Music & Mathematics: Their Intimacy

Gottfried Wilhelm Leibniz, a renowned scientist, mathematician, and philosopher, once said “Music is a hidden exercise in arithmetic, of a mind unconscious of dealing with numbers,". Many eminent mathematicians, physicists or other scientists that research the fields related to mathematics tend to pursue their interest in music too. For instance, Einstein, a Nobel Prize winner for his discovery of the law of the photoelectric effect, was a passionate violinist. Mozart, one of the most accomplished composers of the time, was also a genius mathematician; his music often contained very mathematical patterns.

Music seems to have more connection to mathematics compared to other fields within arts, such as paintings or literature. Why would this potentially be true? To elaborate on this inquiry, music has more similarities than other forms of art. Indeed, music and mathematics are both absolutely precise fields. The paintings would not be essentially altered when one extra color is added, and the line of the song might be improved if it is changed to a word that matches more. However, replacing one note from A flat to A sharp, for example, would lead to a disastrous result, just like changing the plus and minus signs in a mathematical equation. In this sense, music does resemble the underlying principles of mathematics.

The next question that might emerge would be which elements of music make it so similar to mathematics. Looking back, the intimacy between mathematics and music began with the discovery of Pythagoras, an ancient Greek mathematician, and philosopher who is often known for his theorem regarding the right triangles. In addition to the Pythagorean theorem, one of the less-known discoveries he made was closely related to music and sound. In the 6th century B.C., Pythagoras experimented with strings of different lengths and discovered that when the string is divided into two parts so that it produces two distinct sounds but in a stable chord composed of consonants, the length of the divided parts can be expressed as the ratio composed of 1, 2, 3, or 4. During the time of Pythagoras, the consonants were defined as c, f, g, and octave higher c, and he thus explained the combination of these notes with mathematical ratios. Furthermore, by inserting notes in between those consonants and expressing them as ratios, c,d, e, f, g, a, b, and c can be written as 1, 8/9, 64/81, 3/4, 2/3, 16/27, 128/24, respectively. As Pythagoras began his discoveries, the intimacy of mathematics and music continued to intrigue mathematicians and musicians.

Adding on to the contribution of Pythagoras, other technical parts of music make it similar to mathematics. The most notable few are sound waves and patterns. The sound waves the music produces can be analyzed with mathematical equations, especially sinusoidal equations. The patterns within music often resemble mathematics too. Each piece of music has complex patterns that have underlying mathematical principles in it. Mozart is known to be one of the most mathematical composers, and his music contains complicated tone patterns.

Such affinity between music and mathematics can be found in multiple theories and figures. Music and mathematics may seem like two very different areas of subject matter, but they are one of the subjects that has deep connections with each other.