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The Mathematics of Healing: Sound, Frequency, and the Actuarial Case for Music in Medicine

  • Feb 24
  • 12 min read

Updated: Feb 25

A synthesis of ancient mathematics, modern neuroscience, harmonic theory, and healthcare economics

 

  • The body mathematically computes music, it actually consumes music 


  • Sound measurably reduces medical costs 


  • Music improves brains, hearts, medical outcomes 


  • Actuaries can lead the charge to communicate the economic value of music as medicine






Executive Summary

 

What if music weren’t just art or entertainment, but one of the most mathematically precise, neurologically efficient, and economically rational health care interventions? Ancient harmonic ratios, analyzed with the same mathematics that govern brain waves, heart rhythms, and neural prediction, can be a tool towards better health care. The human body is not merely moved by music, it computes and consumes it to modulate stress and balance emotions. Clinical research into the impact of music demonstrates measurable improvements in anxiety, cognition, heart rate variability, and neurological function across populations ranging from ICU patients to individuals with Alzheimer’s disease. Crucially, this physiological impact translates into dollars through reduced sedation and medication use, fewer readmissions, improved cardiac biomarkers, and lower total cost of care. From an actuarial perspective, music therapy emerges not as “wellness culture,” but as a scalable, low‑cost, data‑credible intervention capable of shifting utilization, mitigating chronic disease risk, and improving population health outcomes over time. The conclusion is both ancient and modern. Sound is medicine, and health systems that ignore it are leaving measurable clinical and economic benefit on the table.


 

The History and Science Behind Music as Medicine

 

Several years ago, I was introduced to the Dallas Street Choir. Its members were homeless but their music was extraordinary. I wrote about it, conjecturing about the positive impact the music and choir fellowship had on their self-esteem and ultimately their health. The question is a broader one now, where clinical research demonstrates measurable positive population-level health and well-being impacts from treatments using harmonic frequencies.


The evidence spans twenty-six centuries, from ancient Greece, Himalayan monasteries, eighteenth-century European mathematics departments, Victorian-era physics laboratories, and modern ICU cost-accounting spreadsheets. This is not about meditation or wellness culture. The rigor of mathematics, neuroscience, clinical research, and actuarial data make the case for music as medicine.


The story begins not with a stethoscope but with a blacksmith's hammer. Pythagoras, the sixth-century BCE philosopher and mathematician, observed that hammers of different weights struck an anvil and produced different tones, some tones were pleasing in relation to others, other combinations were not. (Apologies to those of you who know I geek out a bit over Pythagoras, especially in my historical fiction novels). He identified the foundational ratios of musical consonance. The octave vibrates at a ratio of 2:1, or double the frequency, an exact doubling of pitch. The perfect fifth is 3:2. The perfect fourth is 4:3. He furthered applied the concept of natural observed mathematical harmonics to the cosmos and the human body that resonates within it.


Pythagoras called specific frequencies and intervals, basically a stack of pure fifths that aligned human 'being' with a natural mathematical order, the divine tones. Other mathematical constants are found in nature, such as the Schumann Resonance, the electromagnetic frequency of the Earth's ionospheric cavity, and Solfeggio frequencies, a set of tones used in Gregorian chant and traced to ancient sacred music traditions.


Research by Dr. Glen Rein found certain frequencies had measurable effects on DNA structure in laboratory conditions. These frequencies map onto integer-ratio mathematical relationships that Pythagoras identified. Research by Contzen Pereira, published in the International Journal of Science and Research (2016), decomposed Buddhist chant into constituent frequencies, finding that the lower frequencies physically entrains biological tissue as well as stimulates auditory nerves. Pereira's analysis into the formation of sonic wave induced constructive interference patterns looked for measurable effects on cell membrane behavior. Lestar et al. (2013) also looked for modulation of physiological processes within the cells themselves. EEG studies of chanting reveal neurophysiological impact associated with relaxation and creativity, as documented in a 2019 study published in Nature Scientific Reports by Junling Gao and colleagues. A 2023 Singapore Management University review synthesized the available literature of the time and concluded that evidence supports chanting as an effective intervention for stress reduction, anxiety management, and emotional regulation.


Why do specific frequencies, sustained at specific intervals, produce specific physiological outcomes? The answer lies in the mathematics of harmonic decomposition, biological resonance, and the quantification of consonance. It lies with Fourier, Helmholtz, and Euler. The math is certain. There are numerical relationships between frequencies that create constructive resonance that the human body will respond to.


Ancient wisdom and rigorous mathematics converge in ways that should compel the attention of every clinician, healthcare economist, and benefits architect in the country.


