Sonic Geometry: Where Music, Mathematics, and the Architecture of Reality Converge
In August 2012, filmmaker and researcher Eric Rankin had an experience that would redefine his life's work. An insight arrived -- sudden and complete -- telling him to draw a triangle on a whiteboard, write down the sum of its interior angles, and play that number as a frequency.
Sonic Geometry: Where Music, Mathematics, and the Architecture of Reality Converge
The Discovery
In August 2012, filmmaker and researcher Eric Rankin had an experience that would redefine his life’s work. An insight arrived — sudden and complete — telling him to draw a triangle on a whiteboard, write down the sum of its interior angles, and play that number as a frequency.
A triangle’s angles sum to 180 degrees. Rankin generated a 180 Hz tone.
Then he moved to a square: 360 degrees. A 360 Hz tone.
Pentagon: 540 Hz. Hexagon: 720 Hz. Heptagon: 900 Hz. Octagon: 1080 Hz.
When he played these frequencies together — starting from the triangle and adding each successive polygon — the result was not dissonance. It was harmony. Perfect, major-chord harmony. The geometric shapes of mathematics, when translated to sound, produced the most pleasing combinations in music.
This was the birth of Sonic Geometry — the discovery that geometry and musical harmony are not separate phenomena but two expressions of a single underlying structure.
The Mathematics of Polygon-Frequency Harmony
The interior angle sum of any regular polygon with n sides is given by the formula: (n-2) x 180 degrees.
| Polygon | Sides | Angle Sum | As Frequency | Musical Relationship |
|---|---|---|---|---|
| Triangle | 3 | 180 | 180 Hz | Root note |
| Square | 4 | 360 | 360 Hz | Octave above triangle |
| Pentagon | 5 | 540 | 540 Hz | Perfect fifth above square |
| Hexagon | 6 | 720 | 720 Hz | Octave above pentagon root |
| Heptagon | 7 | 900 | 900 Hz | Perfect fifth above hexagon |
| Octagon | 8 | 1080 | 1080 Hz | Octave above heptagon root |
The octave relationship (frequency ratio 2:1) appears between shapes that differ by two sides (triangle to pentagon area, square to hexagon). The perfect fifth (frequency ratio 3:2) appears between adjacent odd-even polygon pairs. These are the most consonant intervals in music — the intervals that the human ear perceives as most harmonious.
The polygon frequency series does not merely contain musical intervals. It generates the most fundamental and universal intervals in music: the octave and the fifth. The circle of fifths — the backbone of Western music theory — is encoded in the geometry of polygons.
This is not a forced correspondence. The mathematics is straightforward and verifiable. The angle sum formula produces numbers that happen to be in exact harmonic ratios. The question is: why?
Pythagoras and the Music of the Spheres
The connection between music and mathematics was first systematically explored by Pythagoras in the 6th century BCE. Pythagoras discovered that the most harmonious musical intervals correspond to simple whole-number ratios: the octave is 2:1, the perfect fifth is 3:2, the perfect fourth is 4:3.
Pythagoras extended this discovery into a cosmological principle. He proposed that the planets, as they moved through space, produced tones determined by their orbital ratios — the “Music of the Spheres.” This was not metaphor for Pythagoras; he believed the cosmos was literally a musical instrument, with the movements of celestial bodies producing an inaudible harmony that governs the structure of reality.
For two and a half millennia, this idea was treated as beautiful but unscientific mythology. Sonic geometry suggests Pythagoras may have been more right than wrong.
Johannes Kepler, in his 1619 work Harmonices Mundi (The Harmony of the World), demonstrated that the ratios of planetary orbital velocities at their closest and farthest points from the Sun correspond to musical intervals. Saturn’s ratio approximates a major third. Jupiter’s approximates a minor third. Earth’s ratio corresponds to a semitone — the smallest interval in Western music. Kepler showed that the solar system’s orbital mechanics produce harmonic ratios with mathematical precision.
Modern astronomy has confirmed and extended Kepler’s findings. The orbital periods of Jupiter’s moons are in simple integer ratios. The rings of Saturn exhibit spacing patterns that follow harmonic series. The cosmic microwave background radiation — the echo of the Big Bang — has been analyzed for harmonic content and found to contain patterns consistent with acoustic oscillations in the early universe.
