## The Geometry of Musical Rhythm (2004)

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Venue: | In Proc. Japan Conference on Discrete and Computational Geometry, LNCS 3742 |

Citations: | 15 - 7 self |

### BibTeX

@INPROCEEDINGS{Toussaint04thegeometry,

author = {Godfried Toussaint},

title = {The Geometry of Musical Rhythm},

booktitle = {In Proc. Japan Conference on Discrete and Computational Geometry, LNCS 3742},

year = {2004},

pages = {198--212},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Musical rhythm is considered from the point of view of geometry.

### Citations

1930 |
Pattern Classification
- Duda, Hart, et al.
- 2001
(Show Context)
Citation Context ... k onsets in a time span cycle of n units by R[k, n]. 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]-=-. The first natural question that arizes is whether there exist any rhythms whose inter-onset intervals have perfectly flat histograms of height one with no gaps. This is clearly not possible with R[5... |

901 |
Algorithms on Strings, Trees and Sequences
- Gusfield
- 1997
(Show Context)
Citation Context ... time? In contrast, if the swap distance is replaced with the Hamming distance, then the cyclic (or necklace) Hamming distance may be computed in O(n log n) time with the Fast Fourier Transform [23], =-=[29]-=-. The work of ÓMaidín [41] and Francu and Nevill-Manning [25] suggests several interesting open problems. In the acoustic signal domain the vertical transposition is continuous rather than discrete. T... |

114 |
String matching and other products
- Fischer, Paterson
- 1974
(Show Context)
Citation Context ...o(n2 ) time? In contrast, if the swap distance is replaced with the Hamming distance, then the cyclic (or necklace) Hamming distance may be computed in O(n log n) time with the Fast Fourier Transform =-=[23]-=-, [29]. The work of ÓMaidín [41] and Francu and Nevill-Manning [25] suggests several interesting open problems. In the acoustic signal domain the vertical transposition is continuous rather than discr... |

113 |
Comparison of musical sequences
- Mongeau, Sankoff
- 1990
(Show Context)
Citation Context ...the edit distance which allows for insertions and deletions of notes. Discussions of the application of the edit-distance to the measurement of similarity in music can be found in Mongeau and Sankoff =-=[40]-=- and Orpen and Huron [42]. A noteworthy more recent generalization is the fuzzy Hamming distance [6] which allows shifting of notes as well as insertions and deletions. Using dynamic programming these... |

53 |
The Structure of Atonal Music
- Forte
- 1973
(Show Context)
Citation Context ...entify, if not explain, cultural preferences of rhythms in traditional music. 2.1 Maximally even rhythms In music theory much attention has been devoted to the study of intervals used in pitch scales =-=[24]-=-, but relatively little work has been devoted to the analysis of time duration intervals of rhythm. Two notable exceptions are the books by Simha Arom [3] and Justin London [36]. Clough and Duthett [1... |

43 |
On a cyclic string-to-string correction problem
- Maes
- 1990
(Show Context)
Citation Context ...ance between two rhythms minimized over all possible rotations of one with respect to the other. Some work has been done with cyclic string matching for several definitions of string similarity [28], =-=[38]-=-, [11]. Consider two binary sequences of length n and density k (k ones and (n − k) zeros). It is desired to compute the minimum swap distance between the two strings under all possible alignments. I ... |

40 |
On measuring the distance between histograms
- Cha, Srihari
- 2002
(Show Context)
Citation Context ...larity between two patterns is measured by a simple template matching operation. More recently similarity has been measured with more powerful and complex functions such as the earth mover’s distance =-=[7]-=-, [64], weighted geometric matching functions [37], the swap distance [61], and the directed-swap distance [15], [13]. 4.1 Swap distance A well known distance measure between two n-bit binary sequence... |

