# Could a 3D sphere of fifths reveal more insights than the 2D circle of fifths?

The circle of fifths is a well-known device for showing relationships between keys, and shows the concept of close vs distant keys.

But is there any way of arranging keys in three dimensions rather than two, and would that give any insights that aren't available from the 2D circle of fifths?

• What are you hoping to visualize with it? The circle of fifths is an illustrative metaphor, like the Guidonian Hand. Maybe a better way to go about it, instead of inventing an illustration and then asking what it illustrates, is to first identify the principles that need illustrating. Commented Aug 13 at 15:30
• @AndyBonner- I wondered if organising the key names on the surface of a sphere instead of on a flat disk might give some fresh insight into their relationship one to another. Commented Aug 13 at 18:12
• ...there might be a pattern that's not immediately obvious needing to be represented in some fresh way for the pattern to become obvious. Commented Aug 13 at 18:56
• A torus would be the proper 3D object. Commented Aug 14 at 10:38
• @Olórin: Or an orbifold. See Science, vol. 313, p. 72 (2006). Commented Aug 14 at 21:14

A circle is actually a one-dimensional figure. The two-dimensional figure that we think of when we hear "circle" is a disk. A circle is one dimensional because, given a circle, you can construct a coordinate system that allows you to identify a point on the circle with a single value. For example, you can choose a point on the circle as the origin and identify other points on the circle by their distance from the origin.

(Similarly, mathematicians use "sphere" to refer to a two-dimensional surface and "ball" to refer to a three-dimensional solid object; see https://en.wikipedia.org/wiki/Sphere for a more detailed discussion.)

I can't actually work out a useful way of displaying Pythagorean fifths and the theoretical infinity of double, triple, quadruple, etc., flats and sharps as a helix. I find a straight line works better:

...-B♭♭♭-F♭♭-C♭♭-G♭♭-D♭♭-A♭♭-E♭♭-B♭♭-F♭-C♭-G♭-D♭-A♭-E♭-B♭-F-C-G-D-A-E-B-F♯-C♯-G♯-D♯-A♯-E♯-B♯-F♯♯-C♯♯-G♯♯-D♯♯-A♯♯-E♯♯-B♯♯-F♯♯♯-...

Because of enharmonic equivalence, we can wrap this line around a circle whose circumference is 12 times the distance between two adjacent elements. It doesn't matter whether that distance represents a ratio of 1.5 or the 7/12 root of 2 or anything else; the only thing that matters is that we can consider (e.g.) A♭ as being enharmonically equivalent to G♯.

Some representations of the circle of fifths (or circle of semitones) use variable distance around the circle to represent different temperaments: 30 degrees represents 700 cents (or 100 cents) and longer and shorter arcs show which fifths (or semitones) are wider or narrower. But this is still one dimensional, and if one extends into multiple sharps or flats one doesn't get a helix but rather (for example) a B♯ that is a bit to the left or right of C♮

To represent these relationships in two or three dimensions, you would need to define some aspect of the relationship that you could represent using a second or third dimension. E major is just twice as distant from C major as D major is: a larger magnitude in the same dimension.

There is a 2-dimensional representation of pitch relations called the Tonnetz, but I suspect that even if it is possible to bend it into a curved surface using enharmonic equivalence, doing so would not yield a sphere.

