|Math 5 Project
June 1, 2007
Modern Mathematical Composition and Analysis
For my project, I researched different mathematical methods of composition that have been employed in the past 100 years. Many composers including Anton Webern, Bela Bartok, and Elliott Carter have used math to compose music. Mathematical analysis of sounds, such as Fourier series and spectrograms, paved the way for mathematical composition by supporting the idea that because music can be represented mathematically, it can also be constructed mathematically. As part of my project a composed and constructed a piece of music using several of the methods which I researched. In addition to analyzing these methods of composition, I will show how these methods both relate to and at times contrast to the ideas about sound and music we have discussed in class.
Although some connections between composition and music are more interesting for mathematicians than musicians, the use of fractals has an important aesthetic impact that has been shown to affect people’s perception of music. For the purposes of musical analysis, self-similar fractals are the most interesting due to their relation to overall musical structure and individual sounds. Fractals are not only found in nature, but also in music in the relation of amplitude in harmonic partials. The relative strengths of musical partials generally approximate inversely proportional relationships with frequency, giving sounds a musical quality. Pink noise, a kind of static similar to white noise but decreasing 3 dB per octave, is an example of fractal noise. In fact, studies have shown that pink noise sounds more like music to the human ear than white noise or brown noise, another kind of static sound. Furthermore, fractal relationships in intensity are also manifested in rhythmic structures; the strongest 1/f relationships that have been found in music are found in frequencies of less than 1 Hz, that is, rhythmic frequencies (Loy). This suggests that musical downbeats have a mathematical structure that reflects the structure of individual sounds.
With the increase of mathematical analysis of noise in the past 100 years, the definition of music has been challenged. In classical music, there has been much departure from traditional harmony based on the harmonic series and low integer ratios. However, these new relationships are not necessarily random or foreign; they are frequently based in natural phenomena, and even in the traditional ideas about sound that have pervaded Western music for hundreds of year. The Hungarian composer Bela Bartok said, “Every art has the right to strike its roots in the art of a previous age; it not only has the right to but it must stem from it” (Lendvai). Bartok used fractal structures based on Fibonacci sequences to organize pieces of music, but also to determine harmony. His harmony is based on opposition and balance, just like traditional Western harmony. However, instead of scales and chords based in the harmonic series, Bartok preferred to use intervals of 2, 3, 5, 8, and 13 semitones, giving him an utterly unique sound (Lendvai). This sharply contrasts the basis of traditional Western harmony on intervals approaching low integer ratios. In terms of structure, Bartok consciously used the golden mean and Fibonacci sequences to anchor the climax of pieces. The golden mean is a ratio found in nature in spiral shells and even human bodies; in a segment of length 1, the golden mean is the point on the segment where (1-x)/x=x/1. The relationship between each Fibonacci number and the preceding number in the series approaches the golden mean the higher the numbers get. Bartok’s “Music for Strings, Percussion and Celeste” frequently uses the golden mean for significant changes in the music. In the same manner, music as early as Mozart and Hadyn can be seen to use golden mean proportions, but the 20th century marked the first clearly conscious use of Fibonacci sequences in music.
Although fractal ideas can easily combine with traditional ideas about the harmonic series and just intervals, more radical music challenges the idea of music through its use of mathematical ideas. The ideas of John Cage and Pierre Boulez, for instance, stem from the fact that sounds can be represented as sums of mathematical functions: they viewed sound as the combination frequency, amplitude, timbre. Cage wrote music entirely dictated by chance operations, eliminating the human element of composition. In some ways, experimental and atonal music allowed composers to explore greater realms of accuracy in terms of representing nature through music. Boulez changed the timbre of instruments in order to represent atonal sounds in the partials themselves; this mirrors the idea of stretched partials, or the idea that our ears can perceive sounds as musical without their signals being exactly periodic. The French composer Oliver Messiaen used partials in a way almost similar to Tuvan throat singers, but more subtle. In his Quartet for the End of Time, the solo clarinet movement “Abyss of the Birds” defies traditional harmonic analysis but ultimately works because the higher harmonics of the clarinet shape a perfect arc over the course of the piece (Hirsch). In addition, Messiaen used his considerable pitch recognition skills to write accurate bird calls into his music.
Like Messiaen’s music, music free from traditional ideas of harmony can more accurately represent natural sounds. Bartok, for example had excellent pitch, and recorded folk music in order to mimic the sounds of the instruments in his music. In his piano sonata, low chords approximate the sounds of drums, which have non-harmonic signals. His use of minor/major chords (chords containing both a minor and major third scale degree) can cause the ear to hear beats or combination tones that sound like pitches in between the two thirds. This could either be seen as an approximation of a just major or minor third, which are in between equal-tempered intervals, or as a more accurate representation of actual folk instruments that do not always play perfectly on pitch.
