Math, Music and Imagination

Math can be experienced as play much as music is—just what’s needed to enlarge the tribe of creative problem solvers in mathematics and other human disciplines

Math, Music and Imagination
Marcus Miller on sax. 

Like most New Yorkers, I tend to work late. My typical evening involves leaving the stage shortly after midnight and then preparing some problems in number theory or combinatorics until about 3 or 4 AM. I am a jazz musician and mathematician. My skill set allows me to interpret musical experience through the language of mathematical structure and creative problem solving.

My practice involves using ideas of mathematical transformations on melodies, rhythms, and harmony. My compositions are developed using relationships between sound frequencies. But, to me, the notion that math and music are the same thing is both terribly poetic and also too reductionist to be useful. Still I believe the two disciplines are connected by an uncanny similarity in the roles creativity and imagination play in both.

I started learning music at nine years old and worked as a musician through my teenage years, but opted to attend a university instead of a conservatory on the advice of my music mentors who encouraged me to learn more about the world. At school, I became enamored of math because of the allure of the elegant theoretical worlds mathematicians built. Upon graduation, however, I decided to spend my twenties back on stage rather than in a graduate school library. Still, I continued reading math texts, as well as tutoring high school and college students, whenever I wasn’t touring around the world.

The uses of math in music are legion. You can find math the construction of modern harmony and counterpoint, development of rhythms, and the proportioning of arrangements. What I would like to see further explored, are the commonalities between the subjective experiences of doing math and music. Although the lifestyles of mathematicians and musicians might seem worlds apart, at least for me, the “thought work” behind them are more closely related than you might think: the magic of engaging with math and music fundamentally changes the way you imagine and create.

Marcus Miller.

Much of this “thought work” can be summarized as first creating, in the senses (and perhaps on a whiteboard or an instrument), a representation of an idea, and then imagining it transformed in creative and useful ways. Developing the representation is thus a form of self discovery, while transforming it is a kind of play. Can you spot the inference to be made here? It deeply informs my life and I would love to see it become more present in our culture:

Math can be experienced as play in much the same way music is.

Let me explain. To improvise or compose one must learn the technique of the instrument and the harmonic and rhythmic language of music. All the while, the fun comes in imagining and experimenting with the technical and linguistic components one has incorporated, in order to invoke a sensation, express an emotion, or tell a story. As the mastery of both the instrument and the underlying language expand, the mind becomes more sensitive to different ideas while the body becomes more competent at putting those ideas in practice. The process thus expands naturally from creative absorption to transformation, and eventually to execution.

Math can work similarly. A student must become adept with numbers and other symbols, various rules of algebra and calculus to manipulate symbols, and several functions. This is the language of mathematics; its grammar, and its technique. Mathematical problems can be viewed as structured opportunities to play with what is already known in order to discover what is not.

Through this process, a mathematician begins to develop a sense of the nature of mathematical ideas and their logical interrelationships, thus becoming sensitive to new ideas while becoming better equipped to manipulate them internally. To think of math as just formulas memorized through rote learning and mechanical thoughtless symbol shunting tragically misses the point.

As with music, everyone incorporates the underlying language, grammar and technique in their senses differently, and thus comes to their own individual understanding that leads them to express ideas in their own unique way. Contrary to the trope of the socially dysfunctional lone genius, mathematicians collaborate for most of their work, which makes them in some sense much like musicians. Expressing our internal worlds through pictures, words, and symbols and sharing them with one another, riffing off of each other’s ideas is how much of modern mathematics is done.

What if the world understood math in this way? What if we educated with the idea of playing with numbers in order to master arithmetic the same way improvising musicians are taught to play with musical notes to learn their scales? What if we honored the unique way that people understand and taught from that space rather than by rote? What if we refined people’s logical aesthetics to the point that mathematics felt more personal, more artsy, and the profound experiences of mathematical “beauty,” “elegance” or “risk” weren’t reserved for an intellectual elite?

Math as self-discovery, math as play. These two ideas may seem foreign at first, but I am convinced this change in paradigm is exactly what’s needed in order to enlarge the tribe of creative problem solvers in mathematics and many other human disciplines—equipping them to “jam” on the world’s toughest challenges.

How Music Helps Resolve Our Deepest Inner Conflicts.

Music unifies the world into a whole. Feliciano Guimarães/Flickr, CC BY-SA
Music unifies the world into a whole. 

