The Connection Between Music and Mathematics
The famous Greek philosopher and mathematician Pythagoras once said, “there is geometry in the humming of the strings, there is music in the spacing of the spheres.” Although some may interpret this statement as mere poetry, Pythagoras was actually making a direct statement on the relationship between music and mathematics. You see, music is entirely intertwined with mathematics, so much so that even a basic major chord can be described mathematically. To further highlight the connection between music and mathematics, lets examine the mathematics in common musical concepts, such as wave frequencies, scales, intervals and tones.
History of Studying Music & Mathematics
It’s common knowledge that music has long played for performance and pleasure, yet the study of music, particularly its relation to mathematics, has been going on for equally as long as music for performance. From Greeks to Egyptians to Indians to Chinese, nearly every ancient civilized culture has examined the connection between music and mathematics. Famous philosopher Plato was known to have an extreme interest in music, particularly harmonies, and helped highlight their importance within both an individual and society. Plato wasn’t the only philosopher who found the importance of studying the relationship between music and mathematics – ancient Chinese philosopher Confucius is said to have stated that within music are a number of fundamental truths.
Wave Frequencies
When we listen to music, we assume that we are hearing a song or a collection of notes, but what our brains are actually processing are sound waves. For example, when a note is played, sound waves travel from an instrument or amplifier and reverberates on our ear drums, and it’s the frequency of this sound wave that tells our brain which pitch or note is being played (e.g. the E above middle C reverberates at approximately 329.63 Hz). Understanding sound waves, particularly the difference between octave notes, requires a bit of mathematics and physics. To find the frequency of a given note, take a constant note (which traditionally is the A above middle C, which contains a frequency of 440Hz) and multiply it by the twelfth root of 2 to the power of the amount of half steps away your desired note is from middle A (if the note is below middle A, make the power a negative). If that confuses you, don’t worry! Below is an example of how to find the frequency of middle C:
- Frequency of Middle C
- = 440Hz * 2 (1/12) to the negative 9th power (middle C is 9 half steps below A)
- =440Hz * 0.59460
- = ~261.625
Intervals & Tones
If you were wondering why certain notes or intervals sound pleasing when played together, there is a mathematical explanation for this as well! As shown above, each note has a unique frequency, yet when combined, not all of these frequencies will make a beautiful harmonic chord. In fact, some note combinations can sound quite piercing and harsh. So what gives? Well, intervals that make a beautiful sounding chord tend to have sound waves that reverberate in similar patterns. Let’s look at a middle A major interval, which is A (440 Hz) and E (659.25 Hz). If examining each sound wave, with A on the bottom and E on the top, it will become clear that the frequency of E is approximately 3/2 larger than that of A, making an easy, digestible fraction. This simple mathematically relationship is largely why the two notes sound so pleasing together, whereas a more abstract fraction would result in a more dissonant, less pleasing sound.
Music & Math in the Classroom
The connections between music and math can appear vast and complex, so to help, here are a few different ways to examine the relationship between music and math in the classroom:
- Pattern activities
- Have students analyze a pattern, then communicate the pattern’s rule
- Next, have them use the rule to predict what’s next
- Add and subtract notes and rests
- Examine compositions to better understand fractions
- Analyze time signatures to understand patterns, fractions, ratios, sorting and combinations
Music teachers can help combine music and math to bring a more holistic take on music and why notes sound the way that they do. Simply teaching the connection in class may inspire a new career or adventures in other kid students
Exploring the Concepts of Music
Music, as described in the Oxford Dictionary, is considered to be sounds combined to achieve “beauty of form, harmony and expression of emotion.”
Although this might be true, the theory within music is more complex, consisting of a number of different concepts. Of these concepts, music scholars have distinguished six which are most key. Concepts included within this essential group include duration, pitch, dynamics, tone, texture and structure.
DURATION
Duration refers to how long or short a particular sound is. Duration does not necessarily refer to how long the entire piece of music is, but rather the length of the sounds being made. The combination of different durations determines the rhythm or beat of music within a piece. For example, a note stretching over two beats, followed by a one-beat note, may indicate a waltz timing, or 3/4-time signature.
Different durations of sounds can change the speed of the beat or tempo within a piece. One phrase of the piece may have many fast, short sounds, which could indicate the tempo allegro or vivace while a piece with longer, drawn out sounds may be more lento.
Students can learn duration by studying different notes and phrases while adding a physical action to each note, such as a stomp and clap. The clap can represent he note while the stomps can represent the beat. Using simple actions like this is a straight forward way to help students recognize different durations and note values.
