Music. Words cannot describe how music touches our core. It is the universal language that can cross borders and give meaning to the mere words we use in our day-to-day lives. Uncovering music, we can see that it is made with some sound, which combines vocals and instruments in a way that takes us to a whole different world. Music is an art that involves the emotional manipulation of the feelings and emotions of the listener and helps artists express their emotions, ideas, and feelings in a more impactful manner.
Well, whatever you do, mathematics never leaves you alone. Although music and mathematics can appear worlds apart, they are quite related to each other. If one peeks into the world of music, one can find plenty of mathematical principles at work. Mathematics serves as the foundation of music, from chords to harmony to sound frequencies, and much more. If you are a musical artist, a mathematics enthusiast, or a geek who wants to learn more about how music and mathematics are related, stick around! Because this blog is going to do just that for you!
What is harmony? The simple lo-fi music you hear on your way to work, while studying, or while reading this blog is nothing but harmony. Harmony is the pleasing arrangement of musical tones. Some may think harmony is merely a simple combination by some experienced musical artists; however, it is a mathematical principle. Chords are the building blocks of harmony, formed by the mathematically correct combination of specific notes from a scale. For example, a major chord is made up of the tonic, major third, and perfect fifth notes of a scale. The frequencies of those chords are mismatched in such a way that they sound harmonically attractive to the ear and provide a form of completion to the musical idea.
Theoretically based harmonies can teach artists how to analyze, write, and interpret complicated historical and contemporary music systems. The arithmetic language of music has endless potential because understanding how to identify patterns and correlations between sounds allows us to unlock the mysteries of harmonic sound combinations. We explore the miracles of harmony in music through the lens of mathematics, revealing the core symmetries and proportions that support the beauty of musical compositions.
Moving on to frequencies, what are frequencies? Frequencies are the vibrations, or cycles per second, that determine the pitch of a sound. In the realm of music, understanding frequencies is really important for bringing out the differences in pitch and creating harmonious combinations of tones. The relationship between the frequencies, however, follows mathematical principles, resulting in the well-known musical scale. When we hear different notes, what is happening is that we hear the sound waves vibrating at distinct frequencies. For example, note A above middle C vibrates at about 440 Hz, whereas note C vibrates at about 261.63 Hz.
In case you’re wondering what Hz is, Hz is the abbreviation for hertz. According to the International System of Units (SI), the standard for measuring frequency is the hertz. Frequency is also measured in hertz. Within the musical scale, certain intervals are created by specific ratios of the frequencies. These ratios are based on mathematical principles that define the various intervals, such as the perfect fifth or the major third. A perfect fifth, for example, is formed by a 3:2 frequency ratio, whereas a major third is formed by a 5:4 ratio. Let’s not go too much into what a perfect fifth is and what a major third even means.
Now that we have talked about something about symmetries, what is symmetry in music? It is the repetition of a specific aspect or component. Why are symmetries important? Well, mathematical symmetries in music are an essential factor that determines rhythm and harmony. Music theory uses the circle of fifths, a 3:2 ratio of 12 chromatic scale notes, to simplify the interpretation and understand key relationships. This symmetry pairs well with melodic and mathematically rich patterns in music.
The golden ratio may be the most well-known example of math in music. The golden ratio is a value of around 1.618 that is linked in numerous ways to the mathematical foundations of music. Many composers used the golden ratio to organize phrases and climax lengths, as well as the overall musical form. Furthermore, intervals with values closest to this ratio sound more harmonious and appealing, while those farther away sound more discordant. While the golden ratio's direct applicability is debatable, it demonstrates the strong connections between the mathematical and aesthetic domains in music.
The best example of how to understand and listen to how music sounds when golden ratio and mathematics are integrated with music is Sumeru’s Battle Music from the game Genshin Impact. According to the composer Yu-Peng Chen,
‘In one of the pieces, I experimented with the Fibonacci sequence to create rich and varied rhythmic changes, which makes it sound very modern.’
Gives the best example of an Asian stereotype: doing mathematics even when making music. Just kidding, but I’d recommend every song in Genshin, and I’m sure you’ll love them too!
To conclude, I’d like to say that mathematics and music come together to weave into a bewildering creation of human creativity. Finally, whether we're mathematicians, musicians, or just someone who enjoys vibing along with their favourite music, we're all part of the same symphony, with each note echoing the universe's eternal pulse and allowing us to express and feel ourselves at the same time.
Authors:
Apoorv Jadhav
Shambhavi Bhalse
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