Robert Edward Grant: The Geometry of Sound
Robert Edward Grant, a polymath whose work spans mathematics, cryptography, music theory, and consciousness studies, has made discoveries that expand Rankin’s sonic geometry into new dimensions.
Musical Notes as Polygon Angles
Grant demonstrated that the 12 notes of the chromatic scale, when tuned to A=432 Hz, produce frequency values whose decimal components correspond to the internal angles of regular polygonal shapes. This is not an approximate correspondence — it is mathematically precise.
For example, at 432 Hz tuning, certain notes produce frequencies that are exact decimal references to the angles of equilateral triangles (60 degrees), squares (90 degrees), pentagons (108 degrees), hexagons (120 degrees), and so on. This geometric-musical correspondence does not manifest at 440 Hz tuning, which Grant cites as evidence that 432 Hz represents a more natural and mathematically coherent tuning standard.
Precise Temperament Tuning
In July 2020, Grant proposed a new tuning system called Precise Temperament Tuning, derived from geometric principles rather than the logarithmic division used in standard Equal Temperament. His system uses a perfect 3:2 ratio for the fifth and 2^(1/3) for the major third, producing a scale at A=432.081 Hz that maintains harmonic alignment with geometric ratios.
Grant’s tuning system represents a fundamental rethinking of how music should be organized. Rather than dividing the octave into 12 equal logarithmic steps (as Equal Temperament does, sacrificing harmonic purity for mathematical convenience), Precise Temperament derives each note from the geometric relationships inherent in polygonal forms. Music tuned this way is literally geometry made audible.
Quasi-Primes and Number Theory
Grant’s mathematical research led to the discovery of “Quasi-Primes” — numbers that occupy positions in prime-number distributions associated with specific polygonal geometries. When numbers are arranged around polygons with sides that are multiples of 6, prime and quasi-prime numbers reveal distinct geometric patterns.
This work connects to the Rodin vortex mathematics in a fundamental way: the number 6 is central to both systems. In vortex math, 6 is part of the 3-6-9 governing triad. In Grant’s number theory, multiples of 6 define the geometric framework within which prime number distribution becomes visible. The suggestion is that the distribution of prime numbers — one of the deepest unsolved problems in mathematics — may have a geometric solution.
The Mod-24 Connection
Grant has identified that DNA structure (often conceptualized as 24-strand in expanded models), the 24,000-year precessional cycle of Earth’s axis, toroidal electromagnetic fields, and prime number distribution all share a common mathematical foundation in Mod-24 logarithmic Mobius spirals. The geometry underlying this recursion is the icosidodecahedron and the cuboctahedron — two polyhedra that Buckminster Fuller identified as fundamental to the architecture of space.
The Fibonacci Sequence and Musical Harmony
The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…) is intimately connected to both sacred geometry and music.
The ratio between consecutive Fibonacci numbers converges on the golden ratio, phi (approximately 1.618). This ratio appears throughout nature in spiral growth patterns, branching structures, and proportional relationships.
In music, the Fibonacci numbers map onto the structure of the octave: the chromatic scale has 13 notes (including the octave), the diatonic scale has 8 notes, and pentatonic scale has 5 notes — all Fibonacci numbers. The most consonant intervals involve frequency ratios of small Fibonacci numbers: the octave (2:1), the fifth (3:2), the fourth (4:3), the major sixth (5:3), the major third (5:4).
The piano keyboard itself is a Fibonacci structure: each octave contains 13 keys, of which 8 are white and 5 are black. The black keys are grouped in clusters of 2 and 3.
The golden ratio also appears in the proportions of the greatest musical instruments. Stradivarius violins, considered the finest ever made, have proportions that approximate golden ratio relationships. Whether Stradivari used these proportions intentionally or intuitively, the result is instruments whose vibrational properties produce unmatched tonal beauty.
The Harmonic Series and Overtones
Every musical note produced by a physical instrument contains not just its fundamental frequency but an entire series of overtones — higher frequencies that sound simultaneously. This harmonic series follows a simple mathematical pattern: if the fundamental is f, the overtones are 2f, 3f, 4f, 5f, and so on.
