More Curiosities: Major-Key Symmetries

 The topic today is another in the series of curiosities. Earlier posts were these: Ascending resolutions of the ninth: link; Extending downward: A curiosity: link; and 5-34 and its hexachords: link1, link2.

The major scale is symmetrical, that is, it has the same interval sequence going up and going down--but not on its tonic note. Instead, it is on scale degree ^2, so perhaps we should call it "Dorian symmetry." 

This is relevant to the major dominant-ninth chord as follows. If you build a ninth chord on the tonic (^1), the result is M9. Taking D4 or ^2 in C major as the axis of symmetry, as above, the inverse is d9, or m9: see the first pair under (b1) below. It follows, of course, that if you build a minor ninth chord, you'll get a M9 as the inverse: the second chord pair, which in this case happen to be the same ones: d9 and CM9. Under (b2) are additional chord pairs: the minor(flat 9) on E4 inverts around D4 to Bø(b9), and vice versa. The point of interest--the curiosity--is that the only ninth chord inverting to itself is V9: see under (b3).

Here are a few extra bits. Under (c1), ninth chords on the seven scale degrees in C major are given in the treble clef. Their inversions around the root of each chord are given in the bass clef. As expected, the only inversion that produces a diatonic chord is the one around scale degree ^2. Under (c2), all the inversions are shown with a root C3 to facilitate comparison.

Simple shifts toward IV happen by changing B to Bb, toward V by changing F to F#. These can also be understood symmetrically, as in (d1). Under (d2), the axis of symmetry and the altered notes are isolated. Under (d3), see the two major-dominant ninths that result: V9/V and V9/IV. A view through the circle of fifths is under (d4), where Bb1 is the fourth note below D4 and F#6 is the fourth note above.

Note: I show symmetry of the major dominant-ninth chord within the harmonic series in the post Harmony at the Ninth: link.