A = 432 Hz · Whole-Number Harmony
HelioSynth is an interactive study of Pythagorean musical theory — whole-number ratios, the classical quadrivium, and the physics of sound in space — rendered as living visual harmonics you can watch and listen to at once.
Side Chapel · Module 2
Greek harmonic theory tied consonance to whole-number division of a string. Choose an interval below: two tones sound together above the chosen root, while their waves — and their sum — draw themselves in real time.
Root fixed at 432 Hz — a modern design choice, not an ancient standard. Every frequency derives from the visible formula: f = root × num ÷ den. The smoother the combined wave repeats, the more consonant the interval.
# ratio -> frequency -> note define :ratio_hz do |root, num, den| root * num.to_f / den.to_f end play hz_to_midi( ratio_hz(432, 3, 2) ) # the fifth: 648 Hz
// ratio -> frequency -> oscillator const ratioHz = (root, num, den) => root * num / den; const osc = ctx.createOscillator(); osc.frequency.value = ratioHz(432, 3, 2); // 648 Hz osc.connect(master).start();
Side Chapel · Module 3
The first four integers — 1, 2, 3, 4 — hold every central concord between adjacent numbers. Tap a row to hear it as a harmonic of the ground tone; the triangle is a contemporary rendering of an ancient idea.
Between neighboring rows live the three relations Greek theory called symphonic — and multiplying them rebuilds the whole system.
The Four Liberal Arts
The Pythagoreans studied number in four aspects. HelioSynth maps each quadrant to a layer of the experience.
Pure ratio — 1:1, 2:1, 3:2, 4:3 — the raw material of the tuning system and every interval on the monochord.
Waveforms, Lissajous curves, and standing-wave patterns: the visible shape of every sounding ratio.
Ratio set in motion — intervals, scales, and rhythm generated live from whole-number relationships at A = 432.
The harmony of the spheres: orbital-style animations where periods and cycles obey the same ratios as the tones.
Whole-Tone Foundations
Every interval as a bar whose length is its frequency, rising from the 432 Hz fundamental to the octave.
Transept · Modules 4–6
The same eight degrees, three systems. Tap any cell to hear it. Tap a degree number to hear Pythagorean and Just together — where their cents differ, you will hear the beating.
No system is ranked best — each is an answer to a different question. Pythagorean tuning optimizes pure fifths; Just intonation optimizes thirds from the harmonic series; Equal temperament trades exactness for uniform modulation. Listen for degrees III, VI, and VII.
Side Chapel · Module 5
Stack twelve pure fifths and you *almost* land back where you began — seven octaves up. Almost. The tiny gap that remains is the Pythagorean comma, the reason no tuning can be perfect in every key at once. Walk the spiral fifth by fifth and hear the doors refuse to close.
Workshop · Sound to Play With
Nothing to read here — just make sound. A drone pad, an overtone sculptor, and a ratio keyboard, all sharing the same safe audio bus and stop control.
Choir · The Twelvefold Circle
Twelve tones on one ring. Tap any tone to ring it. Arrange the ring by semitones and the chain of fifths becomes a twelve-pointed star; arrange it by fifths and the star unwinds into the circle. Same tones, two geometries.
The star inside the ring is the path of stacked fifths — the same walk you took in the Comma chapel, now seen whole.
Choir · Concord in Three or More
A chord is a stack of ratios sounding at once. Choose one: each voice becomes an orbiting body whose speed is its frequency — the chord's consonance is how soon the orbits realign. Sound it as a chord, or let it arpeggiate.
The major triad 4:5:6 realigns quickly — tight geometry, smooth sound. Watch how much longer 10:12:15 takes to close its figure.
A chord is ratios sounding together; an arpeggio is the same chord unfolded in time — each note in sequence, from the Italian arpeggiare, to play as a harp does. Nothing about the harmony changes; only its dimension does.
In HelioSynth terms, arpeggiation moves a chord from the frequency axis to the time axis — the 4:5:6 you hear as color in a triad becomes an order you hear as motion. Same numbers, new art.