37 |
Heuristic and special case algorithms for dispersion problems
- Ravi, Rosenkrantz, et al.
- 1994
(Show Context)
Citation Context ...ly difficult, researchers have proposed approximation algorithms [30], and heuristics [21], [66], for the general problem, and have sought efficient solutions for simpler special cases of the problem =-=[49]-=-, [54]. Fejes Tóth also showed in [55] that in three dimensions four points on the sphere maximize the sum of their pairwise distances when they are the vertices of a regular tetrahedron. The problem ... |

35 | Approximation algorithms for maximum dispersion
- Hassin, Rubinstein, et al.
- 1997
(Show Context)
Citation Context ... it is one of the dispersion problems called the discrete pmaxian location problem [20], [21]. Because these problems are computationally difficult, researchers have proposed approximation algorithms =-=[30]-=-, and heuristics [21], [66], for the general problem, and have sought efficient solutions for simpler special cases of the problem [49], [54]. Fejes Tóth also showed in [55] that in three dimensions f... |

33 | A comparison of rhythmic similarity measures
- Toussaint
(Show Context)
Citation Context ..., of this numerical interval-length representation, are two obvious advantages, but its iconic value is minimal. For a description of additional (more geometric) methods used to represent rhythms see =-=[61]-=-. In this paper several geometric properties of musical rhythm are analysed from the musicological and mathematical points of view. Several connecting bridges between music theory, mathematics, and co... |

27 | Classification and phylogenetic analysis of African ternary rhythm timelines
- Toussaint
(Show Context)
Citation Context ... that the second rhythm is internationally the most well known of all the African timelines. It is traditionally played on an iron bell, and is known on the world scene mainly by its Cuban name Bembé =-=[60]-=-. Traditional rhythms have a tendency to exhibit such properties of evenness to one degree or another. Therefore mathematical measures of evenness, as well as other geometric properties, find applicat... |

24 |
Maximally Even Sets
- Clough, Douthett
- 1991
(Show Context)
Citation Context ...4], but relatively little work has been devoted to the analysis of time duration intervals of rhythm. Two notable exceptions are the books by Simha Arom [3] and Justin London [36]. 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 mathematic... |

24 | G.T.: El compás flamenco: A phylogenetic analysis
- Díaz-Báñez, Farigu, et al.
- 2004
(Show Context)
Citation Context ...been measured with more powerful and complex functions such as the earth mover’s distance [7], [64], weighted geometric matching functions [37], the swap distance [61], and the directed-swap distance =-=[15]-=-, [13]. 4.1 Swap distance A well known distance measure between two n-bit binary sequences is the Hamming distance trivially computed in O(n) time. However, this distance measure is not appropriate fo... |

23 |
The Structure of Homometric Sets
- Rosenblatt, Seymour
- 1982
(Show Context)
Citation Context ...ore recently in DNA sequencing [52]. Two noncongruent sets of points, such as the two different necklaces of Figures 5, are called homometric if the multisets of their pairwise distances are the same =-=[50]-=-. For an extensive survey and bibliography of this problem see [34]. The special cases relevant to the theory of rhythm, when points lie on a line or circle, have received some attention, and are call... |

23 | A mathematical analysis of African, Brazilian, and Cuban clave rhythms, in
- Toussaint
- 2002
(Show Context)
Citation Context ... played with two sticks made of hard wood also called claves [43]. More relevant to this paper, there exist purely geometric properties that may explain the world-wide popularity of this clave rhythm =-=[58]-=-. The Clave Son rhythm is usually notated for musicians using standard music notation which affords many ways of expressing a rhythm. Four such examples are given in the top four lines of Figure 2. Th... |

20 |
African Polyphony and Polyrhythm
- Arom
- 1991
(Show Context)
Citation Context ...the top four lines of Figure 2. The fourth line displays the rhythm using the smallest convenient durations of notes and rests. Western music notation is not ideally suited to represent African music =-=[3]-=-, [18]. The fifth and sixth lines show two popular ways of representing rhythms that avoid Western notation. The representation on line five is called the Box Notation Method developed by Philip Harla... |

