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.

Hahn, Five Little Songs (5 petites chansons)

In 1915, Reynaldo Hahn set poems for children by Robert Louis Stevenson in the Five Little Songs (5 petites chansons). The cycle was first performed and published in 1916. The first song, "The Swing" [La Balançoire], is the subject here.

"The Swing" belongs to a very select group of published compositions in which the major dominant-ninth chord is essential to sound and affect. These include Marion Bauer's "The Epitaph of a Butterfly" (link to my post), Charles Griffes' "The White Peacock" (check this blog post for an abstract and link to my essay: 2020 June 5), and of course the "Russian Dance" from Stravinsky's Petrushka, which I wrote about in the previous post to this blog.

The evocative accompaniment figure in bar 1 is repeated, with slightly shifting harmonies, in all 35 bars, excepting an abbreviated form in bars 32-33. Since the pedal will certainly be held down through each beat, the initial--and frequently repeated--tonic, which I have labeled "I" is just as easily heard as Iadd6. The V11 (or V9sus4) of the second beat "relaxes" into a clear V9 by bar 2, beat 2. In context, the notated Fnat4 in bar 3 sounds like a neighboring E#4 (so G-F#-E#-F# in bars 1-4), but the pitch as Fnat4 will play a role later. The G-F#-E#-F# motion, of course, adds a slightly longer-range swinging figure to the mix.

The poem is in three verses. Each is set slightly differently. Here is verse 1. The letter above each bar indicates the accompaniment figure, as they were labeled in the example above.

Here is verse 2:

And here is verse 3:

Note above that figure C--not figure A or B--appears in the first three bars of this verse. From this point, new twists on the harmony show up in each bar--see below, the score for the second line to the end of the verse. Now the Fnat4 of figure C signals a turn toward IV, which in turn leads quickly to the closing cadence, where the V is again a clear V9.

The piano's brief coda brings a small but interesting play on the ancient post-cadence turn toward IV. In bar 33, Fnat4 hints at a repeat of bars 25ff. and it does go down to E4, but not over IV: instead it's over I to make Iadd6. The arpeggios  to prominent E5 and E6 (arrows) are a delightful (but still safe) "swing" just past the tonic triad.

Stravinsky, "Russian Dance"

In an earlier post (link), I wrote that the final cadence of the "Russian Dance" from Stravinsky's Petrushka "is a famous example [of the bass moving as expected while the upper-voices don't go as neatly as one—traditionally, at least—might like]. The C5 as the top note doesn't budge, nor does the A5 below it, while the bass pounds out a simple ii-V-I."

 

In fact, from the standpoint of this blog's theme, the entire dance is very interesting. Its signature harmony is a prolonged C: V9. Here is the beginning:

True, V11 does resolve to e6 in bar 8, that is, to iii as a functional (third-related) substitute for I. I can't go so far as to imagine e6 as a CM7 without its root but I'll allow that it's not out of the question.

In the return of the opening figure at rehearsal 43, the V9 is unequivocal:

Here's the continuation from that point to the end:

And here's another take on the final bars:

The G11 is still "full diatonic", containing all the thirds above G up to the 11th, that is, not deleting the third (which would be typical and is recommended). Oddly, the ii11 that follows does delete the third (F). The penultimate chord marked V11 is the same "full diatonic" version.

NB: I've based this on the published piano reduction, not the full score.