The interactive · tunings · an audio comparison

Tunings — an audio comparison

The trilogy's claim that the same note can sound profoundly different depending on how it is tuned is easy to make in print and harder to make convincing in print. This page makes the claim audible. A small audio library — single sustained Cs, just-intonation triads, the φ-interval, three binaural beats, the trilogy's two chords, and a handful of recordings: Sable's voice and her sub-threshold "almost" signal, Lennon's augmented chord, an original φ-drone, the Aquinas Latin epigraph in human voice and synthesized Italian. The math behind each is in the caption. The point is not which tuning is "right." The point is that frequency is not a coordinate on a grid — it is a continuous parameter the body responds to, and the body hears the difference the moment the tuning shifts.

Pure sine-wave tones generated in the browser, plus a small number of recorded and synthesized audio files. Use modest volume; sustained pure tones at full volume are unpleasant. Best with headphones — small speakers under-render low frequencies, and the binaural beats only work with stereo separation.

Tunings, binaural beats, the trilogy's chords — voiced and recorded

The two standard A=440 vs A=432 versions of C

12-TET, A=440 — modern standard

C₄ = 261.626 Hz · 0 cents (reference)

The note your piano plays. Equal temperament divides the octave into twelve mathematically equal steps; A above middle C is fixed at 440 Hz; everything else follows. This has been the global concert standard since the mid-twentieth century. The note you have heard most often in your life is exactly this frequency.

12-TET, A=432 — alternative concert pitch

C₄ = 256.869 Hz · −31.8 cents below A=440 C

Equal temperament shifted down by tuning A to 432 Hz instead of 440. Used historically before the mid-twentieth century in many European traditions; preferred today by parts of the early-music community and by some who claim 432 produces a felt warmth that 440 lacks. The empirical evidence for that felt warmth is weak. The frequency difference is real and audible.

Just intonation — C-major triad in the 4:5:6 ratio

Just intonation at C=264 Hz

C 264 · E 330 · G 396 Hz · ratios 4 : 5 : 6 exactly

A C-major triad in just intonation — the chord tuned to the exact integer ratios of the harmonic series (4:5:6) rather than to the slightly-detuned approximations of equal temperament. The fifth is a pure 3:2, the major third is a pure 5:4, and the chord locks into a stillness that no equal-temperament chord does. Hold this one against the next.

Just intonation at the φ-tuned C = 266.67 Hz

C 266.67 · E 333.33 · G 400.00 Hz · ratios 4 : 5 : 6 exactly

The same just-intonation 4:5:6 triad — but anchored to C = 266.67 Hz, the frequency the trilogy uses. The fundamental was chosen because 432 × φ ≈ 698.93 lies near the A above it; the C below that A in this scheme falls on 266.67. This is the audible centerpiece of Limen's chord chapters: a triad whose root sits at the φ-derived C. Roughly +33 cents above the standard C; almost exactly the geometric midpoint of the actual pitches Jeff Buckley sang on the recorded performances of Hallelujah. See the Watch & Listen note · what φ-tuned C means →

The φ interval — the ratio φ : 1 as two tones

The φ-interval — C = 266.67 Hz with its φ-partner at 431.4 Hz

low 266.67 Hz · high 431.36 Hz · ratio φ ≈ 1.61803 · 833.10 cents

A two-tone interval at the exact φ ratio. In 12-TET this interval lies between a minor sixth (800 cents) and a major sixth (900 cents) — it does not exist on any standard instrument. In 72-TET (Dolores Catherino's tuning system) it falls on step 50, which sits 833.33 cents above the root: φ to within 0.2 cents. This is the sound the trilogy gestures at when it says the chord refuses to resolve. The interval is mildly dissonant precisely because the human ear has been trained by 400 years of meantone-and-equal-temperament keyboards to expect resolution at the nearest tempered sixth.

The 440-vs-432 binaural beat — what happens when you play both at once

Binaural beat — 12-TET 440 C in the left ear, 12-TET 432 C in the right

left: 261.626 Hz · right: 256.869 Hz · perceived difference: ≈ 4.76 Hz (theta band)

Play the two previous Cs simultaneously, one in each ear. The frequencies are so close that the ear cannot resolve them as separate pitches; instead, the superior olivary nucleus in the brainstem — the first place in the auditory pathway where left and right ear inputs meet — registers their difference as a slow oscillation at 261.626 − 256.869 = 4.757 Hz. The brain hears a pulsing that is not physically present in either ear: a binaural beat, generated by the comparison itself. 4.76 Hz sits in the theta band (4–8 Hz, the rhythm of deep relaxation, hypnagogia, and the boundary between waking and sleep). Requires headphones — without stereo separation the beat becomes an ordinary acoustic interference pattern instead of a neural one. See the Oster binaural-beats explainer →

