Monday, May 31, 2010

Geometry of Music

A version of the following will appear in the 2011 volume "Encyclopedia of Mathematics in Society".

Geometry is the study of shapes.  Musical information can often be represented naturally by 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, whose Elements of Geometry (300 B.C.E)  is the founding document of modern geometry, also wrote a comprehensive treatise on the mathematics of musical pitches called Theory of Intervals.  The great 18th 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.   Due to the concept of octave equivalence, the twelve 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, particular 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 twelve chromatic half steps within the octave represent uniform frequency scaling:  given a pitch with frequency f, the pitch one half step above has frequency 21/12 f.    In the equally tempered scale, enharmonically spelled notes such as C# and Db represent the same pitch. 

The twelve pitch classes are inherently cyclic.  This is represented in the left view of Figure 1.  Notice that this is identical to an analog clock face, with the traditional “12” replaced by “0”.   The diatonic scale (discussed in the Scales entry of this volume) 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 twelve 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 the American composer and music theorist Howard Hanson.

Figure 1 Top: The twelve pitches of the equally tempered chromatic scale arranged on a circle.  Center: The vertices of the inscribed polygon represent the pitches of the diatonic scale.  The diatonic arrangement is the most evenly spaced distribution of seven vertices in a twelve-sided figure.  Note the symmetry inherent in the Dorian mode, which begins and ends on pitch 2 (D).  Bottom: Diametric reflection of the odd numbered pitches results in the circle of fifths.

Tonnetz:  Representing Harmonic Structure in Two Dimensions

Beginning with the musical writings of the 18th century mathematician Leonhard 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 19th 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 twelve 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 (northeast).  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.

C      G      D      A      E       B     F#     Db    Ab    Eb    Bb         F         C
     Eb    Bb    F      C      G      D      A       E       B     F#      Db     Ab
B       F#    Db   Ab    Eb    Bb     F      C       G      D      A         E          B
      D      A      E       B      F#    Db   Ab    Eb    Bb     F        C          G  
 Bb      F      C       G      D      A       E     B     F#    Db    Ab      Eb        Bb
      Db    Ab    Eb     Bb     F       C     G      D     A       E        B        F#      

Figure 2  The first six rows of a Tonnetz (or Tone Network).  The pitch classes of the circle of fifths are arranged horizontally.  The vertical alignment of the pitch classes is chromatic.  Diagonals in the southeast direction progress by intervals of the minor third.  Northeast diagonals progress by major thirds.  All tonal sonorities are given in this representation by polygons containing adjacent pitches.  For example, major triads are given by triangles with vertex at top and minor triads are given by triangles with a vertex at the bottom, as shown above for the C major (C E G) and A minor (A C E) triads. 

Rhythmic Symmetry

Like the twelve 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 hand-clap 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.

Figure 3  Eighth-note subdivisions of rhythmic units arranged around a circle.  Top: The vertices of the inscribed polygon represent the hand-clap rhythm heard in the Elvis Presley recording of “Hound Dog”.  Center: The vertices of the inscribed polygon represent the well-known clave rhythm heard in Afro-Cuban music.  Bottom: The bossa nova cowbell rhythm heard in Quincy Jones’s “Soul Bossa Nova”.


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Demaine, E. D., Gomez-Martin, F., Meijer, H., Rappaport, D., Taslakian, P., Toussaint, G. T., Winograd, T., and Wood, D. R. “The Distance Geometry of Music,” Computational Geometry: Theory and Applications, (v. 42/5, 2009).

Hanson, Howard.  Harmonic Materials of Modern Music: Resources of the Tempered Scale.        New York:  Appleton, 1960

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Jones, Quincy. The Reel Quincy Jones, Hip-O Records, 1999

Nolan, Catherine. “Music Theory and Mathematics,” in The Cambridge History of Western Music Theory, ed. Thomas Christensen.  Cambridge:  Cambridge University Press, 2002

Presley, Elvis.  Elvis 75, RCA/Legacy, 2010

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