### Musical combinatorial designs

The new edition of Handbook of Combinatorial Designs points to Tom Johnson's musical compositions based on

1. Kirkman's Schoolgirls Problem: "Fifteen young ladies of a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk abreast more than once." [T. P. Kirkman, Query VI. Lady's and Gentleman's Diary (1850), 48.] Johnson's 2005 score for three flutes or solo harp, "Kirkman's Ladies," is based on the solution to the follow-on problem (solved by Denniston in 1974): can all 455 triples from a 15-element set be arranged into 13 disjoint Kirkman Triple Systems of order 15, thereby allowing a walk for each of the 13 weeks of a school term, without any three girls walking together twice?

2. The specific t-(v, k, λ) design given in the Handbook's 4.6 Example. In describing his piece "Block Design for Piano," Johnson gives the definition of a 4-(12, 6, 10) design in musical terms: "There are 12 notes, distributed into 6-note arpeggios, in such a way that every combination of four particular notes comes together exactly 10 times in 10 different arpeggios."

See Recent works by Tom Johnson for more information on the compositions. For more information on the combinatorial designs, see pages 13 and 79 of

Handbook of Combinatorial Designs, 2nd ed., edited by Charles J. Colbourn, Jeffrey H. Dinitz. Boca Raton, FL: Chapman & Hall/CRC, 2007. Mathematics Library Quarto QA166.25 .H36 2007 Link to MNCAT record