For nine different plucked string instruments. Dedicated to Jean-Paul Davalan, the mathematician who computed many of these tilings. One movement, 26 minutes; 15€.

An excellent 2017 recording is available, courtesy of the Partch/Just Strings ensemble and MicroFest Records.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**2016

Flexible instrumentation. One movement, 6 minutes; 10 pages, 10€.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**2015

Flexible instrumentation. A collection of all-interval tetrachords arranged to be played with two instruments. 23 pages, 12€.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**2014

For 2 flutes, oboe, clarinet, 2 violins and viola. All composed with mathematical techniques, but each quite different. Average durations 10 minutes, 10 to 15€.

Chords of five notes cycle around a scale of 11 notes, following a classical block design. Recommended for contemporary music ensembles, both amateur and professional. Duration 12 minutes. Score 10€, parts 10€.

My music is always more concerned with notes than with timbres, and often the instrumentation is not specified, since the music can sound equally clear with many different combinations of colors. In the case of the Septet, however, I decided to score the music specifically for two flutes, oboe, clarinet, two violins, and viola. The music is essentially a long progression of five-note chords, and the sound must be homogenous enough to be able to hear the harmonies clearly, yet subtle color differences greatly improve the musical interest, so it seemed best to solve this problem myself. I intentionally broke all the usual laws of voice leading, and crossed voices very often, so that one would hear the chords independently, without melodic connections.

The chords are constructed on an 11-note scale in a rather narrow range, following a combinatorial design known as (11,5,2), which means that:

- Eleven elements (11 notes) are distributed into 11 subgroups of five elements (11 chords of five notes)
- Each note occurs five times in five of the chords
- Each of the 55 pairs of notes comes together in two of the chords
- Each chord has exactly two notes in common with each other chord.

I simply took the unique solution for this rather amazing symmetrical structure, transformed it into 10 related solutions, as shown on the cover, selected my 11-note scale, and arranged the result for the selected instruments. The total duration is about 12 minutes.

Those wishing to know more about the mathematics of these sorts of structures may consult *The Handbook of Combinatorial Designs*, edited by Charles J. Colbourn and Jeffrey H. Dinitz (second edition, Chapman and Hall/CRC, 2007)

Chords of five notes cycle around a scale of 11 notes, following a classical block design. Recommended for contemporary music ensembles, both amateur and professional. Duration 12 minutes. Score 10€, parts 10€.

My music is always more concerned with notes than with timbres, and often the instrumentation is not specified, since the music can sound equally clear with many different combinations of colors. In the case of the Septet, however, I decided to score the music specifically for two flutes, oboe, clarinet, two violins, and viola. The music is essentially a long progression of five-note chords, and the sound must be homogenous enough to be able to hear the harmonies clearly, yet subtle color differences greatly improve the musical interest, so it seemed best to solve this problem myself. I intentionally broke all the usual laws of voice leading, and crossed voices very often, so that one would hear the chords independently, without melodic connections.

The chords are constructed on an 11-note scale in a rather narrow range, following a combinatorial design known as (11,5,2), which means that:

- Eleven elements (11 notes) are distributed into 11 subgroups of five elements (11 chords of five notes)
- Each note occurs five times in five of the chords
- Each of the 55 pairs of notes comes together in two of the chords
- Each chord has exactly two notes in common with each other chord.

I simply took the unique solution for this rather amazing symmetrical structure, transformed it into 10 related solutions, as shown on the cover, selected my 11-note scale, and arranged the result for the selected instruments. The total duration is about 12 minutes.

Those wishing to know more about the mathematics of these sorts of structures may consult *The Handbook of Combinatorial Designs*, edited by Charles J. Colbourn and Jeffrey H. Dinitz (second edition, Chapman and Hall/CRC, 2007)

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**2007-2010

For jugglers. In preparation.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**20**

Seven kinds of music derived from seven drawings all based on a (12,3,2) combinatorial design. About 20 minutes. The three B-flat clarinets each read from the score. 15€.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**2012

Written for Gandini Juggling, this marks the first time a composer has written for jugglers, and perhaps the first time three jugglers have performed with amazing musical precission. The premier in Amsterdam was with sounding balls developed by Steim researchers. 10€.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**2011

The falling thirds are played by some solo instrument, the drum keeps the beat, and everything is the result of a drawing. Eight minutes, 8€.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**2011

Three percussionists [one cowbell, one wood block, one bongo] seem to be “mocking” one another as they progress from one rhythm to another slightly different rhythm, all nicely charted in graphic illustrations. By the end of seven minutes, each of the rhythms has been played once and none has been repeated. Dur. about 7 min. Score 12€.