The pattern low · high · middle · high, named for Domenico Alberti, carried half the accompaniments of the classical era — a chord kept gently airborne rather than struck.
An arpeggio repeats a short cycle just slower than the ear fuses it — so you hear both the melody of the steps and the ghost of the chord behind them. Two perceptions from one figure.
Side Chapel · Module 7
Two nearly equal tones fold into a slow pulse — the beat is their difference. Drawn against each other, the pair traces a Lissajous figure: a clean ratio holds a still shape, and every hertz of detune sets it precessing.
Side Chapel · Module 8
A polyrhythm is an interval slowed down. Two pulses in ratio a:b drift and lock like the tones of the monochord — and if you spin the cycle fast enough, the clicks fuse into two pitches at exactly that ratio. Rhythm and harmony are the same number at different speeds.
Acoustic Cloister · Module 9
Rooms resonate at their own whole-number modes. Below, a standing wave forms between two walls — the starting point for understanding baffles, absorption, and why spaces sound the way they do.
Room modes — the wave fits the space in whole halves. Higher modes, higher pitch.
Standing Wave Anatomy
Two speakers face each other — one firing up, one firing down. Sweep the frequency: each time the gap holds a whole number of half-wavelengths (L = n·λ/2), the wave locks into resonance and tiny beads suspend in the pressure nodes, spaced exactly λ/2 apart.
Dome / Observatory · Module 10
Some orbits really are locked in whole-number ratios. Jupiter's moons Io, Europa, and Ganymede circle in a true 1:2:4 resonance — celestial octaves — and Neptune and Pluto hold a 3:2, a fifth written in centuries. The mapping below is disclosed: frequency is inverse period, octave-folded into hearing range. The ratios are astronomy; the music is our translation.
Kepler searched for exactly this in Harmonices Mundi (1619), assigning intervals to planetary angular speeds. His data was real; his conclusion — that the heavens sound a chord — remains a beautiful speculation, and is labeled as one here.
Module 11 · The Capstone
Everything in the cathedral was made from whole numbers. Now choose your own: a root, a concord, a rhythm, a voice — then sound the chamber you built and carry it out as Sonic Pi code.
Borrowed live from the Chord Temple — currently Major Triad. Change it there (or build a custom one) and the chamber follows.
Bell Tower · The Patch Collection
Each engine is a full reverse-engineering of one Sonic Pi composition — its live loops become channel strips, its set variables become the desk, and the code panel rewrites the source as you play.
Uplifting drift on E. A six-step breath swells nine layers: harmonic pad in seven chord modes, bell ascents, shimmer, and a euclidean ghost floor.
Steadier breath, drums awake by default with a live snare on 5 and 13, tom-bell accents, and the whole field re-centered on a darker B2.
Five euclidean cycles — 3:8, a 12-ring, 5:16, 4:9, 3:12 — sweep a concentric wheel and only meet again every 36 beats. Module 8, audible.
A two-bar quarter-note gate chops the sub and mid while an industrial rumble grinds and a vast choir alternates Cmaj7 and Cm7 every 24 beats.
Every step lasts 0.33 beats, so a bar is 5.28 — nothing lands square. Sub, mid, and rumble quantize to 10.56; a gong strikes off-grid; and filter cutoffs join the desk for the first time.
Carrier tones from the 432 family, beat rates banded Delta to Gamma, brown and pink noise beds with a tide swell, deep drone and hum voices, and a breathing mandala that paces coherent, box, or 4·7·8 — labeled honestly down to the vagus nerve.
Three macros drive eight generative loops with a seeded Mutate — now with seven genre profiles: Circuit, Goa, Full-On, Progressive, Darkpsy, Forest, and Hi-Tech, each with its own bass engine, scale, and phrase FX.
Each engine is its own HelioSynth artifact. Keep the files
in one folder and the cards link straight through; every desk state can be copied
back into Sonic Pi as a set : block.
Information Architecture
HelioSynth is organized as a building you move through. Chambers glowing gold are open on this page; the rest are under construction, in order.