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€.

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 Klang, the score is not recommended for part-time ensembles. Score 15€, parts 15€.

Introduction

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

Each of the eight sections consists of 9 phrases of 9 notes. These formations are musical equivalents of what are known as Latin Squares, a phenomenon first explained by Leonhard Euler. The music allows you to hear lovely symmetries, all notes equally frequent and to learn something about Latin squares at the same time. Duration about 10 minutes, 10€.

[CD Recording available: Tom Johnson - correct music]

For 2 flutes, oboe, clarinet, 2 violins, viola. 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€. 

Introduction

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)

Tom Johnson, Paris, August, 2007

The Combinations for String Quartet were commisioned by MärzMusik in Berlin, for the premier by the Bozzini Quartet in MärzMusik 2004.

The music is constructed by systematically taking all the combinations of something, but each movement does this in a quite different way.

25 minutes. Score, € (Euros) 15, parts € (Euros) 15.