20 | Distance metrics and indexing strategies for a digital library of popular music
- Francu, Nevill-Manning
- 2000
(Show Context)
Citation Context ...ghth or sixteenth notes). In this context it is desired to compute the minimum area between the two contours under vertical translations and horizontal scaling of the query. Francu and Nevill-Manning =-=[25]-=- claim that this distance measure can be computed in O(mn) time but they do not describe their algorithm in detail.s6 New Open Problems Let us assume that we are given a circular lattice with n points... |

19 |
Two special cases of the assignment problem
- M, Li
- 1975
(Show Context)
Citation Context ... one. Furthermore, in the case where both binary sequences have the same number of “one’s” (or onsets), there is a one-to-one correspondence between the indices of the ordered onsets of the sequences =-=[32]-=-. The swap distance may of course be computed by actually performing the swaps, but this is inefficient. If X has one’s in the first n/2 positions and zero’s elsewhere, and if Y has one’s in the last ... |

19 |
Basic Atonal Theory
- Rahn
- 1987
(Show Context)
Citation Context ...ve high evenness value. How close to optimal is this procedure? The two sequences shown in Figure 5 are the only possible rhythm bracelets with flat histograms, for any values of k greater than three =-=[48]-=-. Therefore in order to be able to generate additional rhythms with near-flat histograms the constraints outlined in the preceeding need to be relaxed. We may proceed in several directions. For exampl... |

19 | A partial digest approach to restriction site mapping
- Skiena, Sundaram
- 1994
(Show Context)
Citation Context ...erpoint distances: given a distance multiset, construct all point sets that realize the distance multiset. This problem has a long history in crystallography [34], and more recently in DNA sequencing =-=[52]-=-. Two noncongruent sets of points, such as the two different necklaces of Figures 5, are called homometric if the multisets of their pairwise distances are the same [50]. For an extensive survey and b... |

18 | L.: Pattern matching in polyphonic music as a weighted geometric translation problem
- Lubiw, Tanur
- 2004
(Show Context)
Citation Context ...le template matching operation. More recently similarity has been measured with more powerful and complex functions such as the earth mover’s distance [7], [64], weighted geometric matching functions =-=[37]-=-, the swap distance [61], and the directed-swap distance [15], [13]. 4.1 Swap distance A well known distance measure between two n-bit binary sequences is the Hamming distance trivially computed in O(... |

18 |
Measurement of similarity in music: A quantitative approach for non-parametric representations
- Orpen, Huron
- 1992
(Show Context)
Citation Context ...llows for insertions and deletions of notes. Discussions of the application of the edit-distance to the measurement of similarity in music can be found in Mongeau and Sankoff [40] and Orpen and Huron =-=[42]-=-. A noteworthy more recent generalization is the fuzzy Hamming distance [6] which allows shifting of notes as well as insertions and deletions. Using dynamic programming these distances may be compute... |

12 | An efficient algorithm for generating necklaces with fixed density
- Ruskey, Sawada
- 1999
(Show Context)
Citation Context ...espect to the underlying beat). The number of onsets in a rhythm is called the density in combinatorics, and efficient algorithms exist for generating all the necklaces with a specified fixed density =-=[51]-=-. 3.1 Rhythms with specified duration multiplicities In 1989 Paul Erdős [19] asked whether one could find n points in the plane (no three on a line and no four on a circle) so that for every i, i = 1,... |

11 |
Vector products and intervallic weighting
- Block, Douthett
- 1994
(Show Context)
Citation Context ...he books by Simha Arom [3] and Justin London [36]. 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 add... |

11 |
A comparison of p-dispersion heuristics
- Erkut, Ülküsal, et al.
- 1994
(Show Context)
Citation Context .... In operations research it is studied under the umbrella of obnoxious facility location theory. In particular, it is one of the dispersion problems called the discrete pmaxian location problem [20], =-=[21]-=-. Because these problems are computationally difficult, researchers have proposed approximation algorithms [30], and heuristics [21], [66], for the general problem, and have sought efficient solutions... |