Billions of people enjoy music; many feel that they can’t live without it.


It’s a question that has puzzled scientists and philosophers for centuries. 2,400 years ago Aristotle wondered, “Why does music, being just sounds, remind us of the states of our soul?”

In the 19th century Darwin tried to decipher if our ability to create music evolved by natural selection. Of all human faculties, only music seemed beyond understanding; flummoxed, he came to the conclusion that “music is the greatest mystery.”

More than 200 years ago, Kant declared music useless. And near the end of the 20th century, celebrated psychologist Steven Pinker – also unable to comprehend its purpose – called music “auditory cheesecake.”

A few years ago, the respected journal Nature published a series of essays about music. Their conclusion? That it’s impossible to explain what music is and why it affects us so strongly – and that it’s not even clear if music can serve “an obvious adaptive function.”

But my recent research suggests otherwise: music is an evolutionary adaptation, one that helps us navigate a world rife with contradictions.

The crippling effect of cognitive dissonance

Music’s effect on our brains is closely related to what’s been dubbed “the greatest discovery in social psychology” of the 20th century: cognitive dissonance. Cognitive dissonance is the idea that people experience unpleasant feelings when they either possess contradictory knowledge, or are confronted with new information that opposes existing beliefs.

One way we alleviate dissonance is through suppressing or rejecting this contradictory knowledge.

Aesop’s fable “The Fox and the Grapes” illustrates this common human response. In the tale, the fox is distressed over the fact that he can’t reach a bunch of grapes. Even more unpleasant is the dissonance he experiences: the grapes are so tempting and so close – yet unattainable.

‘If I can’t have it, I don’t want it’: the fable ‘The Fox and the Grapes’ illustrates cognitive dissonance, a core human response to conflicting information. 

As a result, the fox attempts to alleviate the dissonance by rationalizing, “Oh, you aren’t even ripe yet! I don’t need any sour grapes.”

During the 20th century hundreds of experiments confirmed this common psychological response. When faced with dissonant thoughts, children, teens and adults all responded the same way: if I can’t have it, then I don’t need it.

A manifestation of cognitive dissonance is the rejection of new knowledge. Even some great scientific discoveries have had to wait decades for recognition and acceptance, because they contradicted existing beliefs that people didn’t want to surrender. For example, Einstein didn’t receive a Nobel Prize for his Theory of Relativity – now considered one of the greatest discoveries in the history of mankind – because it contradicted our core beliefs about space and time.

Music helps us grapple with dissonance

So if people are willing to deceive themselves or ignore new information, how has human culture evolved? After all, the foundation of culture is the accumulation of new knowledge – much of which contradicts existing knowledge.

Consider language: when language emerged in our species, every new word was a nugget of new information that contradicted an existing idea or belief. A powerful mechanism of the mind must have evolved to enable our ancestors to overcome these unpleasant dissonances that split their world, and allowed them to keep contradictory knowledge – to absorb new words rather than immediately discarding them.

Could it be that this ability was enabled by music? While language splits the world into detailed, distinct pieces, music unifies the world into a whole. Our psyche requires both.

Several experiments have proven music’s ability to help us overcome cognitive dissonances and retain contradictory knowledge.

For example, in one experiment, an experimenter gave a group of four-year-old boys five popular Pokemon toys. Playing with each boy individually, she had them rank, one by one, their preferences for the five toys. Then the experimenter told each subject that she needed to leave for few minutes, and asked him not to play with his second-ranked toy. When she returned, she re-initiated play and found that the formerly second-ranked toy was entirely ignored. When confronted with conflicting information (“I like this toy, but I shouldn’t play with it”), each boy apparently rejected his initial preference for it.

But when the experimenter turned on music when leaving, the toy retained its original value. The contradictory knowledge didn’t lead the boys to simply discard the toy.

In another experiment, we gave a group of fifteen-year-old students a typical multiple choice exam, and asked them to record the difficulty of each question, along with how much time it took them to answer each one.

It turned out that more difficult questions were answered faster (and grades suffered), because students didn’t want to prolong unpleasant dissonance of choosing between difficult options. However when Mozart’s music played in the background, they spent more time on the difficult questions. Their scores improved.

Life’s big choices become more informed

Beyond multiple choice tests, we’re constantly confronted with choices in our day-to-day lives – from the mundane (what to buy for lunch), to the major (whether or not to accept a job offer). We often use both intuition and pragmatism when evaluating complex situations, but we also incorporate emotion.

