Vector Products and Intervallic Weighting

The Geometry of Musical Rhythm

“…Clough and Duthett [12] introduced the notion of maximally even sets with respect to scales represented on a circle. According to Block and Douthett [5], Douthet and Entringer went further by constructing several mathematical measures of the amount of evenness contained in a scale (see the discussion on p. 41 of [5]). One of their measures simply adds all the interval arc-lengths (geodesics along the circle) determined by all pairs of pitches in the scale.…”

“…For example, the two four-onset rhythms [ Figure 3, and discussed in [58], have an equal evenness value of 48, and yet the Rumba clave is clearly more uneven than the BossaNova clave. The use of interval chord-lengths (as opposed to geodesic distances), proposed by Block and Douthet [5], yields a more discriminating measure, and is therefore the preferred measure of evenness.…”

“…The evenness measure of Block and Douthet [5], which sums all the pairwise straight-line distances of a set of points on the circular lattice, brings up the question of which configurations of points (rhythms) achieve maximum evenness. In fact, this problem had been investigated by the Hungarian mathematician Fejes Tóth [55] some forty years earlier without the restriction of placing the points on the circular lattice.…”

“…The discrete version of this problem, of interest in music theory [5], is a special case of several problems studied in computer science and operations research. In graph theory it is a special case of the maximum-weight clique problem [22].…”

“…In other words R[k, n] consists of all n-bit cyclic binary sequences with k 1's. Thus all the 4/4 time clave-bell patterns in Figure 3 belong to R [5,16].…”