# Why is there only a circle of fifths?
The circle of fifths in one of the most important and fundamental concepts in western music.
It links every note in the chromatic scale in a sequence of successive fifths,
starting and ending on the same given note.
It's doubly important because, other than the unaltered chromatic scale itself, it's the only
way to organize notes this way: there's no circle of seconds, or thirds or anything.
So why is the circle of fifths unique? It all comes down to math. To explain this concept,
we have to abstract a bit. Instead of imagining a circle of fifths,
think instead of any circle with any number of items in it. If you want
to visit every item in the circle by moving by a certain number of spaces a certain direction,
the size of the step can't share any prime factors with the number of items in the circle.
This, of course, includes every number that divides the number of items in the circle evenly.
For example, if you move by 7 spaces along a circle with 20 items, you'll visit every
item once without repeating because 7 and 20 don't share any prime factors. In fact,
7 itself is prime. If a line is drawn that traces the path from point to point as you step,
a star-like shape will be produced with each point corresponding to one item in the circle.
However, if you pick a number that divides twenty equally, like five,
you'll see that the "star" will return to the starting item without having visited all of them.
In fact, it only visited 4; the "star" is actually a square. If you want to visit every
item, you would need to have five separate squares starting at five different points.
Another thing to mention: moving by the total number of items minus the step length will create
the same pattern as that step. As it relates to my first example, moving by thirteen spaces
will create the same pattern because twenty minus seven is equal to thirteen. Thirteen is equal to
minus seven in modulo twenty; more simply, moving by thirteen steps clockwise is equivalent to
moving by seven steps counter-clockwise. Thus, they create the same pattern, just in reverse.
Now, we can apply this knowledge to the chromatic scale, which contains all 12 notes
used in twelve-tone equal temperament. In order to create a sequence that visits every note once
by jumping by a constant interval, the number of semitones in that interval can't share any
factors with 12. Unfortunately, 12 has a lot of factors; in fact, it's known as a highly
composite number because it has more factors than any positive number smaller than it.
To illustrate this, we'll go through all of its factors.
First, 1. 1 Isn't technically a prime number, so its doesn't share prime factors with 12, and
thus does create a sequence containing every note. However, it's just the notes in order; it's known
as the chromatic circle. This isn't any different than the chromatic scale, so we'll move on to 2.
2 and 12 share the prime factor 2. Because 12 divided by 2 is 6, skipping by 2 creates
a hexagon on the chromatic circle. In order to cover all of the notes, you would need 2 hexagons.
These two hexagons correspond to the two possible whole tone scales,
because each note in the scale is two semitones, one whole tone, apart.
3 also divides 12 evenly, creating a set of 3 squares on the chromatic circle.
These 3 squares correspond to the 3 possible diminished 7th chords.
4 divides 12 into a set of triangles. These 4 triangles represent the 4 possible
augmented triads, which contain three notes each.
Now, we get to five. Five is the first prime number that doesn't share any factors with 12,
and is thus our first circle. However, you might have noticed that five semitones doesn't make a
perfect 5th, but rather a perfect 4th. So, ya boi messed up the title, huh? Well, not really:
the perfect 4th and 5th are equivalent intervals: the circle of fifths can also be thought of as the
circle of fourths, but in reverse order. This is also strengthened by the fact the total of
12 semitones minus 5 semitones is 7 semitones, which is the interval of the perfect fifth.
So, jumping by 5 or 7 leads to our circle of fifths. Doing so by the number between them,
6, creates 6 biangles, which represent the 6 possible pairs of tritones.
Now, anything higher than 6 already has an equivalent jump: 5 and 7 are equivalent, as are 4
and 8, 3 and 9, 10 and 2, and 11 and 1. So, we've reached our conclusion: there's only two sequences
of every note separated by a constant interval: the chromatic scale[], and the circle of fifths[].
However, this only applies to our current tuning system, which divides an octave into
12 equal pieces. In 24 equal temperament, where every whole tone is divided into 4 instead of 2,
there are four distinct circles. There's the circle formed by jumping by 5 quarter tones,
which is between a second and a third, a circle formed by jumping by neutral thirds,
or 7 quarter tones, an 11-quarter tone circle, and of course the 24-tet chromatic circle.
Of course, there's still the circle of fifths and the 12-tet chromatic circle,
but they no longer cover every note, and exist in two variants.
Moreover, in tuning systems with prime divisions of the octave, like the popular 19 and 31-tet,
jumping by any interval will result in a circle because prime numbers,
by definition, don't have any prime factors smaller than themselves.
However, using highly composite numbers as octave divisions can also be useful as it creates the
possibility of dividing the octave evenly. Though each system has advantages and disadvantages, it's
unlikely that 12-tone equal temperament, and the circle of fifths, are going anywhere anytime soon.
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