And then there are choices related to two universal themes of our existence – love and death – which are inherently steeped in contradictions.

With love, we’d like to fully trust it. But we know that to fully trust is dangerous – that we can be betrayed and disappointed. With death, one of the most difficult contradictions of all is our longing to believe in spiritual eternity and our knowledge that our time on Earth is finite.

Is it any coincidence, then, that there are so many songs about love and betrayal? Or that we are drawn to sorrowful songs in times of mourning?

The idea is that music – which can convey an array of nuanced emotions – helps us reconcile our own conflicted emotions when making choices. And the more diverse, differentiated emotions we possess, the more well-founded our decisions become. Whether it’s choosing to play with a toy or deciding to propose to a boyfriend or girlfriend, our research shows that music can enhance our cognitive abilities.

Thus, because we constantly grapple with cognitive dissonances, we created music, in part, to help us tolerate – and overcome – them.

This is the universal purpose of music.


The Link Between LSD and Music

LSD and music are always linked, and it is now known that music alters the experience of taking the drug. Imperial College neuroscience student Mendel Kaelen has been studying music and the drug, and a big challenge is the type of music.

It has been Mendel Kaelens job to set the ideal playlist for a trip, and as it must fulfill research regulations, it has not been easy. The intent is to find a therapeutic use for the drug, and the initial attempt is to find a way to treat depression. As all therapy trials use music now it is vital the right tunes are used.

The Link Between LSD and Music
The areas that contributed to vision were more active under LSD, which was linked to hallucinations. 

Before testing on people, the team tested the theory that drugs such as LSD improve the response to music. Volunteers listened to a few musical tracks, once after taking LSD once after taking a placebo. The tracks were largely unknown so as there would not be an immediate attachment and Brian McBride’s and Greg Haines music was used – along with others. Emotional responses were given to the tracks – were they peaceful, was their tension – and emotions were rated higher after taking LSD not the placebo. Power and tenderness were emotions that scored highly.

Kaelen has also researched using a substance found in magic mushrooms to patients who were resistant to treatment for depression, but the results are not yet available. Here the music ad to last for 6 hours and some tracks were taken from the list of Helen Bonney, a music therapist in the 1960’s.

If You Want To Accelerate Brain Development In Children, Teach Them Music.

Alternative and complementary treatments such as creative art, meditation, and yoga have been proposed to bridge many gaps that conventional medicine cannot. But music, because of its ubiquity in our society as well as its ease of transmission, has perhaps the greatest potential among alternative therapies to reach people in deep and profound ways. Music matters and it heals.

Music instruction appears to accelerate brain development in young children, particularly in the areas of the brain responsible for processing sound, language development, speech perception and reading skills, according to initial results of a five-year study by USC neuroscientists.

We now know through controlled treatment outcome studies that listening to and playing music is a potent treatment for mental health issues. 400 published scientific papers have proven the old adage thatmusic is medicine.”

Research demonstrates that adding music therapy to treatment improves symptoms and social functioning among schizophrenics. Further, music therapy has demonstrated efficacy as an independent treatment for reducing depression, anxiety and chronic pain.

The Brain and Creativity Institute (BCI) at USC began the five-year study in 2012 in partnership with the Los Angeles Philharmonic Association and the Heart of Los Angeles (HOLA) to examine the impact of music instruction on children’s social, emotional and cognitive development.

These initial study results, published recently in the journal Developmental Cognitive Neuroscience, provide evidence of the benefits of music education at a time when many schools around the nation have either eliminated or reduced music and arts programs. The study shows music instruction speeds up the maturation of the auditory pathway in the brain and increases its efficiency.

“We are broadly interested in the impact of music training on cognitive, socio-emotional and brain development of children,” said Assal Habibi, the study’s lead author and a senior research associate at the BCI in the USC Dornsife College of Letters, Arts and Sciences. “These results reflect that children with music training, compared with the two other comparison groups, were more accurate in processing sound.”

For this longitudinal study, the neuroscientists are monitoring brain development and behavior in a group of 37 children from underprivileged neighborhoods of Los Angeles.

Thirteen of the children, at 6 or 7 years old, began to receive music instruction through the Youth Orchestra Los Angeles program at HOLA. The community music training program was inspired by the El Sistema method, one that LA Philharmonic conductor Gustavo Dudamel had been in when he was growing up in Venezuela.

