Geometry of music

Summary: The mathematical principles of symmetry and scaling play important roles in musical composition.

Musical information can often be represented naturally with shapes, allowing insights to be gained from geometric techniques.

One indication of the close connection between music and geometry comes from the fact that Euclid of Alexandria, who wrote Elements of Geometry (300 b.c.e.), a founding document of geometry, also wrote a comprehensive treatise on the mathematics of musical pitches, Theory of Intervals. The eighteenth-century mathematician Leonhard Euler also developed geometric tools for music analysis.

Symmetry is one of the most powerful ideas in geometry. No less so in the geometry of music, where symmetries abound. Geometric techniques can be applied to musical scales, chords, and melodic lines. Because of the concept of octave equivalence, the 12 pitches of the equally tempered chromatic scale are inherently cyclic in nature. Thus, the geometric theory of cyclic groups plays a major role in the mathematical description of scales and chords. Similarly, geometry can play a role in the analysis of musical rhythm, particularly in musical forms based upon a repeating rhythmic motif. In twentieth-century atonal music, geometric ideas have been proposed as unifying theoretic structures to fill the role once played by tonal harmonic concepts.

Symmetries in the Twelve Pitches of the Equally Tempered Scale

Two fundamental principles of modern musical analysis are “octave equivalence” and “equal temperament.” Octave equivalence refers to the perception, believed to be universal in developed music cultures, that two pitches separated by an octave are members of the same “pitch class.” Equal temperament refers to the system of musical intonation by which the 12 chromatic half steps within the octave represent uniform frequency scaling—given a pitch with frequency f, the pitch one half step above has frequency 2^1/12 f. In the equally tempered scale, enharmonically spelled notes, such as C& A, B, and C♯ and D♭, represent the same pitch.

The twelve pitch classes are inherently cyclic. This principle is represented in the left view of Figure 1, which is identical to an analog clock face, with the traditional “12” replaced by “0.” The diatonic scale is represented by the vertices of the inscribed polygon in the center view of Figure 1. This arrangement of the seven diatonic pitches is the most even spacing possible for seven pitches in the 12-tone octave. The evident symmetry about the 2–8 axis puts the complicated diatonic sequence of half steps and whole steps into a simpler conceptual framework. The figure illustrates that the Dorian Mode (which begins and ends on the second diatonic scale degree, given here as “D” or “2”) is unique among the diatonic modes in that it follows the same sequence of intervals both ascending and descending.

The six pairs of diametrically opposite pitch classes in the clock representation are separated by the interval of a “tritone,” so named because it contains three whole steps. In tonal music, the tritone is considered the most dissonant-sounding interval. If the three odd-numbered pitch class pairs on the clock face are reflected diametrically, the result is the “circle of fifths” shown in the right view of Figure 1. The circle of fifths is familiar to music students as a mnemonic device for learning the musical key signatures: the number of sharps increases by one (or alternatively, the number of flats decreases by one) at each step in the clockwise direction, while the number of flats increases (or sharps increase) at each step in the counterclockwise direction. The circle of fifths is used extensively as an analytical tool for twentieth-century music in the work of American composer and music theorist Howard Hanson.

Representing Musical Structure in Geometric Spaces

Beginning with the musical writings of Euler and continuing at least through the work of the influential music theorist Hugo Riemann (not to be confused with the mathematician Bernhard Riemann) in the nineteenth century, the representation of harmonic concepts in a two-dimensional array called a “Tonnetz” (Tonal Network) has guided the understanding of tonal harmony. In the tonnetz shown in Figure 2, the rows are simply the entries of the circle of fifths, while the columns are the 12 diatonic pitch classes arranged chromatically (by half steps). The result is that the diagonals are made up of pitch classes separated by minor thirds (in the southeast direction) and major thirds (in the northeast direction). In this arrangement, the sonorities of tonal harmony can be represented by polygonal groupings of the adjacent symbols: triangles for major and minor triads, parallelograms for major and minor seventh chords, and similar structures for diminished, augmented, and dominant seventh chords. The musical theory of “modulation” (changing from one tonal center to another in the course of a musical composition) is aided by the geometric perspective of a Tonnetz. Tonal networks such as the one shown here are precursors of the contemporary musical theory of “pitch class spaces.”

Recently, chords have been modeled as points in geometric spaces called “orbifolds.” Music theorists analyze the symmetry of chords inside of the space with respect to translation, reflection, or permutation and look at short line segments between structurally similar chords.

Rhythmic Symmetry

Like the 12 pitch classes, the metrical organization of music in time is also highly cyclic, allowing similar geometric techniques to be applied to rhythm. The left view of Figure 3 shows the eighth-note subdivisions of a 4/4 measure. The vertices of the inscribed polygon represent the rhythmic placement within the measure of the handclap rhythm from the iconic 1956 Elvis Presley recording of “Hound Dog.” This complicated rhythm has a simple symmetric structure when viewed geometrically. Similarly, the center view in Figure 3 shows the clave rhythm familiar to listeners of Afro-Cuban music, with its line of symmetry. The left view of Figure 3 shows a characteristic bossa nova rhythm (which can be heard on the cowbell in Quincy Jones’s “Soul Bossa Nova”) and its line of symmetry.

Bibliography

Archibald, R. C. “Mathematicians and Music.” American Mathematical Monthly 31, no. 1 (1924).

Demaine, E. D., F. Gomez-Martin, H. Meijer, D. Rappaport, P. Taslakian, G. T. Toussaint, T. Winograd, and D. R. Wood. “The Distance Geometry of Music.” Computational Geometry: Theory and Applications 42, no. 5 (2009).

Hall, Rachel Wells. “Geometrical Music Theory.” Science 320 (2008).

Johnson, Timothy. Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Lanham, MD: Scarecrow Press, 2008.