PITCH
Pitch is crucial to developing a tune as it refers to how high or low the notes are. Music is rarely just one pitch – it would be hard on the ears to hear the same note over and over again, even if it was varying in duration and dynamics. The pitch of the note also includes their tone and the key signature. A note that is played even slightly off pitch will stick out and cause even untrained listeners to cringe!
Pitch also refers to intervals, triads, major and minor scales. Scales – both minor and major are made up of tones and semi-tones. These variances in pitch can change the mood of a piece of music. An interval may sound happy – a major scale – or sad, a minor scale. A diminished triad may leave a sense of suspense that there is more to come. Pitch can also dictate mood.
Teaching pitch is a crucial aspect of music education as this concept adds variety and complexity to a piece of music. Playing chords and encouraging students to determine whether they are major, minor, diminished or augmented will help them understand how pitch works to create atmosphere and harmony.
DYNAMICS
Dynamics is a crucial aspect of musical expression. It refers to how soft or loud a sound is – known as piano or forte in a musical score. It includes crescendo or diminuendo – the process of increasing or decreasing the loudness of the sound.
Dynamics create drama and atmosphere within a piece of music. Sounds can be long and smooth, called legato, or short and sharp, staccato. Dynamics also refer to the way a piece of music is played. For example, a string instrument may play pizzicato which is when they pluck the strings with their fingers, rather than using the bow to make the sound.
Accents are also part of the concept of dynamics, where emphasis is placed on certain notes within a phrase. Accents help create rhythm.
Dynamics are relatively straight forward to teach. A music phrase can be played loud or soft for students then to repeat. If they are able to recognise the dynamics in the piece they will be able to repeat it as instructed. If they struggle to create a loud of soft sound themselves, they may not be able to understand different dynamics and may require further instruction.
TONE
Tone can be considered as the colour of a sound, or quality. Tone depends on the instrument being played. For example a violin might be considered as having a bright, sharp tone while a tuba may be described as having a warm and rich tone.
Different tones play an important part in the dynamics of a piece of music, and can help create atmosphere or moods. A harp might be described as having a magical tone, and therefore creating a mystical atmosphere.
In vocal training, the tone of a singer’s voice can vary from person to person. Tone may change depending on a person’s breath control and from which part of their body they are using to produce the sound. A person singing through their nose or head can sound ‘tinny’ or ‘nasal’ but singing using full and deep breaths gives resonance and a rich tone. Students can test this out by trying to sing firstly with an exaggerated nasal-sounding tone by singing through their nose, followed by singing while breathing out and pushing out their stomach muscles to help them understand the difference.
TEXTURE
The texture of music is about how thick or thin the sounds are and takes into account the number of sounds there are.
There can be single notes played one at a time, or notes played together to form a chord, which produces a more complex harmony. Sounds can be played in unison or may be layered. This could be in the form of a ‘call and response’ or in a round – where the same tune is repeated by different groups, starting at different points in the phrase.
A solo is when one person plays an instrument or sings on their own, carrying the main melody. In a group setting, the lead instrument or voice may carry the main tune, but others may form background sounds and tunes to compliment the melody.
Educators may wish to play different music with different textures to students to see if they can pick out how many instruments or voices they can hear, and whether it is ‘thin’ or ‘thick’. Students need to be thinking about how different sounds can be brought together to create harmony.
STRUCTURE
Structure is how a piece of music is put together, or the order of the parts of the song. There are many different structures to music, but one of the most popular forms is a chorus/verse form, with an intro and bridge also forming the overall structure.
Binary form is another type of musical structure commonly used. Binary form is where there are two different parts to the song. The first part may be repeated before the second part of the song is played, rather than alternating like the chorus/verse structure. Ternary is a three-part music structure. There are a number of music structures – and some pieces of music have no structure whatsoever.
Students should listen to music pieces carefully and decide what kind of structure the piece has. Teachers can play certain pieces and have students say which part is the introduction, the bridge, the verse and so forth.
Learning these concepts will help music students understand their own practice and performance. Students should be encouraged to consider all these concepts with composing or performing a piece of music. A memorable piece of music will have distinct aspects from each of these concepts to create something unique and expressive.