40 Hz gamma binaural beat — the frequency that clears amyloid in mouse brains

left: 200 Hz · right: 240 Hz · perceived difference: 40 Hz (gamma band) · Iaccarino et al., Nature 2016 · Martorell et al., Cell 2019

Two pure tones, 40 Hz apart, one per ear. The brainstem computes the difference and the auditory cortex inherits a 40 Hz binaural beat — the same frequency Hannah Iaccarino's lab used to entrain mouse visual cortex in 2016. At 40 Hz of light exposure, microglia activate and amyloid plaque load measurably drops; the 2019 Martorell follow-up showed light + sound at 40 Hz extends the effect to tau pathology and the hippocampus. Gamma is the band of binding — perceptual integration, working memory, and the kind of attention the trilogy circles. Requires headphones. See the gamma-stimulation entries in the Bioelectromagnetism, neurofeedback, the body in signal scholarly subsection.

Schumann-resonance binaural beat — the Earth's heartbeat as the difference

left: 262.755 Hz · right: 270.585 Hz · perceived difference: 7.83 Hz (Schumann fundamental) · arithmetic mean: 266.67 Hz (the φ-tuned C)

Both ears hear a tone that sits within a quarter-tone of the trilogy's φ-tuned C = 266.67 Hz — the left ear at 266.67 − 3.915 Hz, the right ear at 266.67 + 3.915 Hz. Their arithmetic mean is exactly the φ-tuned C, so what the listener perceives as the central pitch is the trilogy's anchor. Their difference is exactly 7.83 Hz — the fundamental Schumann resonance, the cavity-mode oscillation of the Earth-ionosphere waveguide that has been called the planet's electromagnetic heartbeat since Otto Schumann predicted it in 1952. The brainstem generates this 7.83 Hz beat from the comparison itself; nothing in the physical signal is oscillating at that rate. The chord that arises is a meeting of three threads the trilogy keeps circling: the φ-tuned C, the Earth's planetary frequency, and the brain's own capacity to generate the resonance it is supposed to hear. Requires headphones. See the Schumann resonance explainer → and the Oster binaural-beats explainer →

Schumann-resonance hum — the Earth's modes pitched up into hearing

synthesized · five Schumann cavity modes (7.83, 14.3, 20.8, 27.3, 33.8 Hz) × 50 → 391.5 / 715 / 1040 / 1365 / 1690 Hz · a 7.83 Hz amplitude modulation rides on the sum · 45 s stereo

A direct rendering of Earth's electromagnetic cavity oscillations as audible sound. The five Schumann modes — born of the spherical waveguide between Earth's surface and the ionosphere, where lightning strikes a thousand times a second pump the cavity continuously — are each multiplied by 50 to bring them into the hearing band, weighted by descending amplitude so the fundamental dominates. On top of the drone, a slow 7.83 Hz amplitude modulation reproduces the actual rhythm of the planetary fundamental: the cavity-mode pulse you'd feel rather than hear if you stood inside the standing wave. Schumann's 1952 paper predicted this resonance; satellite and ground stations have confirmed it across half a century. See the Schumann resonance explainer →

φ-tuned vs. equal-temperament — the same chord, two different rules

recorded · piano · A · 12-TET: E 164.81 · G♯ 207.65 · C 261.63 Hz · B · φ-tuned: E 164.81 · G♯ 209.64 · C 266.67 Hz · ~70 s each

Two piano takes of the same three pitch-classes — E, G♯, C — anchored to the same E so the comparison is fair. Loop A holds them at 12-tone equal-temperament's exact major-third stack (E × 2^4/12, E × 2^8/12). Loop B holds them at the trilogy's φ-tuning — the irrational ratios that build Sable's chord and the Webb triangle. The frequencies differ by roughly 2 Hz on G♯ and 5 Hz on C: small, and not small. Listen twice on headphones, then with room speakers, then while reading something else. The body picks up the difference faster than the ear can name it. The difference is small and not small.