• Thank you for the thoughtful insight @phoog, I'd wondered if there was any way of mapping 11 points (one per chromatic degree) onto the surface of a ball in such a way that "close" keys are nearer on the surface than "distant" keys. I don't think there are, but you may have other ideas… Commented Aug 13 at 18:09
• A mapping of the Tonnetz to a 2-dimensional surface is toroidal. See Torus § Configuration space. Commented Aug 13 at 18:35
• @BrianTHOMAS I think your comment misses a lot that phoog's saying about dimensionality. Currently the circle of fifths has one measure of "distance" between keys: the number of fifths it takes to get from one to the other. Do you have a second way of ~measuring~ distances between keys? If so, then arranging on a 2d surface--the interior+perimeter of a circle, the surface of a torus, the surface of a sphere, points in a plane--might make sense. If you come up with a third ~distance~ between keys, we could look at placing keys in a 3d volume, &c. up to as many dimensions as you can conceive. Commented Aug 14 at 23:55
• @nitsua60 - I don't have a new way of determining "distance" between keys. I was hoping that representing the keys as points on the surface of a sphere might reveal some new "neighbour" relationship that's not apparent when they're laid out on a straight line or on the circumference of a circle. For instance, I can imagine keys a minor third apart being represented as the four vertices of a tetrahedron - wondered if that could be extended up to a sphere. Commented Aug 15 at 8:49
• @BrianTHOMAS The 12 points of the enharmonic chromatic scale could be mapped onto a torus, as Theodore pointed out, using major third and minor third relationships in addition to fifths, as in the Tonnetz. This could help visualize certain kinds of "distance" between notes or keys, but it's not clear that would be more useful than the Tonnetz. Since a Tonnetz is flat, you can see it all at once... The toroidal shape would however let you avoid repetition of enharmonically equivalent notes while still showing all their relationships. Commented Aug 15 at 16:13

Assuming enharmonic equivalence, any mapping of (perfect) fifths will eventually wrap around onto itself.

As such, in a two-dimensional representation, distance isn't always as clear as one might hope since the mappings of a key onto itself are, by definition, in another location. Starting at x, the distance to y is clear enough; but the distance to z, which is equivalent to x, may not initially be obvious if one isn't primed to look for multiple x equivalents.

In a three-dimensional representation, however, distance could be made more clear in the right circumstances. If the representation were a sphere with only one instance of each key, for instance, showing maximum distance (i.e., the pole) is immediately obvious and musically very meaningful.

• The whole point here is that the mapping of fifths wraps onto an enharmonic of itself. It will never wrap onto itself as such but only onto more and more exotic enharmonics (C, B#, Ax#, Gxx#, Fxxx#, Exxxx, Dxxxxx, Cxxxxxx, Bxxxxxx#, and so on). Commented Aug 13 at 14:01
• "Assuming 12-tone equal temperament": any 12-tone temperament will do; the crucial element is enharmonic equivalence, not equally-spaced semitones. Commented Aug 13 at 15:51
• @Divizna I would guess that 99.9999% of music written in the European classical tradition or a derived tradition either (1) is insufficiently chromatic for enharmonic equivalence to be an issue or (2) depends on enharmonic equivalence. Do you know of any examples that would fall in the estimated 0.0001%? Commented Aug 13 at 15:56
• @phoog The issue lies not in the sound itself but in the confusing frame of reference to describe it. The circle of fifths, as popularly drawn, tells you that if your tonic is F# then the dominant is Db. Try making sense out of a bunch of back and forth chord changes notated like that. Of course an even slightly advanced musical theoreticist won't fall for it but that isn't the person who has any use for looking up the circle anyway. The person that's going to refer to the circle is a beginner, and running into a sudden seam that throws them off isn't really helpful. Commented Aug 13 at 17:44
• @phoog Yes, you're exactly right. (As is Divizna.) Thanks for the catch! Commented Aug 14 at 10:01

A 3D model of fifths would not be a sphere.

A friend of mine says that this is what bothers him about the "circle of fifths": That it isn't in fact a circle; it's a helix. He'd like that fact acknowledged.

Pretending it actually is a circle is a lie-to-children, and like all lies-to-children, it can mislead or get confusing.

I'm not sure if he'd actually want to see the helix of fifths represented with a 3D model rather than a flat image, though. That seems kind of unwieldy for practical purposes (reproduction, transporting, easy reading).

But it does seem to me that a more representative visual than a circle would be a spiral in which for instance Gb and F# would be in the same direction from the centre, but drawn distinctly, one on the inside and the other on the edge, each connected to their own (and not one another's) mates.