The harmonic series is nature’s own chord. It is not a human invention but a physical law. And the intervals between harmonics are the same intervals that sound most pleasing to the human ear: octave (2:1), fifth (3:2), fourth (4:3), major third (5:4), minor third (6:5).
This means that the foundations of musical harmony are not cultural conventions but reflections of physical law. The human preference for consonant intervals is not arbitrary — it reflects our attunement to the mathematical structure of vibration itself.
Sonic geometry extends this insight by showing that these same ratios are encoded in geometric forms. The universe does not merely vibrate harmonically — its very shape is harmonic. Geometry and music are two languages describing the same reality.
Eric Rankin’s Genesis Structure
In his ongoing research, Rankin has been exploring what he calls the “Genesis Structure” — a geometric pattern that he believes reveals the fundamental matrix of the quantum universe. Working with physicists, mathematicians, and audio engineers, Rankin has been conducting experiments at the Integratron in California’s Mojave Desert, a structure built by George Van Tassel in the 1950s specifically for acoustic and electromagnetic research.
The Integratron is a 38-foot-high, 55-foot-diameter dome constructed entirely without metal fasteners, designed to function as a giant resonance chamber. Rankin’s experiments there explore how geometric forms respond to specific frequencies in an acoustically optimized environment, seeking to identify the fundamental geometric-harmonic patterns from which more complex structures emerge.
The Genesis Structure concept proposes that just as simple geometric shapes produce harmonic frequencies, there exists a foundational geometric pattern — a “seed geometry” — from which all more complex geometric and harmonic relationships derive. This would be the geometric equivalent of the fundamental frequency from which all overtones emanate.
Implications for Consciousness
If the universe is structured as a harmonic-geometric system, then consciousness — as a product of the universe — should also exhibit harmonic-geometric properties. Several lines of evidence suggest this is the case:
Brain wave harmonics: The brain’s electrical activity organizes into frequency bands (delta, theta, alpha, beta, gamma) that have harmonic relationships with each other. Meditative states are characterized by increased coherence — different brain regions vibrating in phase with each other, producing something analogous to a musical chord rather than noise.
Cardiac coherence: The HeartMath Institute has documented that the heart’s electromagnetic field becomes more coherent (more harmonically structured) during states of positive emotion and meditation. This coherent field extends several feet from the body and can influence the heart rhythms of nearby people.
Cymatics of thought: If thoughts are electromagnetic events, and electromagnetic events produce cymatic patterns, then every thought literally shapes the field around the thinker. Coherent thoughts produce coherent fields; chaotic thoughts produce chaotic fields. This is not metaphor — it is physics applied to neuroscience.
Resonance and empathy: The phenomenon of “entrainment” — the tendency of oscillating systems to synchronize — may underlie human empathy, social bonding, and collective consciousness. When people spend time together, their brain waves, heart rhythms, and breathing patterns tend to synchronize. We are vibrational beings living in a vibrational universe, and resonance is the mechanism of connection.
The Unified Vision
Sonic geometry, in its fullest expression, offers a vision of reality as a unified harmonic-geometric system:
Mathematics is music. The relationships between numbers are the same relationships between tones. Number theory and music theory are dialects of the same language.
Music is geometry. The intervals that sound harmonious correspond to the angles and proportions of geometric forms. A chord is a shape. A shape is a chord.
Geometry is physics. The forms of nature — from atoms to galaxies — follow geometric patterns that are expressions of the vibrational processes that create and sustain them.
Physics is consciousness. The vibrational patterns that organize matter are the same patterns that organize thought, emotion, and awareness. Mind and matter are not separate substances but different octaves of the same vibration.
This vision is not proven. It is a hypothesis — a direction of inquiry. But it is a hypothesis supported by verifiable mathematical relationships, reproducible physical experiments, and the convergent testimony of contemplative traditions spanning thousands of years.
Sonic geometry does not ask us to believe. It asks us to listen — to the numbers, to the shapes, to the sounds, and to the silence from which they all emerge.
“Geometry is frozen music.” — Johann Wolfgang von Goethe
“Music is liquid architecture; architecture is frozen music.” — Attributed to both Goethe and Frank Lloyd Wright
The universe is neither frozen nor liquid. It is vibrating. And in its vibration, geometry and music are one.