Looking at Numbers

Tom Johnson (1939) is considered a minimalist, since he works with simple forms, limited scales, and generally reduced materials, but he proceeds in a more logical way than most minimalists, often using formulas, permutations, predictable sequences and various mathematical models. He is particularly well known for his operas. The Four Note Opera (1972) has been presented in 11 different languages since its premier 40 years ago. Riemannoper has been staged more than 30 times in German-speaking countries since its premier in Bremen in 1988. Often played non-operatic works include Bedtime Stories, Rational Melodies, Music and Questions, Counting Duets, Tango, Narayana's Cows, and Failing: a very difficult piece for solo string bass. His longest work is the Bonhoeffer Oratorio for four soloists, two choruses and orchestra, presented in Maastricht (1996), Berlin (1998) and New York (2000). The book Conversations avec Tom Johnson, in collaboration with Bernard Girard, was published in 2011. Franck Jedrzejewski is a researcher in mathematical physics at the CEA Saclay and has also a PhD in philosophy and musicology. He teaches at the INSTN as well as in the University of Paris Sud and Versailles-Saint-Quentin.

"T. Johnson and F. Jedrzejewski's book Looking at numbers presents a dual perspective (musical and mathematical) on making sense of a large set of drawings created by Johnson to illustrate self-discovered patterns in finite collections of numbers. ... Overall, for a non-expert in the overlap of music and mathematics, the book is interesting and fairly well written." (Guadalupe I. Lozano, Mathematical Reviews, June, 2015)

• Introduction.- 1. Permutations.- 1.1 Symmetric Group.- 1.2 Bruhat Order.- 1.3 Euler Characteristic.- 1.4 Group Action.- 1.5 Permutohedra and Cayley Graphs.- 1.6 Coxeter Groups.- 1.7 Homometric Sets.- 2. Sums.- 2.1 Integer Partitions.- References.- 3. Subsets.- 3.1 Combinatorial Designs.- 4 Kirkman’s Ladies, a Combinatorial Design.- 4.1 Steiner and Kirkman Systems.- 5. Twelve.- 5.1 (12,4,3).- 6. (9,4,3).- 6.1 Decomposition of Block Designs.- 7. 55 Chords.- 7.1 Chords and Designs.-8. Clarinet Trio.- 8.1 Strange Fractal Sequences.- 9. Loops.- 9.1 Self-Replicating Melodies.- 9.2 Rhythmic Canons.-10. Juggling.- 10.1 Juggling, Groups, and Braids.- 11. Unclassified.- 11.1 Some Other Designs.- A Figures.- References.