The idea of atonal music seems to directly contrast the Helmholtz theory of dissonance, which states that intervals are pleasing to the ear if the higher partials do not create beat frequencies. Much of atonal music is based on the exact opposite idea, that is the idea that music is created based on opposition. Set theory, a musical concept based on mathematical set theory, uses groups of notes and intervals placed in opposition to one another; rather than traditional ideas of keys, composers use these sets to compose music. For instance, if one passage of music focused on 6 particular scale degrees, the resolution and balancing conclusion to this passage might focus on the 6 scale degrees not used in the beginning of the passage. Bartok’s axis theory, although more anchored in his unique harmonic language, similarly placed different keys in contrast to one another to create musical interest. Spel Against Demons, the piece I composed/constructed based on a poem by Gary Snyder, uses set theory as well as Fibonacci numbers; after the climax, which occurs at the golden mean, the harmonic structure shifts to the opposite chromatic tones that were initially used.
The more music is used to represent bird calls, echoes, train whistles, and similar phenomena, the more the line is blurred between music and sound. John Cage, in particular, was important for his philosophical explorations into the nature of music: his piece 4’33’’ consisted of a musician sitting at a piano for four minutes and 33 seconds without playing a single note. Instead, the ambient noise and resonance of the hall became the music. Although we analyzed the resonance of rooms in terms of closed-closed pipes, it can be difficult to think of a room as a musical instrument. Similarly, much of our analysis in the class has supported the idea that many sounds have inherently musical qualities. Vocal formants even in speech behave like musical instruments; speech analysis can in fact be used to compose music (Eldenius). In Spel Against Demons, I used sounds clips of myself reciting Gary Snyder’s poem as background rhythmic noise.
After analyzing modern compositional methods, I made sure to connect my composition reflect the overall struggle in music between music based on the harmonic series and just intervals, and music based on other mathematical ideas. For instance, the piece opens with pink noise shifted in intensity in order for it to sound vaguely rhythmic. In the 13 seconds of pink noise, the first second is 13 times the intensity (11.14 dB greater) than the rest of the noise, while several syncopated beats are 3 or 5 times the intensity of the rest of the beats. The overall structure of the piece uses golden ratios and Fibonacci ratios; the piece is 377 seconds long, with the climax coming 233 seconds into the piece. Similarly, the rhythmic sections interspersed throughout the piece use rhythms of 3, 5, or 13. Another technique I used was the convolution of signals through Praat, which takes the integral of two signals after reversing and shifting one of them. Through this technique, I was able to further blur the line between music and speech, as well as being able to create echo sounds and sounds that do not sound pianistic, even though the only instrument used in the piece is the piano.
As the piece progresses, it begins to rely more heavily on harmonic structures, but also stretches the capabilities of the piano strings. As discussed in class, the vibration of piano strings, because they are hit with hammers, is imperfect vibration that activates more modes than just where it is hit. By plucking the strings directly, close to their center, I tried to achieve an almost harp-like sound from the piano. In addition, in the section leading up to the climax of the piece, I drummed with my hands directly on the lower strings of the piano in order to get a rhythmic, somewhat chaotic sound that seems more like an airplane flying overhead than a musical sound. The final section of the piece uses ideas based on combination tones and the harmonic series. When Barton Workshop visited the class, they discussed how even two instruments playing together can cause the ear to hear faint lower notes based on the harmonic series. By essentially approximating the ‘missing fundamental’ for intervals played in the right hand, I determined the chords that would be played in the bass. The last chord, a G-sharp minor chord alternating with an E-major chord, stems from the fact that G sharp and B, when played loudly, cause the ear to perceive a low E, while G sharp and D sharp cause the ear to perceive a G sharp an octave down and B and D sharp create the illusion of a lower B. Similarly, the section immediately following the golden mean of the piece alternates between E minor and C major.
Throughout the term, we have analyzed both musical and non-musical sounds, discovering that many non-musical sounds possess musical properties. In the past century, several composers have sought to blur the line between music and noise, representing natural phenomena through mathematical relationships in tone and rhythm. Bartok used fractals to determine structure and harmony, while Cage took structures such as star distribution and mapped them onto his violin etudes. Even though avant-garde and atonal music seems to go against the use of the harmonic series and just intervals, it often relates directly to traditional music. For instance, Bartok’s use of minor/major chords to approximate real-life pitches as well as several mathematicians’ mapping of vocal patterns onto pitch exemplify the musical possibilities of sound analysis (Eldenius). Ultimately, the analysis of sound using mathematical language reveals the possibility of composition through mathematical means; these methods often have a fundamental basis in natural and musical phenomena that can hopefully expand our definitions and perceptions of music.
Benson, David J. Music : a mathematical offering New York : Cambridge University Press, 2007.
Eldénius, Magnus. Formalised composition on the spectral and fractal trails Göteborg University, School of Music and Musicology, c1998.
Hirsch, Evan. Professor of Music, Brandeis University, Individual Instruction Program at Dartmouth College. Personal interview. May 30, 2007.
Lendvai, Erno. Béla Bartók: an analysis of his music; with an introd. by Alan Bush London, Kahn & Averill, 1971
Loy, D. Gareth. Musimathics : the mathematical foundations of music Cambridge, Mass: MIT Press, c2006-
Temperley, David. Music and probability .Cambridge, Mass: MIT Press, c2007.