Learning the Violin

The children earn to play instruments, such as the violin, in ensembles and groups, and they practice up to seven hours a week.

The scientists are comparing the budding musicians with peers in two other groups: 11 children in a community soccer program, and 13 children who are not involved in any specific after-school programs.

The neuroscientists are using several tools to monitor changes in them as they grow: MRI to monitor changes through brain scans, EEG to track electrical activity in the brains, behavioral testing and other such techniques.

Within two years of the study, the neuroscientists found the auditory systems of children in the music program were maturing faster in them than in the other children. The fine-tuning of their auditory pathway could accelerate their development of language and reading, as well as other abilities – a potential effect which the scientists are continuing to study.

The enhanced maturity reflects an increase in neuroplasticity – a physiological change in the brain in response to its environment – in this case, exposure to music and music instruction.

“The auditory system is stimulated by music,” Habibi said. “This system is also engaged in general sound processing that is fundamental to language development, reading skills and successful communication.”

Ear to Brain

The auditory system connects our ear to our brain to process sound. When we hear something, our ears receive it in the form of vibrations that it converts into a neural signal. That signal is then sent to the brainstem, up to the thalamus at the center of the brain, and outward to its final destination, the primary auditory cortex, located near the sides of the brain.

The progress of a child’s developing auditory pathway can be measured by EEG, which tracks electrical signals, specifically those referred to as “auditory evoked potentials.”

In this study, the scientists focused on an evoked potential called P1. They tracked amplitude – the number of neurons firing – as well as latency – the speed that the signal is transmitted. Both measures infer the maturity of the brain’s auditory pathways.

As children develop, both amplitude and the latency of P1 tend to decrease. This means that that they are becoming more efficient at processing sound.

At the beginning of the study and again two years later, the children completed a task measuring their abilities to distinguish tone. As the EEG was recording their electrical signals, they listened to violin tones, piano tones and single-frequency (pure) tones played.

The children also competed a tonal and rhythm discrimination task in which they were asked to identify similar and different melodies. Twice, they heard 24 melodies in randomized order and were asked to identify which ones differed in tone and rhythm, and which were the same in tone and rhythm.

Children who were in the youth orchestra program were more accurate at detecting pitch changes in the melodies than the other two groups. All three groups were able to identify easily when the melodies were the same. However, children with music training had smaller P1 potential amplitude compared to the other children, indicating a faster rate of maturation.

“We observed a decrease in P1 amplitude and latency that was the largest in the music group compared to age-matched control groups after two years of training,” the scientists wrote. “In addition, focusing just on the (second) year data, the music group showed the smallest amplitude of P1 compared to both the control and sports group, in combination with the accelerated development of the N1 component.”

Are You Deficient In Vitamin M?

Are You Deficient In Vitamin M?

Are you suffering from low Vitamin M? Here are some of the symptoms:

  • depression
  • lethargy
  • high stress
  • poor circulation
  • weak social or community health
  • isolation
  • poor mobility

Are you suffering from a deficiency of Music?

Music has been demonstrated to have benefits as treatment for illness. Sometimes it is dismissed as if it were ‘anecdotal evidence’. In truth, music is a healthicine. Anecdotal evidence is one of the most uninformed and mis-used phrases by the commercial medical interests – but that should be the subject of another blog post.

A GreenMedInfo search for medical benefits of music therapy shows a wide variety of situations where music is beneficial. I believe it is beneficial because it increases healthiness – as opposed to ‘treating disease’. A subtle but important distinction not typically studied by medical researchers.

Music activates healthy minds, lifts healthy spirits, and builds healthy communities.

Music on the SidewalkBut be careful of synthetic alternatives. Synthetic music – the worst is elevator music, can actually make you feel bad, and in extended periods can feel like become torture.

The most organic music, is music you create. But, the healthiest music is not music you create yourself, it’s music you create in a community. Even if you are only helping put out chairs in the community hall, and tapping your toe to the ‘real musicians’, your participation is enhancing your health, and the health of your communities.

Listening to music on your iPod, or a CD, can be healthy, healthier if it activates you to movement. It might lift your spirits, but it does little for your community. Music in isolation is limited in health benefits and can lead to more isolation from your community, resulting in poor community and spiritual health.

Listening to the radio can be pure commercial poison – or, if you have a good community radio station that is more interested in music than in ‘sales’ – it can be a health benefit. My favorite radio station, the best radio station in the known and unknown universes – is CKUA.

