SPECIFIKAL ESIKA
ACEPTACION DE HERENCIAS
I
n The Birth of Tragedy, Friedrich Nietzsche wrote: “Music is like geometric figures and numbers, which are the universal forms of all possible objects of experience.”*Traditionally, musicologists have analyzed music from the numerical, historical, sociologi- cal, psychological, anthropological, neurological, as well as music theory and praxis points of view. The geometric approach used here permits a new kind of analysis of rhythms that yields novel insights, and thus augments the traditional tools employed by musicologists.†
This is not to imply that geometry has not been utilized in the past as a music-theoretic tool. Indeed, geometric images have served multiple purposes for illustrating a variety of musical concepts since antiquity.‡ The circular notation for cyclic rhythms goes back at
least to the thirteenth century Baghdad.§ In modern times, geometric structures in two
and higher dimensions are applied to a variety of different aspects of music analysis with increasing frequency.¶ Furthermore, the visualization of rhythms as cyclic polygons allows
instant recognition of many structural features of the rhythms that are more difficult to perceive with standard Western music notation or even box notation. For example, sup- pose we want to know whether the clave son has the palindrome property: that it contains an onset from which one can start playing the rhythm either forward or backward so that it sounds the same. With Western music notation, the novice requires some reflec- tion to come up with the answer. On the other hand, with polygon notation, the answer
* Johnson, I. (2009), p. 16.
† See Toussaint, G. T. (2003, 2004a, 2005a,b) for a more detailed and deeper discussion of computational geometric tools
that may be exploited by musicologists.
‡ Christensen, T. (2002), p. 280. § Liu, Y. and Toussaint, G. T. (2010a).
¶ Tymoczko, D. (2011), Bhattacharya, C. and Hall, R. W. (2010), Hall, R. W. (2008), Hook, J. (2006), Rappaport, D. (2005), and
Yust, J. (2009) explore the geometry of harmony. McCartin, B. J. (1998), Don, G. W., Muir, K. K., Volk, G. B., and Walker, J. S. (2010), Hodges, W. (2006), Cohn, R. (2000, 2001, 2003), and Andreatta, M., Noll, T., Agon, C., and Assayag, G. (2001) explore rhythmic canons. See also Mazzola, G. (2002, 2003), Honingh, A. K. and Bod, R. (2005), and Wild, J. (2009).
34 ◾ The Geometry of Musical Rhythm
is instantly revealed. Consider the six distinguished timelines described previously, and pictured in polygon notation in Figure 8.1.
The son polygon has a solid line connecting the pulse at position three with the pulse at position 11. This is an axis of mirror symmetry for the polygon. The polygon looks the same on both sides of this line. Therefore, the son rhythm has the palindrome property when started on the second onset (at pulse position three). Note that among the six time- lines, two other rhythms have the palindrome property, the shiko and the bossa-nova, both of which have mirror symmetry about the vertical line connecting pulses zero and eight. This example highlights the ease with which humans perceive spatial symmetry, especially with the polygon notation, compared to Western music notation. On the other hand, without these visual aids, “it is extremely difficult to perceive temporal symmetry.”*
The rhythm polygons in Figure 8.1 contain two other markers of noteworthy geometric and musical properties. All but the rumba have a dashed line connecting some pairs of onsets: the shiko, son, and soukous each has one such line, the gahu has two, and the bossa- nova has three. Geometrically, these lines determine isosceles triangles, together with the two adjacent edges on the polygons. Musically, such a triangle indicates that there are two adjacent inter-onset intervals of the same duration. Three of the rhythm polygons contain
* Handel, S. (2006), p. 188. 0 1 2 3 4 5 6 7 8 Shiko 9 10 11 12 13 14 15 Son 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 4 12 Rumba 0 1 2 3 5 6 7 8 9 10 11 13 14 15 Soukous 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 3 4 5 Gahu 0 1 2 6 7 8 9 10 11 12 13 14 15 Bossa-nova 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
The Distance Geometry of Rhythm ◾ 35
vertices that make an interior right angle (90°), indicated by the small squares. The shiko and soukous have a right angle at their first onset at pulse zero. The gahu has a right angle at the fourth onset at pulse 10. Geometrically, a right angle indicates that there exists a pair of onsets diametrically opposite to each other.* Musically, this means that there are two onsets
that break the cycle into two equal half cycles, introducing a certain degree of regularity, an important musical property that we shall return to in more detail later in the book.