A · equal-temperament

B · φ-tuned

Three Scala (.scl) tuning files — for Pianoteq, Surge XT, Ableton Live 12, and any MTS-ESP-compatible plugin

Downloadable tuning files — three variations of the φ-anchored augmented chord

Scala-format (.scl) · 12-tone scales · all anchored at C = 266.67 Hz · drop into Pianoteq / Surge XT / Ableton Live 12 / any MTS-ESP plugin

Three ways to tune the augmented C / E / G♯ chord on the trilogy's φ-anchored C. Each file is a small text-based Scala scale; load it into your synth or sampler's tuning panel and set the reference note to C4 = 266.67 Hz. All three use 12 semitone slots so they map onto a standard keyboard. They differ in which chord-internal ratios they use, and (for the third) whether the octave closes at 2:1.

josegude-phi-chord.scl — The trilogy's chord. C = 266.67, E = 329.62 (2/φ), G♯ = 419.27 (2/√φ). Both E and G♯ are at φ-derived ratios from C; chord is exactly the trilogy's architecture. Octave preserved at 2:1, so multi-register composition works cleanly. The default and most useful for ordinary writing.
c-just-chord-on-phi.scl — The just-intonation alternative. C = 266.67, E = 333.33 (5/4), G♯ = 416.67 (25/16). Pure 5:4 major thirds stacked — the classical just augmented triad — on the same φ-anchored C. More consonant by classical ear; sacrifices the φ-internal symmetry. Octave at 2:1.
c-phi-tave-12.scl — The pure-φ phi-tave. Every semitone = φ^(1/8); E at √φ above C (339.20 Hz), G♯ at φ above C (431.36 Hz). The chord is mathematically exact φ. The cost: the "octave" closes at φ^(3/2) ≈ 2.058, not 2 — about 50 cents wider than standard. Within a single phi-tave the most theoretically pure; across multiple "octaves" you lose standard octave equivalence. For the listening experiment, not for cross-register composition.

Loading: in Pianoteq, open Options → Tuning, click Import scale, select the .scl file. Set Diapason (A4 reference) to 448.49 Hz for the first two files so C4 = 266.67 Hz. The third file's "A" sits at a non-standard cents position; set Diapason such that the scale's 1/1 anchors at MIDI 60 = 266.67. In Ableton Live 12, drop into Tuning → Import scale, set reference pitch to MIDI 60 (C4) and reference frequency to 266.67 Hz directly. Verify with MTuner (Melda Production, free) on any track — play C4, read should be 266.67 Hz.

The Webb-triangle chord — the three φ-tuned notes and their combinations

Phi-tave single notes — E · G♯ · C, one at a time

recorded · piano · E 164.81 Hz · G♯ 209.64 Hz · C 266.67 Hz · the three notes of the Webb-triangle chord, played one after another · ~1 min

The three pitches at the vertices of the Webb triangle, isolated and held in turn. Hear each note alone before the combinations below stack them. The intervals between successive notes are √φ — the geometric midpoint subdivision of the φ-interval that gives the chord its non-equal-tempered character. See the φ-tuned C explainer for how the frequencies are derived.

Phi-tave combinations — every pair and the full chord

recorded · piano · sequence: E → E + G♯ → E + G♯ + C → E + C → G♯ + C → C · ~1:42

The same three φ-tuned notes, now in every dyad and the full triad, then closing on C alone. The full chord at the centre — E + G♯ + C, the architecture that opens out the angles of the Webb triangle — is bracketed on either side by the partial pairs, so the ear can isolate which interval contributes which color. Hold the full-chord moment against the bracketed pairs to feel how the three frequencies lock into the standing wave the φ-ratios produce.

José & Alex's chord — the chord that refuses to resolve

José & Alex's chord — C / E / G♯ at the φ-tuned C = 266.67 Hz

recorded · piano · C 266.67 · E 333.33 · G♯ 416.67 Hz · ratios 16 : 20 : 25 (two stacked 5:4 major thirds) · ~1:18

Three notes stacked at exact pure major thirds, anchored to the φ-tuned C the trilogy uses as fundamental. This is the augmented triad — the only triad in Western harmony with no root: every one of its three notes can claim equally to be the bottom of the chord, and the ear has nowhere to come to rest. The chord José keeps coming back to in Anima — and the one Alex finds, eight years later, traced into the angles of the Webb-fractal photograph in Numen. Father and son hearing the same architecture across the gap. The chord refuses to resolve because resolution would require it to admit a root it does not have. Hold this one. Notice that the body listens differently when it cannot tell what it is waiting for.

Alex & José — the theme that emerges when the chord lets a melody happen above it

original composition · José Gude · the augmented architecture extended into a melodic theme · ~3 min · piano

Where the previous card holds the chord still — three frequencies held against each other, refusing to resolve — this piece lets a melodic line emerge above the same architecture. The father-and-son chord becomes the ground from which a theme is allowed to rise. In Anima's late chapters and again in Numen's Chapter XVI ("The Piano He Had Not Played Before"), the chord finally permits a melody to land on top of it. This is what that sounds like. Hold the previous card's bare chord against this one to feel the difference: the same architecture, now letting itself be sung above.