Joseph Fourier identified a mathematical principle to describe how a complex periodic signal, however irregular in appearance, can be decomposed into a sum of simple sinusoidal waves at discrete frequencies. This is the Fourier theorem. It is the single most important mathematical foundation in this discussion for its direct application to music and health care, like an EEG reading. Measuring heart rate variability with a power spectral density plot, separating the cardiac signal into its low-frequency and high-frequency autonomic components for clinical interpretation, is Fourier analysis applied to the cardiovascular system. Malliani, Lombardi, and Pagani's paper on power spectral analysis of HRV, published in Circulation (1991), established that the frequency-domain analysis of heart rate variability that anchors modern autonomic medicine is Fourier mathematics applied to the beating heart. In other words, the universe of complex, irregular, apparently chaotic signals coursing through our bodies are composed of pure, simple, mathematical harmonics. This includes brain waves, cardiac rhythms, respiratory cycles, circadian oscillations. Music, which is organized sound, provides Fourier-structured acoustic input to biological systems that process information in Fourier-structured ways. Further, this implies the Pythagorean frequency ratios correspond to the simplest, lowest-frequency harmonics that the body encounters first and most naturally in any complex acoustic environment because they are the mathematical backbone of the harmonic series itself.


Leonhard Euler in 1797 produced an analysis of why certain frequencies are experienced as physiologically pleasant and others as physiologically distressing. He devised a mathematical method to gauge the nervous system's differential response to certain acoustic inputs. His approach was analyzed further by Cannas and Polo in a 2022 peer-reviewed paper, including analysis with an MRI, and found to be very accurate indicator of  physiological response.


Hermann von Helmholtz in 1863 demonstrated that the basilar membrane of the cochlea, the fluid-filled coiled structure of the inner ear, functions as a precise biological frequency analyzer, where different regions of the basilar membrane resonate at different frequencies. It does not transmit a raw acoustic signal to the brain. It performs a physical Fourier decomposition of incoming sound, separating the complex waveform into its component frequencies before the neural signal is even generated. The body does not passively receive music. It actively decomposes it, at the anatomical level. Helmholtz also determined that consonant intervals such as those with simple Pythagorean frequency ratios, produce smooth, low-frequency beating between their harmonics, creating a physiological state of acoustic smoothness that the nervous system processes as pleasant and relaxing, producing slow, regular heart beats.


Helmholtz argued that the study of music is inseparable from the study of the human body. To understand why music affects health must begin with the physics of sound and the anatomy of hearing.



The philosopher and mathematician Gottfried Wilhelm Leibniz hypothesized that the brain performs continuous mathematical computation on incoming acoustic signals, comparing frequency ratios, detecting consonance, measuring temporal intervals, without any conscious awareness. The emotional and physiological response to music is the inherent and natural felt, emotional, consequence of this unconscious mathematical processing. The body experiences pleasure, tension, calm, or agitation based on what the brain has computed as mathematical order, complexity, resolution, or conflict.


 

Medical Research Behind Music as Medicine

 

Modern cognitive neuroscience, specifically the predictive coding framework developed by Karl Friston and colleagues, substantiated Leibniz's intuition. They found that the brain is a hierarchical Bayesian inference engine that continuously generates and updates mathematical predictions about incoming sensory data. Musical pleasure, surprise, tension, and resolution are the result of that predictive mathematical process operating in real time on acoustic input structured by Pythagorean ratios, Eulerian consonance gradients, and Fourier-decomposable harmonic series.


John Butt's peer-reviewed chapter in The Cambridge Companion to Bach, titled "A mind unconscious that it is calculating: Bach and the rationalist philosophy of Wolff, Leibniz and Spinoza", establishes that this philosophical connection is not only academic but has direct association with therapeutic solutions.


Music's most direct pathway into the body runs through the limbic system, the brain's emotional processing center, tightly coupled to the amygdala, the hippocampus, and the hypothalamic-pituitary-adrenal (HPA) axis that governs the stress response. Harvard-affiliated music therapist Lorrie Kubicek of Mass General Cancer Center noted music's connection to the limbic system means it can access the relaxation response, slow respiration, modulate autonomic tone, and shift the brain's attentional anchor from ruminative fear toward present-moment engagement. The limbic system does not distinguish between therapeutic pharmaceutical molecules and therapeutic acoustic ones. It responds to mathematical input delivered through the cochlear Fourier architecture Helmholtz described, processed by the unconscious arithmetic Leibniz named, and evaluated against the consonance gradients Euler quantified.


A 2021 research review published in Psychiatry Research documented music therapy's consistent impact to reduce anxiety across diverse medical settings and patient populations. A 2023 review in Alzheimer's Research and Therapy demonstrated improvements in memory, attention, and temporal orientation in Alzheimer's patients receiving music therapy, suggesting that even in the context of severe neuro-degeneration, musical memories and their associated neural pathways remain accessible and can be activated therapeutically long after other cognitive functions have failed.


