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.
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