For some time I have enjoyed calculating graphs in order to see how one could progress through a whole group of chords, always following the most subtle voice leading possible. Only one voice is permitted to move one degree up or down. In the case of *Mocking* I did the same thing with rhythms. To have two notes in the time of eight beats, for example, the possibilities are to play on beats 1-2, 1-3, 1-4, 1-5… up to 7-8. There are a total of 28 different two-note rhythms, and the ways in which these can be joined with minimal differences can be seen in the graph on the first page. The subsequent sections of the music follow the same procedure with rhythms of three, four, five and six notes. So by the end we have a little catalogue of all 256 rhythms possible in 8 beats, omitting the eight rhythms of one note and the eight rhythms of seven notes, which both musically and mathematically are almost as trivial as the remaining rhythm with zero notes.

The clearest way of organizing this little catalogue was to follow the graphs from left to right, playing each rhythm once, and the clearest way of hearing them was just to pass them from one simple percussion instrument to another. As I began scoring the sequences for cow bell, wood block and bongo, and listening to the results in my head, the three percussionists seemed to be competing with one another, correcting one another, making fun of one another, or in a word, Mocking one another. Thus the title.

*Tom Johnson, Paris 2009*

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**2009

*Vermont Rhythms* is written for 2 saxophones (tenor and baritone), trombone, percussion, guitar and keyboard.
Two Vermont mathematicians constructed a remarkably symmetrical list that includes all the 462 six-note rhythms playable in a measure of 11 beats. This permitted the composer to write 462 measure of syncopations that are as different from jazz as from Stravinsky. Written for the professional Dutch sextet Ensemble Klang, the score is not recommended for part-time ensembles.
Score 15€, parts 15€.

CD recording available by Ensemble Klang.

This piece is called Vermont Rhythms because it would never have been written without the cooperation of two Vermont mathematicians, working at the University of Vermont in Burlington. In answer to a question of mine, Susan Janiszewski, with her advisor, Professor Jeffrey H. Dinitz, constructed a remarkable list of all the 462 six-note rhythms possible in an 11-beat period. Their impressive list distributes the rhythms in 42 groups of 11, each group forming an 11 by 11 square. The first square, the first 11 measures of music, is shown on the cover, so that one can better appreciate the symmetry of these squares. All 42 squares contain six elements in each row and six elements in each column, giving maximal rhythmic variety within the 11 phrases of each square. Each six-note rhythm has exactly three beats in common with each of the 10 others, and mathematicians will appreciate additional symmetries in these configurations.

My primary interest was the 462 rhythms, but I soon realized that I could choose pitches by employing the 462 six-note chords possible on an 11-note scale at the same time, so I did that too. Of course, much of this organization will not be heard consciously, even by very astute listeners, but some of it will be quite clear to everyone, and it is satisfying to know that many unheard symmetries are also present, reflecting one another in the background.

As the piece became clearer in my mind, I realized it would be particularly effective played by Klang, an ensemble in The Hague that had recently done an amazing interpretation of Narayana’s Cows. They agreed to premier the work, which explains why it is scored for two saxophones, trombone, guitar, percussion, and piano. The music has little to do with instrumental color, however, so the instrumentation may be varied somewhat to be more suitable for other ensembles.

*Tom Johnson, Paris, December 2008*

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**2008

Rational harmonies in five voices for ampliﬁed ensemble (solo 1, solo 2, keyboard, guitar and bass). 20 minutes. Score 12€, parts 15€.

CD recording available by Ensemble Klang.

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.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**2005

Rational harmonies in three voices, preferably for three flutes, or solo harp. Score: 13€.

Available on a CD recording by Manuel Zurria.

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.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**2005

288 three-note chords with sums of 72 (middle C = 24), preferably for violin, viola, violoncello. Score and parts 13€.

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.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**2005

The music is constructed by systematically taking all the combinations of something, but each movement does this in a quite different way. Commisioned by MärzMusik in Berlin, for the premier by the Bozzini Quartet in MärzMusik 2004. 25 minutes. Score 15€, parts 15€.

