature won't exceed 6. Instead of 7 sharps with `C#` major we jump at F#/Gb to flats which decrease each step until we get back to plain `C`.

Why doesn't it continue in a pure, logical fashion as `F# C# G# D# A# E# B# Fx Cx..?`

That would be logical and the accidental counts will increase regularly to 8, 9 , 10... sharps.

Obviously we don't because those aren't practical key signatures.

An that brings us right back around in a circle (pun intended) to say "if the point is to explain key signatures, don't list enharmonic equivalents that don't share key signatures, don't bother with a cirle, just list up to 7 accidentals then stop for sanity's sake."

The equal tempered circle of fiths is really a circle, there are only 12 possible major keys. Theoretically, you have an infinite number of possible names for these keys, as long as you allow double (triple, quadruple...) flats and sharps, but there's no musical difference. The reason for the usual cutoff at 15 key names, that is, 12 keys plus 3 enharmonic equivalents (B/Cb, F#/Gb, B#/C) is that this is all the possible keys that have no double flats or double sharps in their key signature.

That's it. It's just a convention and says nothing about the structure of music.

I have right now written a "circle of fifths" in a form of a line:

These are the usual equivalents

I will use this template to note the progression of the prelude in C# major below it. By the way: in this prelude the chords go till B# and e# minor. (So I actually wondered why Bach didn't write this prelude in Db major. But now I know!)

When I started with music I argued a bit like Lawrence: they teach it because it's nice to look at and it's a stuff that can be easily controled in tests.

But today I knwo that it is one of the moste important things for the use of classical and pop music, jazz as well!

Why don't they show all? It would be too much stuff to show all equivalents and beginners would have a bad overview. I for myself often draw only a sector of it above a bow to see it clearer when I make a mind map to memorize the progression.

a circle of fifths in German spelling:

C o.V (ohne Vorzeichen) means C without sign.

• the circle of fifths with sharps and flats it is identical with the English except B = H. But instead of saying flat or sharp we ad an is for # to the root tones of the scale C,D,E,F,G,A,H,C So the circle and also the accidentals are F#,C#,G#,D#,A#,E#,H#, are called: Fis,Cis,Gis,Dis,Ais,Eis,His. However the flat keys are not so regularly spelt: Fes,Ces,Ges,Des,As,Es,Hes, Apr 12, 2019 at 16:29
• I believe you are mistaken about either the meaning or the translation of o. V. It means ohne Vorzeichen, which translates as without key signature. This meaning is clear from the context, because the analogous notation for each other node on the circle describes the associated key signature. Also, how can there be a Hes when adding a flat sign to H yields B? Apr 13, 2019 at 4:47
• Of course, you are absolutely right. H flat is B. But Hbb (B double flat) is called Heses as Fx (F double sharp) is spelled Fisis (equivalent to A and G). B.t.w. I had to break the circle in the 2nd line above between Cisis and Heses as it usually is at F#/Gb ... Apr 13, 2019 at 5:52
• @AlbrechtHügli - wie macht man aus einem E moll Dreiklang einen E dur Dreiklang? Vergiss es. Apr 13, 2019 at 10:15
• ich weiss nicht was du meinst? worauf bezieht sich deine Frage? (wie macht man aus einem E moll Dreiklang einen E dur Dreiklang?) Apr 13, 2019 at 12:25

For those who are still picky about the reasoning behind the circle, even after several responses, I think the best way to understand it is through constructing it yourself. The circle of fifths, apart from the obvious reason (only the key for the fifth interval in each point) is made to satisfy grammatical rules, being:

• filtering out keys that will include unnecessary accidentals.
• filtering out a key with any accidentals at all when it just becomes a natural.

Take a look at this table:

If you try to map out all the keys to build up their major scales, including those that are enharmonically equivalent (C#/Db, F#/Gb, and so on), you will find that those creating double accidentals are being left out, as well as those that are just a natural. All the others are part of the circle. So, it does have a grammatical and practical logic.

Hope this helps.

The answer is simple. Those 3 keys are listed as enharmonics because they are the only ones that can be represented with key signatures that only use sharps or flats, and not double sharps or double flats. Traditional notation only uses key signatures with all sharps or all flats, not a mixture of both or doubles, etc. Also Cflat (7 flats) is very rare since most composers would use B maj (5 sharps) instead.

This would be the full chart:

C (no sharps or flats) = B# (5 double sharps, 2 sharps)
F (1 flat) = E# (4 double sharps and 3 sharps)
B flat (2 flats) = A# (3 double sharps and 4 sharps) E flat (3 flats) = D# (2 double sharps and 5 sharps)
A flat (4 flats) = G# (1 double sharp and 6 sharps)
D flat (5 flats) = C# (7 sharps)
G flat (6 flats) = F# (6 sharps)
C flat (7 flats) = B (5 sharps)
F flat (1 double flat 6 flats) = E (4 sharps)
B doubleflat (2 double flats 5 flats) = A (3 sharps)
E doubleflat (3 double flats 4 flats) = D (2 sharps)
A doubleflat (4 double flats 3 flats) = G (1 sharp)
D doubleflat (5 doubleflats and 2 flats) = C (again) completes the circle!

Looks like the author of this chart made the fairly sensible decision to place the cutoff at "single sharps or flats only, no doubles".

Sure, they could have chosen a higher cutoff and included more.

If we go on beyond C# major or A# minor with their seven sharps, we find ourselves at G# major or E# minor with a signature of 8 sharps total: One double sharp at Fx (F##) and six single sharps everywhere else. G# major is enharmonic with Ab major and E# minor is enharmonic with F minor.

Then we can make the next step and discover D# major and B# minor with 9 sharps (2 double, 5 single) in the signature, enharmonic with Eb major and C minor respectively.

And so forth.

If we go the other way and leave behind the keys with 7 flats (Cb major and Ab minor), we arrive at Fb major or Db minor: 8 flats (1 double, 6 single), enharmonic with E major / C# minor.

Next there comes Bbb major or Gb minor, the keys with 9 flats. (Yes, that's a double flat for the tonal centre.)

We could go on until we'd come back to the top with an enharmonic equivalent of C major (B# major with 12 sharps or Dbb major with 12 flats), we could pass through that point and continue further, eventually coming into the realm of triple sharps or triple flats, and we don't have to stop there, we don't have to stop ever. Clockwise or counterclockwise, we could march around the circle, round and round, adding more and more in the key signatures, without end. Any key shown on the chart has an unlimited number of enharmonic equivalents.

The thing is, keys with double sharps or double flats are vanishingly rare, and triple or higher are downright unheard of except as a purely theoretical exercise. There isn't much reason to use them when we have a simpler and more user-friendly enharmonic alternative.

In principle, you could go and compose a tune in Ax# major with its 24 sharps (three quadruple and four triple) if you wanted to. (Wait, should I write that as Ax# or A#x? Triple sharps are so rare even as accidentals that I have no idea what the convention here is, if there's any established.) You could go and write something in Cxxxxxx major (84 sharps), for that matter.

In practice, you aren't going to see Cxxxxxx major unless it's meant as a joke, you aren't going to come across Ax# major either, and you're very unlikely to run into B# major.