These two beginnings are similar in several respects:
- As shown with an enclosure, each uses a progression of four chords that are grouped together by repetition and/or silence.
- In each four-chord progression, one treble-clef voice changes to a different note from Chord 1 to Chord 2 and Chord 3 to Chord 4, and all notes change to different notes from Chord 2 to Chord 3. In accord with this -- more voices typically change to different notes at moments of greater metrical accent -- Chords 1 and 3 are more metrically accented than Chords 2 and 4.
- The top voice of each begins on G#4/Ab4 and rises by step to end on B4: Wagner entirely by half step, and Hanson with a whole step then a half step.
- The second-to-lowest voice of each sounds B3 for Chords 1 and 2, and then G#3/Ab3 for Chords 3 and 4, creating a voice exchange with the top voice.
- In contrary motion to the top voice's rise, the bottom voice of each descends by step from its note of Chords 1 and 2 to its note of Chords 3 and 4.
- Each progression is soft, slow, and -- not shown in the reduction -- features the woodwinds of the orchestra.
Wagner's progression uses four voices, while Hanson's uses five. However, in the eleventh measure of the symphony, Hanson removes his second-to-lowest voice: the one with B3 and Ab3, marked with little blue dots in the notation above. This slimmed-down progression reveals other connections to Wagner's music.
These two four-voice progressions are shown below, more abstractly. Each voice, along with each chord, has been numbered: the highest voice is Voice 1, the second highest voice is Voice 2, and so so forth. In Wagner's music, Chords 1 and 4 are conventionally tertian: specifically, Chord 1 is a half-diminished seventh chord and Chord 4 is a major-minor seventh chord. In Hanson's music, Chords 2 and 3 are conventionally tertian: specifically, Chord 2 is a half-diminished seventh chord and Chord 3 is a major-minor seventh chord. Although the two remaining chords in Wagner's music do not similarly match the two remaining chords in Hanson's music, this difference could nonetheless be described with a permutation: Chords 1 and 2 switch places, and Chords 3 and 4 switch places. This can be represented with the notation (12)(34).
The graphic below takes a closer look at the half-diminished (ø7) and major-minor seventh (Mm7) chords from each progression. Each colored arrow measures the number of semitones between the two notes spanning the arrow as if these two notes were transported by octaves to put them as close as possible. For example, in Wagner's Chord 4, the top voice's B and the bottom voice's E are separated by a perfect twelfth, which is seventeen semitones. However, if the top note was lowered by two octaves (or the bottom note was raised by two octaves), they would span a mere five semitones. This five-semitone span is represented by the color white. The correspondence between each arrow's color and the number of semitones of its span is also shown in the graphic below.
The double lines in the graphic above single out the red (three-semitone) arrows between Voice 1 and Voice 3 in all four chords, and the blue (two-semitone) arrows between Voice 2 and Voice 4 in all four chords. In Wagner's chords, the notes in Voices 2 and 4 (D# and F) move in parallel motion to new notes (D and E), while the notes in Voices 1 and 3 (G# and B) switch places (disregarding octaves), as shown with the crossed diagonal arrows. This switch in Wagner's progression could be labeled as a (13)(2)(4) permutation. In Hanson's chords, the notes in Voices 1 and 3 (Bb and Db) move in parallel motion to new notes (B and D), while the notes in Voices 2 and 4 (F and G) switch places (disregarding octaves), as shown with the crossed diagonal arrows. This switch in Hanson's progression could be labeled as a (1)(24)(3) permutation.
These two scenarios flip when we consider each note as labeled by its intervallic environment within its chord. For example, the G# in Wagner's Chord 1 (first chord) and Voice 1 (top voice) is three semitones, three semitones, and five semitones away (disregarding octaves) from the other three notes in Chord 1, as represented by the two red arrowheads and one white arrowhead in the G# cell of Wagner's Chord 1. Therefore, in Wagner's Chord 1, G# can be labeled as two parts red and one part white, which is the distribution of colors on the Austrian flag. In the graphic below, the G# in Wagner's Chord 1 is positioned on the Austrian flag.
I have chosen the colors so that the other notes can be positioned on other country's flags -- Germany (black, red, yellow), Estonia (blue, black, white), and Armenia (red, blue, yellow) -- although, with apologies especially to Armenia, the colors have been standardized to the same primary or secondary hues. The four intervallic environments of the four notes of a major-minor seventh are the same as the four intervallic environments of the four notes of a half-diminished seventh, as shown by the same four flags in each vertical column. However, although Wagner's Chord 1 and Hanson's Chord 2 assign the same flags to the same voices, the registral (vertical) ordering of the flags for Wagner's Chord 4 and Hanson's Chord 3 are different from this and each other. The arrows of the graphic below show how the assignment of each flag to each voice permutes in the progression from ø7 to Mm7 for each composer's work. From this vantage point, Wagner's permutation is (1)(24)(3), because the flags of Voices 2 and 4 switch places, while those of Voices 1 and 3 do not. Hanson's permutation is (13)(2)(4), because the flags of Voices 1 and 3 switch places, while those of Voices 2 and 4 do not. These permutations are swapped from those shown earlier.
These two voice permutations also share a relationship with the aforementioned (12)(34) chord permutation, as shown below: the latter -- called an automorphism -- transforms one voice permutation to the other.
Henry Klumpenhouwer's 1991 dissertation from Harvard inspired this analysis.
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