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.