11 | Reconstructing sets from interpoint distances, DIMACS
- Lemke, Skiena, et al.
(Show Context)
Citation Context ...al problem of reconstructing sets from interpoint distances: given a distance multiset, construct all point sets that realize the distance multiset. This problem has a long history in crystallography =-=[34]-=-, and more recently in DNA sequencing [52]. Two noncongruent sets of points, such as the two different necklaces of Figures 5, are called homometric if the multisets of their pairwise distances are th... |

10 |
ethnomathematics. The logic underlying orally transmitted artistic practices
- Ethnomusicology
- 2002
(Show Context)
Citation Context ... properties of evenness to one degree or another. Therefore mathematical measures of evenness, as well as other geometric properties, find application in the new field of mathematical ethnomusicology =-=[9]-=-, [62], where they may help to identify, if not explain, cultural preferences of rhythms in traditional music. 2.1 Maximally even rhythms In music theory much attention has been devoted to the study o... |

10 | An algorithm for computing the restriction scaffold assignment problem in computational biology
- Colannino, Toussaint
- 2005
(Show Context)
Citation Context ...easured with more powerful and complex functions such as the earth mover’s distance [7], [64], weighted geometric matching functions [37], the swap distance [61], and the directed-swap distance [15], =-=[13]-=-. 4.1 Swap distance A well known distance measure between two n-bit binary sequences is the Hamming distance trivially computed in O(n) time. However, this distance measure is not appropriate for rhyt... |

9 | Maximum dispersion and geometric maximum weight cliques
- Fekete, Meijer
- 2003
(Show Context)
Citation Context ..., 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 operations research it is studied under the umbrella of obnoxious facility location theory. In particular, it is one of the dispersion problems called the discrete pmaxian location problem [20], ... |

9 |
From Polychords to Pólya: Adventures in Musical Combinatorics
- Keith
- 1991
(Show Context)
Citation Context ...2 4 3 10 6 5 7 6 (a) (b) 9 8 11 0 1 5 6 2 3 Fig. 5. Two flat-histogram rhythms. 4 1 5 2 4 3sA cyclic sequence such as [x x . . x . x . . . . .] is an instance of a necklace with “beads” of two colors =-=[33]-=-; it is also an instance of a bracelet. Two necklaces are considered the same if one can be rotated so that the colors of its beads correspond, one-to-one, with the colors of the other. Two bracelets ... |

9 |
On the sum of distances determined by a point set
- TÓTH
- 1956
(Show Context)
Citation Context ...he 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 thesShiko Son Soukous Rumba Bossa Gahu Fig. 3. The six fundamental 4/4 time clave and bell patterns in box notation. points on the circular... |

8 |
combinatorial problems in computational music theory
- Algorithmic
- 2003
(Show Context)
Citation Context ...ast two fundamental ideas: what should be measured, and how should it be measured. There exists a wide variety of methods for measuring the similarity of two rhythms represented by strings of symbols =-=[59]-=-. Indeed the resulting approximate pattern matching problem is a classical problem in pattern recognition and computer science in general [16]. Traditionally similarity between two patterns is measure... |

7 |
On the sum of distances between n points on a sphere
- Alexander
- 1977
(Show Context)
Citation Context ... the vertices of a regular tetrahedron. The problem remains open for more than four points on the sphere. Some upper and lower bounds on the maximum value that the sum may attain are known. Alexander =-=[1]-=- proved an upper bound of (2/3)n 2 − (1/2). It has also been shown that the points must be well spaced in some sense. Stolarsky [53] proved that if n points are placed on the sphere so that the sum of... |

7 | A positive-evidence model for classifying rhythmical patterns
- Eck
- 2000
(Show Context)
Citation Context ...estern notation. It is also convenient for experiments in the psychology of rhythm perception, where a common variant of this method is simply to use one symbol for the note and another for the pause =-=[17]-=-, as illustrated in line six. In computer science the clave Son might be represented as the 16-bit binary sequence shown on line seven. Finally, line eight depicts the interval length representation o... |