A 2025 comprehensive review in Frontiers in Digital Health documented a direct relationship between music therapy and brainwave synchronization, including AI-driven biofeedback. This finding is important because the implication is frequency-based acoustic intervention can be used as a type of precision medicine, where digital (acoustic) therapeutics are calibrated to individual patient autonomic profiles to trigger specific health related outcomes.


None of the above matters to a health plan CFO, a Medicare Advantage medical director, or a benefits manager unless it translates into dollars.


The pivotal study is a 2018 economic evaluation published in Critical Care Medicine, authored by Linda Chlan and colleagues, examining a patient-directed music intervention for ICU patients receiving mechanical ventilatory support. Music therapy significantly reduced patients' need for sedation and anxiolytic medications, producing a net cost savings of approximately $2,000 per patient in reduced pharmacy expenditure alone, against an intervention cost of a few hundred dollars.


The National Institute for Health and Care Excellence (NICE), conducted a 2023 comprehensive evidence review of music therapy for stroke rehabilitation. They found sufficient clinical and cost-effectiveness evidence to warrant inclusion of music therapy in post-acute care pathways.


A 2023 systematic review published in BMC Public Health examining arts and creativity interventions for older adults found consistent evidence of healthcare cost reduction through decreased medication utilization, reduced emergency department visits, and improved mental health outcomes across multiple longitudinal studies.


 

The Economics and the Actuarial Implications of Music in Healthcare

 

From an actuarial perspective, there are implications for shifts in utilization, reducing expensive pharmaceutical utilization in high-cost settings, increasing preventive health with improved stress management, with a potential unquantifiable impact along the lines of social determinants of feeling more positive, of higher self-esteem leading to better self-care and lower utilization.


Stress is the precursor to cardiovascular disease, type 2 diabetes, autoimmune dysfunction, and a significant proportion of behavioral health admissions. Interventions, such as music therapy, can durably reduce chronic stress over a sufficient time horizon, and potentially reduce the incidence of the chronic conditions that drive the largest share of insurance claims expenditure. The actuarial challenge is modeling the lag between the behavioral intervention and the claims impact, but that lag does not negate the effect.


For the cardiac patient population, actuarial modeling and analysis starts with a measurable biomarker, like  heart rate variability (HRV), the beat-to-beat variation in the cardiac cycle governed by autonomic nervous system tone. This is a clinically accepted predictor of adverse cardiovascular outcomes. Patients with depressed HRV carry materially higher expected claims costs, as much as 35–50% above age-sex-condition-adjusted peers. Music therapy, rhythmic auditory stimulation delivered at tempos calibrated to synchronize and arrest the cardiac cycle, has produced statistically significant improvement in HRV in multiple controlled trials across post-MI, chronic heart failure, and hypertensive populations. A durable shift in HRV could reduce heart failure readmission, costing $13,000–$16,000, or reduce onset of AMI, costing $25,000, plus pharmacy based savings from less sedatives, reduced use of antidepressants, lower incidence of depression comorbidity and all its complications, readmission risks and elevated total cost of care. Assuming actuarial incurred claims experience 6% annual medical trend, a credibility-weighted cardiac cohort of sufficient size could produce enough internal experience within two to three plan years to generate statistically valid HRV-linked claims savings. Music therapy intervention could be priced into the health risk model as a formal, trend-eligible cost-management lever.


 

Conclusion

 

Music has impact at the molecular level, at the anatomical level, at the cellular and systems level, at the neurological level, at the psychological level. Combining the observations into practical clinical pathways implies these findings, built from centuries of observation and analysis, can also have economic impact in our health care system.


Healthcare is not what happens in the doctor's office. It is the condition of an entire human being at the molecular, neurological, psychological, and social levels, existing in a state of coherent, resonant, and mathematically organized, complexity. Music does not just support that condition, it is foundational and fundamental. Twenty-six centuries of evidence, from Pythagorean integer ratios to ICU sedation cost data, compel us to acknowledge that sound is medicine, mathematics is its mechanism, and the healthcare system that continues to ignore it is paying a price that can be calculated with today's technology.


Healthcare actuaries can take the lead on addressing that, possibly through risk and rate adjustments. Plus, selfishly, I think we could all use a bit more jazz in our lives.

