This time I’ll keep the axis and the distance the same, and switch direction. Each illustration will compare a note with its mirror twin, its reciprocal.
First up is the strongest polarity flip there is, the perfect fourth and fifth. One divides the tonic by 3, the other multiplies it by 3.
The 4 is clearly unstable, it wants to move. The 5 is clearly stable. If a song ends with this interval, I will feel completely satisfied.
The next matchup is the b7- and the 2. The b7- is the crucial note that provides the tension in dominant-type seventh chords and makes their resolution so satisfying. Here it is in undiluted form.
The 2 is fairly stable. Quite a few songs end on this note, and there is a pretty good sense of resolution, maybe with some wistfulness mixed in.
The two notes are about equally harmonious, and of opposite polarity. This is the same pattern as the 4 and 5, only weaker.
Moving outward, we get the b3- and 6+ pair:
The pattern continues — now both notes are rather dissonant, with the b3- weakly unstable and the 6+ weakly stable. It would be rather unsettling to end a song on the 6+, but maybe you could get away with it.
Here are the next two:
These are interesting. They are dissonant, all right, and the b6- is unstable and the 3+ is stable. But I actually hear the polarity a little more strongly than the last pair.
I think my ear is trying to interpret these notes as out-of-tune versions of the b6 (a strongly unstable note) and the 3 (strongly stable).
How is my ear to interpret this 3+ note, the Pythagorean major third? Can I even hear a ratio of 81/64? Maybe not well enough to really recognize it.
Perhaps the ear “decides” that it’s simpler to read this strange note as a badly tuned version of a simpler interval, one I am familiar with. So I hear it as an out-of-tune 5/4 instead of an in-tune 81/64.
This is why equal temperament works, as Mathieu demonstrates so well in Harmonic Experience. A painting doesn’t have to be exactly straight on the wall for the eye to interpret it as straight. Thank goodness! In the same way, a note doesn’t have to be exactly in tune to be heard as that note. The ear is willing to accept “close enough” and hear it as the real thing, though the consonance will not be as good.
Maybe the part of the mind that processes this stuff is like a quantum computer, taking in the sound, trying out all possibilities at once, and spitting out the “most likely” interpretation, which would be the solution with the lowest “potential energy,” the one that is closest to the center, just like real gravity.
We’re probably too far out now to really recognize these intervals as what they are, but for the heck of it:
Suitably nasty, and now the sense of polarity is pretty much gone, I can’t hear it.
The Pythagorean spine, the sequence of fifths, has come full circle — almost. The two notes are 24 cents apart, a Pythagorean Comma. All that remains of tonal harmony at this distance is a generic sort of dissonance. I hear no polarity at all. The tonal gravity field is too weak to detect.