Perfect Rhythmic Tilings

PERFECT RHYTHMIC TILINGS JEAN-PAUL DAVALAN PREFATORY NOTES HE ALGORITHM FOR COMPUTATION of perfect rhythmic tilings presented herein and the corresponding computer program were both devised in November 2004 after I read Delahaye (2004) and decided to look for perfect rhythmic tilings with index 4, thereby answering Tom Johnson’s question regarding their existence. Delahaye (2005) published the results obtained with this algorithm. All examples of perfect rhythmic tilings given in this paper were computed with this program. Table 1 (see section 1.2.1) sums up the results for indexes 2 to 50 and orders 1 to 60. I had already written a first draft when I learned, in 2007, about former works on kindred topics. For the sake of completeness, this paper includes brief references to those results of other researchers that came to my notice. To round it off, a quick survey of classical rhythmic canons and an algorithm are given at the end of the article. The same T Perfect Rhythmic Tilings 145 tree-traversal technique is applied to two very different types of tilings, and can be compared at the reader’s leisure. This paper was edited and adapted in English by E. Amiot. Some slight emendations were effected for the purpose of this special issue of Perspectives of New Music. 1. INTRODUCTION 1.1. PERFECT TILINGS 1.1.1. THE SQUARE A perfect tiling of a rectangle, or better, of a square, is a tiling by way of smaller squares with different side lengths, without overlap (only edges are shared), and that cover the whole initial rectangle square. In 1938, Smith, Stone and Brooks managed to cover a square with 69 squares; in 1939 Sprague used only 55; and the quest was carried on until 1978, when Duijvestijn discovered a tiling in only 21 squares (Example 1). C. J. Bouwkamp, A. J. Duijvestijn, and P. Medema have shown, using computers, that there is no perfect tiling of a square (with non-nil side) with twenty squares or less (except, of course, the trivial tiling with one square) (Bouwkamp 1992). Integer sequence A006983 in N. J. A. Sloane’s On-Line Encyclopedia of Integer Sequences enumerates the number of different solutions—1, 8, 12, 26, 160, 441, 1152, . . .—for tilings in 21, 22, 23, 24, 25, 26, 27, . . . EXAMPLE 1: PERFECT SQUARE TILING WITH INDEX 21, BY A.J.W. DUIJVESTIJN 146 Perspectives of New Music smaller squares, respectively. Gambini’s PhD dissertation (1999) relates the story of these perfect square tilings and shows in detail how to construct them. 1.1.2. TILING RANGES It is by transposition of the notion of perfect square tilings to tilings of a range, with subsequent application to music composition, that Tom Johnson created a new type of tiling that he christened “perfect rhythmic tilings” (Johnson 2004). In this new type of tiling, the small square is replaced with a finite arithmetic progression; i.e., several points, regularly spaced. Example 2 shows five arithmetic sequences of three terms (0,7,14), (3,8,13), (1,5,9), (2,4,6), and (10,11,12), with respective ratios 7, 5, 4, 2, and 1, which are augmentations turning one element of a triplet into the next. These ratios represent the durations of the rests between elements. (Tom Johnson and others use the term “tempo” in place of “ratio.” This article will use the term “ratio.”) Three boxes on the same line indicate a triplet whose ratio is denoted on the left side. This particular tiling seemed especially remarkable to Johnson, who used it as the backbone of his composition Tileworks for Piano, released in 2003. In his monthly column for Pour la Science, J.-P. Delahaye (2004; 2005) elaborated on the musical use of perfect rhythmic tilings. In Tiling for String Instrument, a mature and original composition for electric guitar or mandolin, Dean Rosenthal (2011) used the second tiling k=4, n=15 (Example 6). The style of the composition is obviously deeply influenced by his friend and teacher, Tom Johnson. EXAMPLE 2: TOM JOHNSON’S PERFECT RHYTHMIC TILINGS WITH LENGTH 15 Perfect Rhythmic Tilings 147 In order to build a perfect rhythmic tiling, we have the following conditions...