7 |
Foundations of diatonic theory: a mathematically based approach to music fundamentals
- Johnson
- 2003
(Show Context)
Citation Context ... equidistant from three others. A musical scale whose pitch intervals are determined by points drawn on a circle, and that has the property asked for by Erdős is known in music theory as a deep scale =-=[31]-=-. We will transfer this terminoly from the pitch domain to the time domain and refer to cyclic rhythms with this property as deep rhythms. Deep scales have been studied as early as 1967 by Carlton Gam... |

7 |
Hearing in Time
- London
- 2004
(Show Context)
Citation Context ...s used in pitch scales [24], but relatively little work has been devoted to the analysis of time duration intervals of rhythm. Two notable exceptions are the books by Simha Arom [3] and Justin London =-=[36]-=-. 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 constr... |

7 |
A Geometrical Algorithm for Melodic Difference
- O`Maidin
- 1998
(Show Context)
Citation Context ...e with a simple scan. Therefore O(n) time suffices to compute dSW AP (U, V ), resulting in a large gain over the linear or dynamic programming algorithms. i=1 5 Introducing Melody into Rhythm ÓMaidín =-=[41]-=- proposed a geometric measure of the distance between two melodies modelled as monotonic pitch-duration rectilinear functions of time as depicted in Fig. 7. ÓMaidín measures the distance between the t... |

7 |
A mathematical measure of preference in African rhythm
- Toussaint
(Show Context)
Citation Context ...erties of evenness to one degree or another. Therefore mathematical measures of evenness, as well as other geometric properties, find application in the new field of mathematical ethnomusicology [9], =-=[62]-=-, where they may help to identify, if not explain, cultural preferences of rhythms in traditional music. 2.1 Maximally even rhythms In music theory much attention has been devoted to the study of inte... |

6 | Computing a geometric measure of the similarity between two melodies
- Aloupis, Fevens, et al.
- 2003
(Show Context)
Citation Context ... a circle as in Figure 1. In music theory this spectrum is called the interval vector (or fullinterval vector) [39]. For example, the interval vector for the clave Son pattern of Figure 1 is given by =-=[0,1,2,2,0,3,2,0]-=-. It is an 8-dimensional vector because there are eight different possible duration intervals (geodesics on the circle) between pairs of onsets defined on a 16-unit circular lattice. For the clave Son... |

6 |
Distances with specified multiplicities
- Erdős
- 1989
(Show Context)
Citation Context ...he density in combinatorics, and efficient algorithms exist for generating all the necklaces with a specified fixed density [51]. 3.1 Rhythms with specified duration multiplicities In 1989 Paul Erdős =-=[19]-=- asked whether one could find n points in the plane (no three on a line and no four on a circle) so that for every i, i = 1, ...n−1 there is a distance determined by these points that occurs exactly i... |

6 |
Prelude to musical geometry
- McCartin
- 1998
(Show Context)
Citation Context ...t durations are present in a rhythm. Again we assume rhythms are represented as points on a circle as in Figure 1. In music theory this spectrum is called the interval vector (or fullinterval vector) =-=[39]-=-. For example, the interval vector for the clave Son pattern of Figure 1 is given by [0,1,2,2,0,3,2,0]. It is an 8-dimensional vector because there are eight different possible duration intervals (geo... |

5 |
de Bruijn. Sorting by means of swapping
- G
- 1974
(Show Context)
Citation Context ...nterchange of a one and a zero that are adjacent to each other in the binary string. Interchanging the position of elements in strings of numbers is a fundamental operation in many sorting algorithms =-=[14]-=-. However, in the sorting literature a swap may interchange non-adjacent elements. When the elements are required to be adjacent, the swap is called a mini-swap or primitive-swap [4]. Here we use the ... |

