Pythagoras might have been speaking for numerous others when he said that he found music in the spacings between the planets and geometry in the sounds of strings. Plato wrote of harmonies in mathematics and how they parallel harmony in a just society. Confucius also found numerous eternal truths in the unfolding of pieces of music.
These ancient philosophers grasped truths about the interconnectedness of music and mathematics that have become even more clear over the centuries.
Here are only a few insights, based on the experiences of musicians and mathematicians, about this close relationship:
1. Activation of analogous skills
Music students, when tested, tend to show more skill in mathematics than their non-musical peers. High levels of cognitive processing ability and executive function—which involves self-regulation and self-management in order to achieve a goal—are essential for success in both fields.
Research also supports the notion that executive function, even more so than overall intelligence, has been shown to influence academic achievement. Learning math ties into the development of executive function by calling on a child to analyze, identify key concepts, and proceed through a series of logical steps. Likewise, learning to play a musical instrument enhances this capacity by, among other factors, drawing on the ability to calibrate motor movements in response to changes of time signature and key.
2. A beautiful symmetry
Some mathematicians explain their field by focusing on how they work to extract the essential elements of any given thing and study the characteristics and interactions of those elements on an abstract plane. This type of learning can help students to understand music and can lead to a deeper engagement with the essential elements of a musical composition.
Music can inspire students to learn more about mathematics through studying, for example, the properties and manifestations of sound. Innovative mathematics teachers have even brought opera singers into their classrooms to show students how the patterns of mathematics are part of the essence of music.
3. Simplicity within complexity
Every note a composer writes or a musician plays is involved in an intricate web of harmony, rhythm, and mathematical patterns.
These patterns tend to be built around elements of symmetry. For example, just as the shapes of regular geometric figures remain the same when rotated, a musical tune can be transposed to another key in a composition such as a fugue.
In a Mandelbrot set, a famous fractal, a smaller replica of the entire patterned set can always be found hidden at the core of any other image in the set. So, we might also say that a musical fractal occurs when one theme harmonizes with a slower version of itself. Johann Sebastian Bach, for example, showcased a talent for repeating his themes numerous times throughout a variety of permutations.
4. A composition made possible by math
In fact, thanks to an extraordinary mathematical insight, Bach had the tools he needed to compose The Well-Tempered Clavier in 1722. The piece consists of a set of masterful preludes and fugues, one in each of the major and minor keys.
But Bach could not have created this much-loved work without mathematics. In 1636, the French monk and mathematician Marin Mersenne successfully solved a difficult problem by deriving the twelfth root of the number 2, thus paving the way for the division of the octave into 12 equal semitones.
Before this division and the associated method of equal temperament of musical instruments, pieces transposed into new keys often sounded uneven and unpleasing. But after Mersenne’s achievement, musicians were able to work with a 12-part octave, evenly spaced and divided into ratios. They could then write music in every key and transpose easily from one key to another. Bach’s The Well-Tempered Clavier was the first noteworthy example of this musical revolution.
5. How math determines pitch
A discussion of pitch is only one way to demonstrate how math undergirds sound.
Pitch is based on wave frequencies. All audible sounds are produced by changes in the air pressure of the pockets surrounding a sound wave. The frequency that hits the human ear translates into the perceived pitch. Each note possesses its own individual frequency.
For an example of sound waves in action, think of a train whistle. Notice that the sound seems higher-pitched as the train approaches. But after the train goes by, the sound seems lower. As the train speeds toward the listener, the forward movement compresses the arriving air pockets against each other, thus pushing them forward more frequently. As a result, the sound seems higher-pitched. Then as the train recedes into the distance, the air pockets slow in their arrival to the ear, giving a lower pitch.
We perceive the most pleasant-sounding chords when we combine notes with sound waves that reverberate in analogous patterns. The mathematical ratios of the intervals between notes give the means of calculating which note combinations produce harmony and which create discord.
Frequency is measured in terms of hertz, and notes with higher pitch have a higher frequency. Middle C has a frequency of approximately 262 hertz. This means that, when middle C sounds on a piano, the sound waves that reach a listener’s ear consist of 262 pockets of higher air pressure striking against the ear every second. As a comparison, the E just above middle C sounds at approximately 329.63 hertz.
Building an understanding of the physics and mathematics behind pitch also leads students to a fuller understanding of octaves, chords, and other musical elements.
6. Pairing music and math in the classroom
When teaching music in the classroom, teachers can incorporate math in a multitude of ways. One is to ask older children to identify the parts of a musical pattern, then to restate the rule governing that pattern. They can go on to use their analysis of patterns to make predictions about the future direction of a composition. An exploration of time signatures and chords can also be the basis for lessons in how math and music work together.
Before younger children even learn the formal concepts of mathematics, they learn through experience about rhythm, repetition, and proportional relationships among musical concepts. They can clap out the syllables of their names, and then see if they can match the number of syllables in their own names to those in other students’ names. They can also echo their teacher, with voice or movement, as he or she calls out and varies notes, beats, and tempos.