Monday, May 11, 2015

Rare Sibling Harmony at the End of Holst’s Solar System

Let us say that a three-note chord’s sibling is a transposition or inversion of the chord; that is, siblings have the same three intervals between their three pairs of notes, allowing for change of octave. For example, F-A-C and C-Eb-G are in the same family—each contains a minor third, major third, and perfect fourth—but C-E-F is in a different family. There are twelve three-note-chord families. Let us further say that the relationship between two (non-identical) siblings in the same family is harmonious if there is no half step, allowing for change of octave, that exists between a note in one chord and a note in the other.

The family of major and minor triads has been shown to be special for many reasons. Here is one more: of the twelve three-note-chord families, the percentage of harmonious types of sibling relationship among the family of major and minor triads is, perhaps surprisingly, the smallest. (In truth, it is tied with the family to which C-E-F belongs.) Shown below are the twenty-three possible types of relationship a major triad can have with its siblings, up to transposition and inversion. The top system shows all eleven non-zero transpositions, the second system shows six inversions around C, and the third system shows six inversions around C/C#. A notehead is filled in if its pitch forms a half step, allowing for change of octave, with a note in the other chord in the same measure: the two clashing notes have the same notehead shape.

Only two out of these twenty-three relationship-types are harmonious: they are the measures without any filled-in noteheads. If you combine together the two triads in each pair into a richer harmony—an F9 chord, and a C#m7 (or EMadd6)—you have the last two chords of Gustav Holst’s The Planets. These are the chords, sung by an offstage female chorus, that alternate with one other until they fade out to silence, that is, unless Colin Matthews’s Pluto, the Renewer follows on their heels, a piece that premiered fifteen years ago today. Pluto also ends with the same choir singing essentially the same C#m7 chord.

For more on this, see Táhirih Motazedian and Scott Murphy, "Holst’s Planets, The Final Frontier:
Interplanetary Voyage as Intrapersonal Escape" Intégral 36: 1–17.