 



Sources for further reading and study

 

Ancient and Classical Sources


  • • Pythagoras of Samos (c. 570–495 BCE). Pythagorean tuning system and musical cosmology. Primary sources reconstructed in: Hall, M.P. (1928). The Secret Teachings of All Ages. Philosophical Research Society. Available: sacred-texts.com/eso/sta/sta19.htm

  • • Leibniz, G.W. (c. 1712). Letter to Christian Goldbach. "Musica est exercitium arithmeticae occultum nescientis se numerare animi." Translated and discussed in: Butt, J. (1997). "A mind unconscious that it is calculating: Bach and the rationalist philosophy of Wolff, Leibniz and Spinoza." In The Cambridge Companion to Bach. Cambridge University Press. doi.org/10.1017/CCOL9780521450690.006


Mathematical and Scientific Foundations


  • • Euler, L. (1739). Tentamen Novae Theoriae Musicae ex Certissimis Harmoniae Principiis Dilucide Expositae. St. Petersburg Academy. Available: scholarlycommons.pacific.edu/euler-works/33/

  • • Cannas, S. & Polo, M. (2022). "Euler's 'Tentamen': Historical and Mathematical Aspects on the Consonance Theory." In Mathematics and Computation in Music. Springer. doi.org/10.1007/978-3-031-07015-0_6

  • • Fourier, J.B.J. (1822). Théorie analytique de la chaleur. Paris: Firmin Didot.

  • • Helmholtz, H. von (1863). On the Sensations of Tone as a Physiological Basis for the Theory of Music. (English translation, 1875, A.J. Ellis.) London: Longmans, Green. Available: babel.hathitrust.org/cgi/pt?id=mdp.39015000592603; physicsvirtualmuseum.ufop.br/literature

  • • Leibniz, G.W. (1695). New System of the Nature of Substances and their Communication. Translated by Jonathan Bennett. earlymoderntexts.com/assets/pdfs/leibniz1695c.pdf



    Buddhist Chant and Sacred Frequency Research


  • • Pereira, C. (2016). "Frequencies of the Buddhist Meditative Chant – Om Mani Padme Hum." International Journal of Science and Research, 5(4). Available: ijsr.net/archive/v5i4/NOV162732.pdf

  • • Gao, J., et al. (2019). "The Neurophysiological Correlates of Religious Chanting." Nature Scientific Reports. doi.org/10.1038/s41598-019-40200-w

  • • Yang, X. (2023). "A Review on the Effects of Chanting and Solfeggio Frequencies on Well-Being." Singapore Management University. ink.library.smu.edu.sg/cgi/viewcontent.cgi?article=10963&context=sis_research

  • • Maschio, M., et al. (2024). "Chanting of Nam-Myoho-Renge-Kyo in the Context of the Buddhist Liturgy of Nichiren Shoshu: Study of Sound Frequencies, Brain Activity, and Microbial Metabolism." Journal of Neurology & Stroke Knowledge. medcraveonline.com/JNSK

  • • Lestar, M., et al. (2013). Modulation of physiological and pathophysiological processes in cells via acoustic standing wave mechanisms. Referenced in Pereira (2016), above.


    Heart Rate Variability and Fourier Analysis


  • • Malliani, A., Lombardi, F., & Pagani, M. (1994). "Power spectrum analysis of heart rate variability: a tool to explore neural regulatory mechanisms." Circulation. Available: heart.bmj.com/content/71/1/1

  • • Ziemssen, T., et al. (2019). "Spectral Analysis of Heart Rate Variability: Time Window Matters." Frontiers in Neurology. doi.org/10.3389/fneur.2019.00545

  • • Ewing, D.J. (1987). "Spectral analysis of heart rate fluctuations: a non-invasive, sensitive method for early diagnosis of autonomic neuropathy in diabetes mellitus." Diabetic Medicine.

  • • Hudspeth, A.J. (2014). "The physics of hearing: fluid mechanics and the active process of the inner ear." Reports on Progress in Physics, 77(7). ui.adsabs.harvard.edu/abs/2014RPPh...77g6601R


Neuroscience and Clinical Music Therapy


  • • Salamon, M. (2024). "Music as Medicine." Harvard Health Publishing. health.harvard.edu/mind-and-mood/music-as-medicine

  • • Kubicek, L. Co-director, Katherine A. Gallagher Integrated Therapies Program, Mass General Cancer Center. Quoted in Salamon (2024).

  • • Research review (2021). Music therapy and anxiety reduction across medical settings. Psychiatry Research. Referenced in Salamon (2024).

  • • Research review (2023). Music therapy for memory, attention, and orientation in Alzheimer's disease. Alzheimer's Research and Therapy. Referenced in Salamon (2024).

  • • Jiao, D., et al. (2025). "Advancing personalized digital therapeutics: integrating music therapy, brainwave entrainment methods, and AI-driven biofeedback." Frontiers in Digital Health. doi.org/10.3389/fdgth.2025.1552396

  • • Dial, H.R., et al. (2022). "On the Role of Neural Oscillations Across Timescales in Speech and Music Processing." Frontiers in Computational Neuroscience. doi.org/10.3389/fncom.2022.872093

  • • Rein, G. Effect of conscious intention on human DNA. Referenced in literature on 528 Hz solfeggio frequency; requires replication at scale.


Actuarial and Health Economics


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