If you want to understand Western music theory, the circle of fifths is an invaluable tool. For one thing, it can help you understand how key signatures work. But it also helps explain how the major scale and diatonic modes relate to each other, and gives a possible explanation for why they sound good.
Here’s the C major scale on the circle of fifths:
The purple notes are the ones that form “perfect” intervals above the root C: unison, octave, fourth and fifth. The green notes form major or “natural” intervals above the root. The numbers refer to the scale degrees.
This kind of circular representation is called a scale necklace (or sometimes scale bracelet). It’s easy to transform the major scale into its parallel modes by adding and removing notes from the ends of the necklace. For example, to change the C major scale into C Lydian mode, you remove the F, the furthest counterclockwise end of the necklace, and replace it with F-sharp, adding it onto the furthest clockwise end of the necklace.
To change the C major scale into C Mixolydian mode, you remove the B at the clockwise end of the necklace and replace it with B-flat at the counterclockwise end.
(In this diagram and all the ones that follow, blue notes are flat or minor intervals above the root.)
To change C Mixolydian mode into C Dorian mode, you remove the E and replace it with E-flat.
To change C Dorian mode into the C natural minor scale, you remove the A and replace it with A-flat.
I don’t feel like making the diagrams, but you do the same procedure to change the C natural minor scale into C Phrygian mode: remove the D and add D-flat. To change C Phrygian mode into C Locrian mode, you remove the G and add G-flat.
Notice a pattern here: as you rotate around in a clockwise direction, adding more “blue” notes (flat/minor intervals above the root), the modes get “darker.” And as you rotate around in a counter-clockwise direction, adding more “green” notes (natural/sharp/major intervals above the root), the scales get brighter. You can think of scale degrees 2, 3, 4, 5, 6 and 7 in a diatonic mode as switches or toggles that can be flipped in the brighter or darker direction. Sorting the modes along the circle of fifths the way we’re doing here also conveniently sorts them from brightest to darkest.
You can rotate scale necklaces around the circle of fifths to change their root note. For example, to change the C major scale into the G major scale, you rotate it one slot clockwise.
This has the same effect as changing C major into C Lydian mode, because G major and C Lydian are comprised of the same notes.
To change C major into F major, you rotate its necklace one slot counterclockwise.
This has the same effect as changing C major into C Mixolydian, because F major and C Mixolydian contain the same notes.
You can change the C natural minor scale into the A natural minor scale by rotating its necklace three steps clockwise. Notice that A natural minor is the same pitches as C major, but with different scale degree labels.
The modes of natural minor work the same way as the modes of major. To change A natural minor into A Dorian mode, you remove the F and add F-sharp.
In all the scale necklaces we’ve talked about so far, the notes have been contiguous (touching each other, with no gaps.) In the course of making lots of necklace diagrams, I noticed a pattern: the more contiguous a scale is on the circle of fifths, the more “consonant” or “normal” it sounds to Western ears. Conversely, the more gaps it has, the more “strange” or “exotic” it sounds. For example, take a look at a scale that’s common in the Middle East but rare in Western music, the C Phrygian dominant scale. It has a couple of big gaps.
You could think of the C Phrygian dominant scale as being the fifth mode of F harmonic minor. Western classical music does actually use this scale, but only in the specific circumstance of dominant chords in minor keys. In F minor, you would use C Phrygian dominant over the dominant chord, C7. In this context, harmonic minor is supposed to feel unstable. The E natural is an outlier, a dissonance, a tension that needs resolving. By contrast, Middle Easterners (including my Jewish ancestors) don’t find Phrygian dominant to be unstable at all. They use it as a resolved sound over tonic chords, and will cheerfully play it for the duration of entire tunes.
Why do Western people prefer their scales to be contiguous on the circle of fifths? It’s probably because we consider the fifth to be the most consonant interval, aside from the octave. The fifth is low in the natural overtone series, and notes that are a fifth apart therefore share a lot of overtones. Western people take this to mean that notes separated by fifths are “related” to each other. If you construct an entire scale from fifths, then by Western standards, it will hang together logically. The major scale has seven consecutive fifths, whereas Phrygian dominant only has four.
When we were figuring out how to organize the scale picker in the aQWERTYon, we considered ordering the scales from “least weird” to “most weird” based on how contiguous or gapped they are on the circle of fifths. That measure of “weirdness” doesn’t exactly align to everyone’s intuition, but we couldn’t think of any better or more objective system for organizing scales.
It’s interesting to visually compare the circle of fifths to the chromatic circle. The image below shows the C major scale in both representations.
My math friends tell me that you transform the circle of fifths into the chromatic circle and vice versa through a process called involution. Specifically, you take every alternating note and reflect it through the center of the circle. In other words, you take one of the whole tone scales and swap all the tritones with each other. Interestingly, the C whole tone scale looks the same on both circles; only the D-flat whole tone scale notes change places. I don’t know what this means in practical music making terms, but it’s fun to think about.