• Just because your friend thinks it’s a helix doesn’t make it a helix. The circle of 5ths is a model and like all models is not the actual thing, just a way to view the thing. A model is what it is, it can’t be “wrong” because we already know it’s only a model. It might be less helpful than another model, but if it has persisted then it’s probably somewhat helpful and therefore has validity in the body of thought about the topic. Your friend might prefer the “helix of fifths”, but that doesn’t make the helix any more accurate than the circle. They are both models. Commented Aug 13 at 13:31
• @ToddWilcox Just because you think it's a circle doesn't make it a circle either (if we really must go ad hominem here). Seeing it as a circle equates enharmonics as strictly indentical. A helix represents their relation properly. That's why it's a helix, regardless of who points it out. Commented Aug 13 at 13:52
• @ToddWilcox is (partly) right. Whether it's a circle or a helix depends entirely on what you're modeling. If you're modeling theoretical Pythagorean tuning without any enharmonic equivalence, then it can be thought of as a helix (which is also a two-dimensional figure, I hasten to add), since the Pythagorean system does not close. If you're modeling the closed 12-tone system with enharmonic equivalence (or any system that closes on the octave) then it's a circle, regardless of your preferred temperament (or lack thereof). ... Commented Aug 13 at 15:42
• ... The implication that it is improper to recognize enharmonic equivalence is similarly misguided. Much music requires it. The claim that A flat isn't really the same thing as G sharp is true in some contexts and absolutely false in others. But even then the "helix" image isn't really consistent with the way the circle of fifths is usually laid out. Each pitch of the scale is mapped to the circle by choosing an angular coordinate based on the pitch and a radial coordinate that is constant. It is impossible to generate a helix this way; a helix's radial coordinate must vary. Commented Aug 13 at 15:46
• @phoog: I like the helix view, because it it makes clear that going from C# to Db leaves one at the same angle, but at a position that's somehow "different" from where one started. Even if acoustically notes that are separated by a revolution may sound the same, music should still generally be notated with notes that are close together on the same "thread", and if it becomes necessary to jump a thread there should be a clear logic to when one does so. Commented Aug 13 at 21:16

Note that the "circle" of fifths is schematically a polygon (dodecagon), in which we can model the keys (or pitches) as vertices, and the relationships between them as edges. (Whether the edges are straight or curved doesn't matter.) I'm assuming enharmonic equivalence, without which no cyclical model makes sense.

So to extend this to another dimension, mapping key relationships onto a sphere would schematically mean mapping onto a polyhedron, in which vertices represent keys and edges represent relationships. The OP mentioned a tetrahedron, which would be an example of this.

In order to represent all 12 enharmonic pitches, you would want a polyhedron with 12 vertices. And you would want the vertices to be symmetric to each other; e.g. every vertex should have the same number of edges. (It wouldn't make sense for some keys to have more relationships than others.)
This basically gives you Platonic solids, Archimedean solids, prisms, and antiprisms.

Of the first two, only the icosahedron, truncated tetrahedron, and cuboctahedron have 12 vertices. The vertices of these polyhedra each have 5, 3, or 4 edges respectively. So if you want each key to have a relationship to 4 other keys, you might try a cuboctahedron.

In a prism, each vertex has 3 edges, and in an antiprism, 4. Imagine an antiprism in which the top is a ring of 6 vertices: A - B - C# - D# - F - G - A. The bottom is a ring of the other six keys: D - E - F# - G# - A# - C - D. Each top note is connected to two bottom notes that are a 5th away from it, sort of like this:

```|--A---B---C#--D#---F---G-|--A
| / \ / \ / \  / \ / \ / \| /
|D---E---F#---G#--A#--C---|D--
```

(showing A and D twice to illustrate where they connect on both sides).

Does this give any new insights? Well, it illustrates the "closeness" of whole steps, in addition to the relationship of 5ths. You could choose other relationships instead.

For example, if each horizontal ring is a cycle of minor 3rds (A--C--Eb--Gb--), then you need 3 such rings:

``````|--A--C--Eb--Gb--|

|--A#--C#--E--G--|

|--B--D--F--Ab--|
``````

But then if you still want to connect the rings to each other in a cyclical way, you can no longer use a prism or antiprism, which only allow 2 rings. To connect a grid cyclically in two different dimensions, you need a torus.

So, as a couple of people have said, you could wrap your grid around a torus. If you do that, your grid could be basically like the Tonnetz. Or you could emphasize different relationships/intervals. But it's hard to imagine how you would improve on the Tonnetz.