# Is the circle of fifths misnamed?

Is the circle of fifths misnamed?

The circle of fifths would actually seem to be a circle of accidentals with zero accidentals at noon while a circle of fifths would seem to start at F as the set of diatonic note names begins with F, being F,C,G,D,A,E,B.

That is:
F(flat),C(flat),G(flat),D(flat),A(flat),E(flat),B(flat) followed by...
F,C,G,D,A,E,B followed by...
F#,C#,G#,D#,A#,E#,B# ... followed by double sharps
... (and double flats to the 'left' of the flatted set but I couldn't format it to look understandable).

Furthermore it seems a 'circle of fifths' is not strictly possible as a circle of fifths would not use enharmonic equivalents and instead spiral off from B to F# to C# to G#, etc., instead of B to F#/G(flat) to C#/D(flat).

• What do you mean "the set of diatonic note names begin[s] with F"? – Richard May 13 '18 at 15:32
• Are you proposing that the Circle of Fifths (or the Circle of Fourths) be instead called the Circle of Accidentals? – David Bowling May 13 '18 at 15:36
• Well if it is indeed misnamed then that's certainly one option - unless of course I'm misunderstanding what it is. – Randy Zeitman May 13 '18 at 15:37
• As far as I know, does the set of diatonic names begin with c. Apart from that, it's a circle, so it has no beginning. Apart from that - it is misnamed, because the circle it is actually a helix, because it is not closed. – tommsch May 13 '18 at 15:45
• "slightly" off-topic,but I far prefer this circle of fifths: ![CircleOfFifths](i.stack.imgur.com/9jTX8.jpg) – Carl Witthoft May 14 '18 at 13:30

The "fifth" in "the circle of fifths" refers to an interval that is (close to) the ratio of 3/2, which gots its name because if you play five "normal" scale steps, you traverse this interval- say, C D E F G. The "circle" part is that if you jump enough fifths- twelve in our usual 12 tone equal temperament, you come back to the note you started with- albeit at some number of octaves away from your starting point. For instance (jumping down) B, E, A, D, G, C, F, Bb, Eb, Ab, Db, Gb, back to Cb=B. In equal temperament, enharmonic equivalents have the same pitch

As has been noted, you don't have a "circle" if the "fifths" are actually the ratio of 3/2, because no power of two is also a power of three. Thus, the "circle of fifths" is indeed a misnomer, or at least not very precise: it should be called a helix of fifths, or a circle of almost-fifths. The accidentals are an artifact of how we notate, though, and calling it "a circle of accidentals" wouldn't be an improvement.

• It's not a helix if you consider the circle to be about pitch classes regardless of octave and not about exact pitch. You could of course play B, E, A, D, G, C, F, Bb... etc. all in the same octave, and it would still be a circle of fifths progression. – Todd Wilcox May 14 '18 at 1:45
• Mr. Mathematician here points out that the even-tempered sequence is either a helix or at the least a multivalued function with branch cuts every few octaves. You never return to middle-C even though you get back to an upper-octave C after a while. – Carl Witthoft May 14 '18 at 13:33
• @CarlWitthoft I'd say that depends on how we are viewing pitches. If we consider "pitches" to only mean 12 elements where going up one half-step from the "last" or "highest" element puts you at the "first" or "lowest" element, like Z_12 (the group of integers 0 - 11 under traditional addition modulo 12), then it can be seen as a circle. – Todd Wilcox May 14 '18 at 14:33
• @scott, it might be more precise to say that the "circle of fifths" is less like a geometric circle, and more like a cyclic group (like what Todd mentioned). Still under the math umbrella though =) – The Chaz 2.0 May 14 '18 at 15:45
• @TheChaz2.0 - yes, cycle does fit better than circle. In any case, I'm all for educating musicians on the mathematics of music as soon as possible. Not that you have to know it to be a good musician, but the information should be offered. I didn't learn anything about it in school, I had to go to the library to learn that there was even a connection between math and music. – Scott Wallace May 14 '18 at 15:54

All the 'circle of fifths' means is that if you start on any note in the chromatic scale of 12 notes, then repeatedly going up a fifth (or down a fifth) will take you through each of those 12 notes before returning to the starting point (obviously you will end up many octaves away). That's the essence of the circle of fifths. (this actually only works in some temperaments, but that's another story - it works well in 12-tone equal temperament, which is commonly-used and often assumed to be the 'default' temperament in modern music.)

However, one thing that could be confusing is that people often choose to show other concepts (such as key signatures, or relative keys) in the form of a circle of fifths diagram. If you're noticing that those concepts aren't entirely circular, you're correct.

Another possible point of confusion is that every note in the chromatic scale has a number of ways that it can be 'named', and the notes that are black notes on the keyboard don't have obvious 'neutral' names that don't include an accidental - e.g. C♯ and D♭ are, in equal temperament, different names for the same note. Even the simplest circle of fifths diagram is going to have to find some way of dealing with this issue that will inevitably makes the circle look more complicated than it is.

It's really not quite clear what you think the circle of fifth is or is for, my bet is that your perception is off more than its name. While there are round here discussion of the finer points of pitches and intervals, if we stay inside the tempered system, the circle links notes with their fifths (and fourths in the reverse direction). Since it's outside a tonality, enharmonics are noted where useful, and, yes, it does follow the order of accidentals and neighbouring tonalities, by construction. That's about it.