These geometric properties can provide new explanations that complement certain con- clusions that musicologists have made about rhythms, on the basis of musicological proper- ties alone. For instance, consider the clave rumba and the clave son, which differ only in the position of the third onset. Musicologists agree that the clave rumba rhythm is more complex than the clave son rhythm because the rumba has its third onset on a weak beat (pulse seven) and a silence on a strong beat (pulse six), whereas the opposite is true for the son.† These two
properties of rhythms (the presence of onsets on weak beats, and the absence of onsets on strong beats) are features of the musicological notion called syncopation.‡ Comparing the two
rhythm polygons in Figure 8.1, we observe that, unlike the son, the rumba has no isosceles triangles and no axes of mirror symmetry. Therefore, from a purely geometric point of view, the rumba is less structured and thus more complex than the son.
When the mind is presented with a rhythm such as the clave son that is repeated con- tinuously throughout a piece of music, and that has a cycle that lasts only a few seconds, it is natural to ask whether it perceives durations other than those that occur between adjacent onsets. There exists plenty of evidence, and consensus, that the “conscious pres- ent” (also called “specious present”§) lasts for about 3 s. This phenomenon is known as the
“three-second window of temporal integration.”¶ Therefore, it is most likely that the mind
also perceives (perhaps unconsciously) the durations between all the other pairs of onsets, in rhythms that last less than 3 s.**
A list of all the inter-onset intervals is called the full interval content of the rhythm. Figure 8.2 shows each of the five onsets of the clave son connected to all the others with straight lines labeled with numbers. The line connecting the first onset at pulse zero with the third onset at pulse six has the label “6” attached to it, indicating that the time dura- tion between these two onsets is six units. This number is the shortest distance along the circle that connects pulse zero to pulse six. Note that the clockwise distance along the circle
* Patsopoulos, D. and Patronis T. (2006), p. 59. The theorem asserting that if in a triangle inscribed in a circle one of its
sides determines the diameter of the circle, then the angle opposite that side is a right angle, is attributed to Thales, the pre-Socratic Greek philosopher from of Miletus, who is considered by many to be the “Father of the Scientific Method” for introducing the notion of a mathematical proof by means of deductive reasoning.
† Velasco, M. J. and Large, E. W. (2011), p. 185.
‡ Fitch, W. T. and Rosenfeld, A. J. (2007), Longuet-Higgins, H. C. and Lee, C. S. (1984). § Phillips, I. (2008), p. 182.
¶ Pöppel, E. (1989), p. 86.
** There is as yet no experimental evidence that durations between nonadjacent attacks play a role in the perception of
rhythm similarity. The analog question in the pitch domain has been investigated experimentally using chords, throw- ing doubt on the perceptual validity of some music theoretical assumptions such as octave equivalence: see Gibson, D. (1993). Nevertheless, experiments performed by Quinn, I. (1999) show that the relations between nonadjacent pitch tones (the combinatorial model) do affect the judgments of perceptual similarity, but to a lesser degree than the relations between adjacent tones (the note-to-note model).
