Of course they do. Read my earlier posts in this thread, I’ve already made that point. A descending fifth is the same NOTE NAME as an ascending fourth, that’s the point, and that’s what enables people to “see” a circle of fourths in the circle of fifths. It is as much a circle of fourths as a mandolin is tuned in fourths.
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While octaves matter, INVERSIONS also matter, which is why you speak of roots, thirds and fifths in chord inversions whether going up or down the octave to find the thirds or fifth … and as we ponder these things, we may begin to understand why the music teacher chooses to use some practical expressions without much regard to being “correct” in order to avoid pedanticism.
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I standby either Circle of 5ths or Cycle of Fourths being the correct name for this tool. Though, just to piss everyone off, I may start calling it the "Circle of Skipping 2nds" haha.
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We call the B section of Rhythm Changes a circle of 4ths progression, because that's how we memorized the circle. In Bb, it's 2 bars of D7, 2 bars of G7, 2 bars of C7, 2 bars of F7, and finally resolving to a BbM7. So, they all sort of function as dominants, but not exactly. Octaves matter not at all.
They all function exactly like dominants! They are called secondary dominants. Each chord functions as the fifth (dominant) of the next chord, so it is a series of fifths. There is no reason to call it a circle of fourths.
You are right, octaves don't matter at all for harmonic function. I mentioned that they matter for naming intervals.
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