To round out our collection of important chord images, here are those of A minor as compared to the root A1. Making this made me appreciate how effective music theory already is at describing the phenomena we experience listening to music. Something important to notice here is that these chords here use all the same notes as the key of C major. But if a piece of music has implied to us a key of A minor, then hearing a G major chord (the bVII) we experience tension, pulling up up toward A. It’s as if G is interfering with the ghost of A in our heads.
And that same G is the V of C. If we look at our images (ignoring the diminished ii because that’s really harsh to listen to and not commonly used), we see that bVII is the one chord where one of the early overlaps lines up with the middle of an overtone of the base note. The images from the key of C major showed the same thing. In both cases it was a weak overlap, but it’s kind of like those notes are tightly squeezing at the base note. A difference is in the major key that occurs over one of the octaves of that base note, and here it occurs over one of the fifths.
Based on what I’ve seen, it seems like the amount of overlapping that occurs between the chord and the (possibly only implied, rather than physically present) base note prior to the chord’s own first significant internal overlap, correlates with the feeling of harmonic stability, whereas the amount of partial overlapping with the base note correlates with tension. And if an early chord-internal overlap occurs directly over an overtone of the base note, that creates a strong pull, an expectation of a cadence.
Overall, I am doubting the usefulness of the analysis I’ve done over the past few weeks, and the applicability to composition. I think the traditional tools of composition are well refined and didn’t really need the application of math when music is such a subjective and innate experience. That said, I will be keep the experience close at hand in my mind when composing music in the future. And next week I will still try and see what happens when I look at some other tuning system. I’m thinking to look at 19-tone equal temperament, because I heard this amazing song that uses it.