36 ◾ The Geometry of Musical Rhythm
starting from pulse six and ending at pulse zero is 10. However, we use the shorter of the two distances (be it clockwise or counterclockwise) as the distance between the pair of onsets. In geometry, such a distance is called the geodesic distance.*
Therefore, the full interval content of the clave son contains 10 distances in total, some of which occur more than once. One numerical way to represent the interval content of a rhythm is by listing how many times each possible distance occurs. In the case of 16-pulse rhythms, the possible distances range from one to eight, and therefore the interval content may be written as (0, 1, 2, 2, 0, 3, 2, 0). This is sometimes called the interval vector of the rhythm.† A more visually compelling representation of the interval vector is as a histo-
gram. Figure 8.3 shows the histograms of the six distinguished timelines. Useful infor- mation about the rhythms may be gleaned from the properties of their interval content histograms. For example, shiko uses only four different distances: two, four, six, and eight. On the other hand, gahu uses seven different distances ranging from two to eight. In a later chapter, we shall explore the concept of rhythm complexity, and its application to the cre- ation of music, and compare a variety of measures of mathematical, perceptual, and per- formance complexities. In the present context, one measure of the complexity of a rhythm is the total number of different distances that it generates. Therefore, one would expect the gahu to be more complex than the shiko, and perhaps more challenging to learn as well. The difference between son and rumba is not as pronounced; son contains five different intervals and rumba six. Nevertheless, it is observed again that this property of histograms (their density of occupied cells) suggests, like the previous geometric features, as well as syncopation, that the rumba is more complex than the son. Furthermore, the higher the number of different distances that a rhythm contains, the flatter the histogram will tend
* Points that lie on a circle are called cyclotomic sets in crystallography, where they serve as models of one-dimensional
periodic molecules of crystals. The actual models are straight line segments of one period, but the ends are tied together into a circle to facilitate the visualization of all the geodesic distances between the pairs of points (atoms), Buerger, M. J. (1978). See also Senechal, M. (2008) for related material. Tymoczko, D. (2009) analyses the relationship between three different musical distances and the musico-geometrical spaces they inhabit.
† Lewin, D. (2007), p. 98. The term interval vector is normally used to describe the pitch intervals in chords and scales. The
terms interval-class content, interval function, and pitch class content are also used: see Isaacson, E. J. (1990), Lewin, D. (1959, 1977), Rogers, D. W. (1999), and Block, S. and Douthett, J. (1994). Much additional work has been done exploring interval vector relations in the pitch domain for chords and scales.
0 1 2 3 6 3 7 4 6 2 6 7 4 7 3 4 5 6 7 8 9 10 11 12 13 14 15
The Distance Geometry of Rhythm ◾ 37
to be, since the histogram bins have to spread themselves out. Therefore, the shape of the histogram is relevant to a variety of musical properties of rhythms. We will return to con- sider shape properties of interval content histograms other than flatness later in the book. It is instructive to compare the histograms that contain all the inter-onset intervals shown in Figure 8.3, with the histograms that contain only the adjacent intervals, shown in Figure 8.4. The latter histograms are equivalent to Pearsall’s duration sets.* Although these
histograms have their strengths, here, we see one of their weaknesses. The son, rumba, and gahu all have identical histograms, and thus cannot be distinguished from each other based only on this information. Indeed, these three rhythms are permutations of their adjacent inter-onset intervals. It will be seen, however, that even the full interval histo- grams have some serious drawbacks for characterizing rhythms, and thus, for one of the most important problems in musicology, the measurement of rhythm similarity.
* Pearsall, E. (1997).
1 2 3 4 5
Shiko 6 7 8 1 2 3 4 5Son 6 7 8 1 2 3 Rumba4 5 6 7 8
1 2 3 4 5
Soukous 6 7 8 1 2 3 4 5Gahu 6 7 8 1 2 3Bossa-nova4 5 6 7 8
FIGURE 8.3 The full interval content histograms of the six distinguished timelines.
1 2 3 4 5
Shiko 6 7 8 1 2 3 4 5Son 6 7 8 1 2 3 Rumba4 5 6 7 8
1 2 3 4 5
Soukous 6 7 8 1 2 3 4 5Gahu 6 7 8 1 2 3Bossa-nova4 5 6 7 8
39
C h a p t e r