John F. Link

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" (sorted by all-trichord hexachord string)

The "Link chords" (sorted by all-interval 12-note chord)

The "Link chords" (56K .pdf file)

linkj@wpunj.edu