The piece contains five movements, and each of these contains all the combinations of something. As usual, I wanted the music to know what it was doing, to be correct and complete in a rigorous sense, and this is one way of achieving this. The theory of combinations is a totally explored mathematical discipline, as we have known for more than a century how to calculate all kinds of combinations and probabilities, and how to prove all of this. So what I say about my composition can not have fundamental significance for mathematics. I can, however, demonstrate that new questions arrive when one wants to go inside some set of combinations, to see how they come together, to observe the many symmetries within them, to find the best sequence for them, to consider how they might sound, to turn them into music.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**2003

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. Score and parts 20€.

“Tilework” has to do with fitting together little tiles to fill lines and loops. One can think of this as making mosaics in one dimension, but it is also very much like stringing beads onto necklaces in various patterns. In musical terms, the Tilework series is a collection of compositions in which individual tiles/rhythms fit together into musical sequences without simultaneities, often filling all the available points of a line or a loop.

Since the notes of different motifs come at different moments, it is possible for a single melodic instrument to play several voices at once, so “tiling” was particularly appropriate for unaccompanied melodic instruments. I gradually found so many ways of tiling musical phrases that I could not stop until I had a piece for each instrument of the orchestra. The year of 2002 was devoted almost exclusively to 14 pieces for 14 solo wind and string instruments. *Tilework for Five Conductors and One Drummer*, *Tilework for Piano*, and *Tilework for String Quartet* came later, in 2003.

At first, interlocking one tile with another seemed obvious, sort of like fitting together the pieces of a jigsaw puzzle. As the work went on, however, it became clear that this was not so simple. Sometimes it is not at all obvious how rhythms/tiles link together, and sometimes one can easily see six ways of solving a certain problem, without being sure if some seventh solution might also be possible, and sometimes the discussion can go way beyond the comprehension of musicians. I knew that making a loop of 16 beats by repeating one eight-note rhythm had something to do with group theory, and I was dimly aware that some mathematicians know how to determine the number of unique necklaces of length n possible using beads of m colors, never allowing two beads of the same color to touch, but I certainly hadn’t studied such things. Gradually I was entering a world I didn’t know much about.

As the *Tilework* project continued, I sometimes stumbled across questions that were as new and interesting for mathematicians as for myself. I owe much to the regular meetings devoted to Mathematical Music Theory at IRCAM in Paris, where I met many mathematicians, particularly Emmanuel Amiot, and Andranik Tangian, who gave me solutions to some problems. In the case of *Tilework for String Quartet*, which involves ways of tiling the line with six-note rhythms, I am particularly indebted to Harald Fripertinger and his most useful computer list of “rhythmic canons.”

But of course, composers, interpreters, and listeners do not need to know all this, just as we do not need to master counterpoint in order to appreciate a Bach fugue. As always, one of the wonderful things about music is that it allows us to perceive directly things that we would never understand intellectually.

*Tom Johnson, Paris, October 2003*

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**2003

Variations on a 20-beat loop that turns continually around the eight instruments: flute, clarinet, trumpet, trombone, marimba, violin, viola, cello. Dur. 16 min, score 18€, parts 30€.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**1998

Four logical progressions for four instruments, each playing independently from the others. Clarinet, trombone, piano, and cello. 10-12 minutes, 10€.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**1996

The first Johnson string quartet, premiered by the Brindisi Quartet in 1994. Eight movements, each following a mathematical formula. 20 minutes, score 14€, parts 20€.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**1996

See title. Six minutes, 12€.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**1993

Trio for saxophone, guitar and bass, written for the German ensemble Ugly Culture. The title means “Unison Polyrhythms.” 18 minutes, 14€.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**1992

For three instruments (flexible instrumentation) ; see detailed description.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**1991

For three voices (flexible instrumentation, preferably large ensembles) ; see detailed description.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**1989

Predictable music for violin, cello and piano, 15 minutes. Score, including string parts: 23€.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**1984

The subtle differences between the colors of these two instruments continue for 16 minutes. Harp and piano: 10€.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**1982

Four movements for wind quintet (flute, oboe, clarinet, bassoon, horn). 20 minutes. Musicians read from the score, 12€.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**1980

A suite of short pieces for unspecified upper-register instruments. 20 minutes. Score 16€, parts 20€.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**1979

Nine short movements in eight minutes, always with the same sequence of 60 notes. 10€.

**Author:**Tom Johnson**Publisher:**Editions 75**Year:**1976