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The tilework series continues, rational harmonies begin, and listeners receive a test in aural perception

Rational harmonies in three voices, preferably for three flutes, or solo harp. 10 euros

Thomas Penyngton Kirkman photo Thomas Penyngton Kirkman (1806-1895)

In 1847 Reverend Thomas Penyngton Kirkman, an English pastor who was also an amateur mathematician, proposed several solutions to this problem:

Fifteen young ladies in a school walk out three abreast for seven days in succession; it is required to arrange them daily so that no two shall walk twice abreast. (Ladies and Gentleman's Diary, Query VI, p.48)

His work can be considered the first “block design,” a subject that was to become a serious study in combinatorial mathematics. Of course, the discussion quickly grew to include all sorts of investigations of the possible combinations of sub-groups within larger groups, and even the original 15-ladies problem did not end with Kirkman, as mathematicians began to wonder whether it would be possible for the ladies to continue their daily walks for a complete semester of 13 weeks, so as to include all 455 possible three-lady combinations, once each. It was not until 1974 that R. H. F. Denniston of the University of Leicester published a solution, probably the only one, and thanks to him, I can now give you the music. Each lady/note appears once in the daily phrases of five chords, each pair of ladies walks together once a week, and by the end of the 13 weeks, about 13 minutes in musical time, all 455 possible trios of women have passed by, as have the 455 chords that represent them. I am particularly indebted to Paul Denny, a computer scientist at the University of Auckland, whose correspondence helped me to find this information and to understand it.

I like to imagine my three-note chords played by three flutes, or as a harp solo, though an interpretation with three oboes, three strings or one vibraphone might also be just fine, and since the music is really only notes and numbers, I would not want to prevent one from playing it on the piano or with some other instruments. We should leave the chords in this register though, and always keep the ladies clean and pretty. It seems safe to assume that the sun is always shining - otherwise they would not be taking walks.

844 CHORDS (2005)
Rational Harmonies in Five Voices for Amplified Ensemble

After some months of work and hundreds of experiments, the music that finally became 844 Chords was defined with a few remarkably simple rules: Use only the intervals between the minor third and the octave. Begin with the five-note chord where the intervals between the instruments are 3, 4, 5, and 6 semitones, which give a total of 18 semitones, an octave and a half. Continue with the four chords having a total of 19 semitones: (4,4,5,6), (3,5,5,6), (3,4,6,6), and (3,4,5,7) and ask the computer to continue this logic with the 9 chords having a total of 20, the 16 having a total of 21, and so on. The result was a tonal-atonal mathematical sequence with inevitable regularity, but at the same time, with continually surprising juxtapositions and modulations. Sometimes one even hears reappearances of chromatic harmonies from the era of Franck and Wagner, though their sensuality is essentially arithmetic rather than emotional. The 136 chords having a sum of 31 bring us to a cadence in G major that seems to have been prepared for a long time, and I found it unthinkable to continue the logic beyond this point, chord 844.

At first the piece seemed destined to be for a classical string quintet, but the energy of the harmonic progressions required a driving rhythm and a louder sound, so I scored it for two solo lines, guitar, bass and synthesizer. The piece lasts about 20 minutes.

288 three-note chords with sums of 72 (middle C = 24),
preferably for violin, viola, violoncello
score and parts 10 euros

As in all the music in the Rational Harmonies series, I want to deduce my chords, rather than choosing them according to the usual musical and esthetic criteria. To compose the Trio I defined a chromatic scale with notes numbered 0 to 48 and counted all the combinations of three notes having sums of 72, permitting octave relationships, but not unisons. Then I found a chain connecting all 288 chords, requiring that each chord have one note in common with each subsequent chord, the remaining two voices moving by half steps in contrary motion with no crossing of voices. The performers may wish to make little glissandos as they move from one chord to the next.
The piece seems best as a trio for violin, viola and cello, with a tempo of about mm. 40 and a duration of about seven minutes, although interpreters may transpose and arrange the music in other ways if they wish. Faster tempos may be tried as well, although it is important that we hear harmonies and not melodic motion.

TILEWORK Series of Fourteen Solo Pieces
complete series 35 euros, individual pieces 10 euros.

How can different rhythmic patterns be combined in such a way that no two patterns ever occur simultaneously and every beat is filled? "Tiling the line" is a subject that Johnson and several mathematicians have been investigating (see the text from the IRCAM lecture of February 2002), and the original Tilework series, published in 2003, consists of 14 pieces for solo instruments, that employ these techniques: Tilework for Flute, Tilework for Oboe, Tilework for Clarinet, Tilework for Bassoon, Tilework for Saxophone, Tilework for Horn, Tilework for Trumpet, Tilework for Trombone, Tilework for Tuba, Tilework for Percussion Solo, Tilework for Violin, Tilework for Viola, Tilework for Cello, Tilework for Double Bass.

New pieces of TILEWORK

Tilework for Piano
10 euros

Premiered in June 2004 by John McAlpine, this composition tiles a line of 15 points with five voices in tempos 7 : 5 : 4 : 2 : 1 and lasts about 10 minutes. It is a structure known as a “perfect tiling,” and one may read about this the article about Tom Johnson's music in the magazine Pour la Science (novembre 2004).

Tilework for String Quartet
score and parts 20 euros

Tilework for String Quartet is a compilation of all the ways one can tile a line of 12 points by overlapping a single six-note rhythm. The four musicians play these rhythms in canon for 10 minutes in a rapid music requiring great precision. The work was premiered in a KlangAktion concert in Münich in December, 2004.

Tilework for Log Drums
15 euros

Tilework for Log Drums is in seven sections, showing the seven possible solutions for tiling a line of 18 points with six voices, each voice inserting three notes in one of five tempos: 1 : 2 : 3 : 4 : 5. I wanted to hear the music played on log drums mostly because I like the sound of these instruments, but also because the notes of log drums tile the playing surface a bit the way the rhythmic tiles cover the 18-beat phrases in this piece. I can imagine the music played as a virtuoso solo, though the counterpoint will be clearer, and the rhythm will flow more smoothly, if played by an ensemble of two to six musicians. The score was originally published as Tilework for Five Conductors and One Drummer, no longer available.

12 euros

After each phrase, the listener is asked to decide whether the two things heard were the same or different. This test of perception skills can be presented silently and discreetly, or with vocal audience participation, or with answer sheets and scores. A commission of VPRO radio, premiered in Hilversum in 2004 by the Dutch pianist Dante Oei. About 15 minutes.

In Preparation

360 Chords
Permutations of intervals forming 7-voice chords, for large orchestra, 3333,4331,3,str.

Block Design for Piano
30 minutes of continually changing arpeggios, following a set of mathematically determined combinations, written for pianist John McAlpine.

Kleine Choräle für kleines Klavier
Short sequences of rational harmonies, scored for toy piano, written for Isabel Ettenauer.

Developed gradually by the composer/performer since 2001, a score will soon be available for percussionists who would like to construct their own sets of five pendulums and work on this 45-minute solo piece.