5 |
The discrete p-maxian location problem
- Erkut, Baptie, et al.
- 1990
(Show Context)
Citation Context ...m [22]. In operations research it is studied under the umbrella of obnoxious facility location theory. In particular, it is one of the dispersion problems called the discrete pmaxian location problem =-=[20]-=-, [21]. Because these problems are computationally difficult, researchers have proposed approximation algorithms [30], and heuristics [21], [66], for the general problem, and have sought efficient sol... |

5 |
Some combinational resources of equal-tempered systems
- Gamer
- 1967
(Show Context)
Citation Context ... transfer this terminoly from the pitch domain to the time domain and refer to cyclic rhythms with this property as deep rhythms. Deep scales have been studied as early as 1967 by Carlton Gamer [26], =-=[27]-=-, and it turns out that the ubiquitous Western diatonic scale is a deep scale. Also, the Bembé rhythm mentioned in the preceeding is a deep rhythm since it is isomorphic to the diatonic scale. The que... |

5 |
Efficient dynamic programming alignment of cyclic strings by shift elimination
- Gregor, Thomason
- 1996
(Show Context)
Citation Context ...e distance between two rhythms minimized over all possible rotations of one with respect to the other. Some work has been done with cyclic string matching for several definitions of string similarity =-=[28]-=-, [38], [11]. Consider two binary sequences of length n and density k (k ones and (n − k) zeros). It is desired to compute the minimum swap distance between the two strings under all possible alignmen... |

5 |
Drum Gahu: An Introduction to African Rhythm. White Cliffs
- Locke
- 1998
(Show Context)
Citation Context ...tured in Figure 3. Examination of the six histograms leads to questions of interest in a variety of fields of enquiry: musicology, geometry, combinatorics, and number theory. For example, David Locke =-=[35]-=- has given musicological explanations for the characterization of the Gahu bell pattern (shown at the bottom of Figure 3) as “rhythmically potent”, exhibiting a “tricky” quality, creating a “spirallin... |

5 |
Comments on the paper: ‘Heuristic and special case algorithms for dispersion problems’ by
- Tamir
- 1998
(Show Context)
Citation Context ...ficult, researchers have proposed approximation algorithms [30], and heuristics [21], [66], for the general problem, and have sought efficient solutions for simpler special cases of the problem [49], =-=[54]-=-. Fejes Tóth also showed in [55] that in three dimensions four points on the sphere maximize the sum of their pairwise distances when they are the vertices of a regular tetrahedron. The problem remain... |

5 | The Euclidean algorithm generates traditional musical rhythms
- Toussaint
(Show Context)
Citation Context ... pointed out that can be used for computer composition. We close the paper by mentioning one additional tool for automatically selecting rhythm timelines that can be used for generating new music. In =-=[63]-=- it is shown that the Euclidean algorithm for finding the greatest common divisor of two numbers can be used to generate very good rhythm timelines when the two numbers that serve as input to the Eucl... |

5 |
The maximal dispersion problem and the “first point outside the neighbourhood” heuristic
- White
- 1991
(Show Context)
Citation Context ...n problems called the discrete pmaxian location problem [20], [21]. Because these problems are computationally difficult, researchers have proposed approximation algorithms [30], and heuristics [21], =-=[66]-=-, for the general problem, and have sought efficient solutions for simpler special cases of the problem [49], [54]. Fejes Tóth also showed in [55] that in three dimensions four points on the sphere ma... |

3 |
Timo Raita. Generalized Hamming distance
- Bookstein, Kulyukin
(Show Context)
Citation Context ...of the edit-distance to the measurement of similarity in music can be found in Mongeau and Sankoff [40] and Orpen and Huron [42]. A noteworthy more recent generalization is the fuzzy Hamming distance =-=[6]-=- which allows shifting of notes as well as insertions and deletions. Using dynamic programming these distances may be computed in O(n 2 ) time. The problem of comparing two binary strings of the same ... |