CKUA is all about the music, which ranges – sometimes from moment to moment – from the musician who lives down my street, to blues from the south, jazz from northern Europe, drums from Africa or Japan. There is no world music – they say, just music. The announcers at my radio station choose what they want to play – it’s not commercial music.

Kill your television. Or turn off the sound. Vitamin M on your television can be toxic. Pretending to be real music, but not involving you. Television plays on your emotions, not on your musicality. Television is designed for passive watchers – not active participants. Not as bad if it’s seen but not heard.

Church music is involving. I think it’s one of the greatest strengths of religion. In the forest, even the ugliest bird can sing. And even the poorest song is beautiful. And so it is in church. Everyone sings. It’s about creating music together – not about competing for the next idol. The churches don’t need another idol. I like churches where the people get up and move to the music. Music has power, let it loose. Poetry is music. Prayer is music. Everyone sings. Everyone prays.

What about ‘best music’? The Grammy’s? Billboard music awards? The latest craze? Music from the 50s, 60s, 90s? Rock music? Blues music? Jazz? Concert music? The best music for you is the music that involves you – the music that you can participate in. Awards are important, they encourage us to reach new heights, creation of new music is healthy. But creation of your own music is important too – even for those of us who might never win an award.

Several months ago, I wrote a post titled “Is listening to the Blues Healthy“, and concluded – yes it is. But it’s not just blues – it’s all Music. Music is vital. Vitamin M.

I love, and actively encourage live music. Support your local musicians. Help music grow and flower in your lives, your communities, your children. Music truly is an essential element of healthiness.

Some people have claimed that Vitamin M is ‘movement’. I beg to differ. Movement comes from the inside, not from the outside. Music inspires movement – let the music move you! Note: In actual fact, Vitamin M is assigned to folic acid or folate, presumably because Vitamin F had already been assigned to fatty acids. I prefer think of Vitamin M deficiency as a deficiency of frolic.

Can we dance without music? Not me… Even if a dance is silent, I hear the footsteps. Music in my head. There is, for me, always music in my head. I don’t get ‘earworms’, I enjoy earworms all day long.

I’m off now, to listen to some music, and of course to dance a bit (in my quiet way), to sing along (under my breath), to tap my toes, and of course to take some pictures.

Everyone has a right to life, liberty, and the pursuit of healthiness. And some music too.

Neuroscientists Still Don’t Know Why Music Sounds Good

Music Note

How Music Stimulates the Unconscious Mind

Music plays with the brain in interesting ways. For instance, past studies have shown listening to a familiar, favorite song causes our brain to release dopamine — a chemical associated with pleasure and reward. However, some researchers believe music could be utilized to boost cognition in unconscious minds.


Alexandra Ossola from Braindecoder writes about the curious case of seven-year-old Charlotte Neve. In 2012, she had a had a brain hemorrhage while she was sleeping. Surgeons were able to stop the bleeding, but she had several seizures after and slipped into a coma. Ossola writes about her astounding recovery:

“Charlotte’s mother, Leila, was at her bedside listening to the radio when Adele’s hit ‘Rolling In The Deep’ started playing. Leila and Charlotte had sung the song together many times and, as Leila sang along to her unconscious daughter, she saw Charlotte smile. The doctors were stunned. Over the next two days, Charlotte recovered more of her faculties — she could talk, focus on colors, and get out of bed.”

It’s uncertain if this recovery was caused by the music or if the entire thing was just a coincidence. However, it has become the basis of a recent study where researchers played music to 13 patients — all in comas for different reasons. The researchers split the patients into two groups; in one, the researchers played some of the patient’s preferred music and in the other, researchers played a continuous sound to act as the control. Then, the researchers measured the patients’ brains with an electroencephalograph (EEG) while they called the patient’s name.

The researchers wrote in their paper:

“The cerebral response to the patient’s first name was more often observed in the music condition, than in the control condition.”

These results have led researchers to demonstrate “that music has a beneficial effect on cognitive processes of patients with disorders of consciousness. The autobiographical characteristics of music, that is, its emotional and personal relevance, probably increase arousal and/or awareness.”

It’s possible that this kind of familiar stimuli could help victims with brain trauma repair certain neural pathways. Past instances have also shown calling a patient’s name shows increased brain activity. But it’s uncertain. The unconscious mind is such a mystery — hopefully, one researchers will be able to figure out how to repair in the future.

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