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Readers of the second edition of David Schiff's The Music of Elliott Carter (London: Faber; Ithaca: Cornell University Press, 1998) may have wondered about the "Link Chords," mentioned by Schiff in connection with Carter's tryptich Symphonia—Sum Fluxae Pretium Spei and several other works of the 1990s. The "Link Chords" are all-interval twelve-note chords, each of which contains either one or two instances of the all-trichord hexachord [0,1,2,4,7,8] as a contiguous subset. Both all-interval twelve-note chords and the all-trichord hexachord are well known. In 1965 Stefan Bauer-Mengelberg and Melvin Ferentz published an algorithm for generating all-interval twelve-note chords (see bibliography). I came across their work when I was in college in the early 1980s and got my friend Tim Vontz to write up a version of the Bauer-Mengelberg and Ferentz algorithm in BASIC. (It took about 12 hours to run on my mother's PC!) When I was in graduate school in 1987 Frank Wilhoit and I wrote a version of this algorithm in the 'C' programming language, and in 1992 I wrote a simple 'C' program that searched the list of twelve-note chords for all the interval strings that would produce the all-trichord hexachord.

In June of 1992 I sent a copy of the list I had produced to Elliott Carter. He was working on Partita at the time, and was in the process of collecting exactly the type of all-interval chords I had found. It must have been a laborious process to search for them by hand, and my list provided a welcome shortcut. The next time I saw him he thanked me for the list and said it had been very useful. I later found an autograph sketch dated June 20th, 1992 in the Elliott Carter collection of the Paul Sacher foundation. It is labeled "since Link" and consists of a three-page list of "Link Chords" written out in music notation.

There are 194 all-interval twelve-note chords that contain the all-trichord hexachord as a contiguous subset, assuming equivalence under P, R, I, and RI. Forty-four of these contain two all-trichord hexachords.*

John Link
January, 2005

*My thanks to Hee-Seng Kye for pointing out inaccuracies in an earlier version of this list.

The "Link chords" (.pdf)