While you don't often hear about the "circle of thirds", some musicians regularly refer to a "cycle of thirds". A cycle of thirds that includes all minor intervals can be seen in the diagram below. Each cycle has 5 points to reach back to the root of the chord.
The cycle on the left shows descending thirds while the cycle to the right shows ascending thirds. Note that there are no major thirds in this circle.
For a major third circle, you would have a cycle that has 4 points in order to reach back to the root of the scale.
This is not to be confused with the mix of major and minor thirds found in a major or minor scale that can create chords called diatonic triads.
There are seven total diatonic triads found on any major or minor scale. Diatonic triad intervals are different than the circle of third intervals shown above since they need to fit the major or minor scale of the key.
The triads in the major and minor cycles of thirds are augmented and diminished triads. Augmented triads are made up of 3 notes with major thirds between both intervals.
Diminished triads have minor thirds for each interval. Although there are a few exceptions, these triads do not often appear in the major and minor scales.
The circle of fifths completes a perfect cycle that includes every note on the scale.
While this cycle actually spans over 7 octaves instead of the 1 octave that the circle of thirds cycle spans, note that there are also seven half steps between each perfect 5th. Pretty cool, right?
The half steps in a scale are what make it tricky for a cycle of thirds. Cycling through the major or minor thirds wouldn't hit every note in the scale the way the circle of 5ths can.
The circle of fifths is often a great way to change keys naturally in many genres. Some key changes are done via whole step or half step, but an interesting, unique way to change keys popular in the jazz field uses a cycle of thirds.
A popular jazz saxophonist by the name of John Coltrane often used the major third cycle for his key changes, bringing light to those funky augmented triads.
This method was coined shortly after his death and continues to be used in many jazz songs today.
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