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The Correlation Between Music and Math: A Neurobiology Perspective
Cindy Zhan
I remember the first time I heard the statement "Did you know that listening to classical music enhances your mathematical abilities?" I was both intrigued and excited, intrigued because I did not understand how music and math, two seemingly unrelated subject could possibly affect each other. I was also excited because I began to view classical music as some kind of magical potion that would transform my math skills from decent to extraordinary. When I had the opportunity to write this web paper, I immediately jumped into the topic of music and math. The questions that I wish to answer throughout this paper are; does listening to music really help you do better in math? If so, which part of the brain is controlling the correlation between math and music? In addition, how does music stimulate the brain in a way that enhances mathematical abilities?
It turns out that there is much evidence that supports the positive effects of music on one's ability to do math. Most research shows that when children are trained in music at a young age, they tend to improve in their math skills. The surprising thing in this research is not that music as a whole is enhancing math skills. It is certain aspects of music that are affecting mathematics ability in a big way. Studies done mostly in children of young age show that their academic performance increases after a certain period of music education and training. One particular study published in the journal 'Nature' showed that when groups of first graders were given music instruction that emphasized sequential skill development and musical games involving rhythmn and pitch, after six months, the students scored significantly better in math than students in groups that received traditional music instruction. (1)
The result of this study posed another important question. How does this type of music that emphasized sequential skills, rhythmn and pitch manage to improve children's ability to do math? It turned out that there are two distinguished types of reasoning, spatial temporal (ST) reasoning and Language analytical (LA) reasoning. LA reasoning would be involved in solving equations and obtaining a quantitative result. ST reasoning would be is utilized in activities like chess when one needs to think ahead several moves. The effect of music on math sometimes termed the Mozart effect. The Mozart effect gain its name after the discovery that listening to Mozart's compositions, which is very sequential, produces a short-termed enhancement of spatial-temporal reasoning. Some key reasoning features used in spatial temporal reasoning are
1. The transformation and relating of mental images in space and time
2. Symmetries of the inherent cortical firing patterns used to compare physical and mental images and
3. Natural temporal sequences of those inherent cortical patterns (3).
The same people who conducted the Mozart effect experiment also suggested that spatial-temporal reasoning is crucial in math. The areas of math that require ST reasoning are geometry and certain aspects of calculus, which require transformations of images in space and time. In higher mathematics, the ability to write mathematical proofs is also associated with ST reasoning because proof writing is a task that requires intuitive sense of natural sequences and the ability to think ahead several steps.
As to the question, what part of the brain controls the correlation between math and music, there are also many resources that provide answers. Dr. Gottfried Schlaug, found that certain regions of the brain such as the corpus callosum and the right motor cortex, were larger in musician who started their musical training before the age of 7 (2). As to what happens in that area of the brain when one listens to music, we turn to the experiment performed by Xiaodeng Leng and Gordon Shaw. Gordon and Leng developed a model of higher brain function, which is based on the trion model. The trion model is a highly structured mathematical realization of the Mountcastle organization principle, with the column as the basic neuronal network in mammalian cortex. The column comprises minicolumns called trions. One particular columnar network of trions has a large repertoire of spatial-temporal firing patterns, which can be excited and used in memory and higher brain functions (3). Shaw and Leng performed an experiment in which they mapped the trion model of firing patterns in that particular column onto various pitches and instruments producing recognizable styles of music. This mapping of the trions gaves insight to relate the neuronal processes involved in music and abstract spatial-temporal reasoning (3). It shows that the part of the cortex, which contains the repertoire of spatial-temporal firing patterns, can be excited by music and is utilized in higher brain functions such as spatial-temporal thinking in mathematics.
In conclusion, my research into math and music does seem to suggest that music enhances mathematics skills. Music targets one specific area of the brain to stimulate the use of spatial-temporal reasoning, which is useful in mathematical thinking. However, as to the question of whether or not music is the magical portion that will elevate anyone's ability to do math, the answer unfortunately . . .would be no. Just because most mathematicians are fond of music, dosen't mean that all musicians are fond of mathematics. I found a letter posted on the web written by a fourteen-year-old overachiever to a mathematics professor. The student expresses his fraustration that even though he is an excellent musician, math is one of his weakest subjects. In math, he is not making the grades that he needs to stay in a certain prestigious academic program (4).
This letter seems to suggest that listening to music, or being able to master a musical instrument does not automatically guarantee that one can perform well in math. In other words, there are many musicians who are good in music but not in math. Music is a lot more than notes conforming to mathematical patterns and formulas. Music is exhilarating because of the intricacies of the patterns that occurs. Whether or not these patterns resemble math has no relevance to many musicians. More often than not, musicians are inclined to practice music because of the wonders and awe that they feel for music even if they are not aware of the math that is in music.
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