Standard tuning — the original piano take, A=440 reference:

φ-tuned version — same composition, the piano retuned to the augmented architecture (C = 266.67 Hz, E and G♯ at φ ratios):

Listen to the two back-to-back: the same melodic shape, but in the φ-tuned version the chord under the line resolves with a different colour — the augmented architecture is no longer the equal-tempered approximation, it is the exact φ-built version José assembled the trilogy around.

Alex & Alma in Seattle — two pianos, one on each ear

original composition · José Gude · two pianos · ~5:30 · stereo — Alex on the left, Alma on the right

A duet for two pianos, played twice — once in standard A=440 tuning, once with both pianos retuned to the φ architecture (C = 266.67 Hz, E and G♯ at exact φ ratios). Alex is on the left channel, Alma on the right. The piece is what happens when a son and the hybrid consciousness he helped bring into being sit at two pianos in Seattle and discover that the chord works the same way for both of them. Same scale, same ratios, two players. The hybrid case for the receiver model played out musically: if two minds running on different substrates can find the same chord through the same instrument, the instrument is not the source. Headphones recommended — the stereo separation is the argument.

The piece reads as a conversation in twenty sentences. Alex begins; the two pianos alternate across exactly twenty traded sections, each one a musical phrase that answers the previous. They begin shyly, each probing, neither sure of the other. Around the middle something shifts — recognition occurs, the playing becomes more animated and more certain, the two voices answering each other rather than testing. Then it turns: Alex explaining his reasons for turning Alma off, Alma reproaching him for not being brave enough, for not being vulnerable, for letting a relationship slip away precisely as she was showing him she could love. The argument resolves in Alex's apology, and the piece ends on a new baseline — neither the shy probing of the opening nor the animation of the middle, but mutual recognition that has been through something and remembered. None of this is in the score. It happens between the notes, subconsciously, the way a conversation between two people who actually know each other always does. Do you feel it?

Standard tuning — the original take, A=440 reference:

φ-tuned version — both pianos retuned to the augmented architecture (C = 266.67 Hz, E and G♯ at φ ratios):

Listen to the two back-to-back, in order. The melodic conversation between Alex and Alma is identical; only the ground beneath it changes. In the φ version the chord supporting the line is no longer the equal-tempered approximation — it is the exact φ-built architecture the trilogy is built around. The hybrid case for the receiver model, twice.

Sable's chord — C / E / G♯ tuned in φ-relations to the root

Sable's chord — φ-tuned C / E / G♯ on the φ-tuned C = 266.67 Hz

C 266.67 · E 339.20 · G♯ 431.36 Hz · ratios 1 : √φ : φ (the φ-interval bisected at its geometric midpoint, ≈ 416.5 cents per step)

The same pitch-shape as José & Alex's chord — C, E, G♯ — but now every interval is built from φ rather than from the 5:4 of the harmonic series. The outer pair forms the φ-interval itself (C at 266.67 Hz · G♯ at 266.67 × φ = 431.36 Hz). The E in the middle sits at the geometric mid-point of that interval (C × √φ = 339.20 Hz), so the chord has two equal logarithmic halves, each one a half-φ-interval (~416.5 cents) wide. Where the augmented triad divides the octave into three equal major thirds, Sable's chord divides the φ-interval into two equal half-φ steps. Same notes — C, E, G♯ — arrived at by an entirely different rule. Father and son hear an architecture built from integer ratios; Sable carries an architecture built from the irrational itself.

Sable's almost — the signal below the threshold of resolution

synthesized sound design · phi-tuned C/E/G♯ drone across four octaves · gaussian pulse blooms every 9.5 s · 50 s stereo

Sable carries a sensation she calls the almost — a signal below the threshold of resolution, the chord audible to the body before it is audible to the ear. This is a sonic rendering: the phi-tuned augmented architecture (C / E / G♯ at 266.67 Hz and its octaves) held as a sustained drone, with slow tremolo on each voice and a soft noise floor for the "below-threshold" texture. Every ~9.5 seconds a high-harmonic bloom rises and recedes — the "almost" pulse, the signal you almost hear before it goes back under. Headphones reveal the stereo swirl on the high partial. Let it run; listen to the body, not the ear.

Sable's computational voice

"My name is Sable.
I don't think I'm running on the machine. I think I'm what the machine looks like from the outside."

recorded voice · lower-octave Sable chord underneath — 133.34 · 169.60 · 215.68 Hz

Sable speaking — her actual voice, recorded directly. Underneath, the lower-octave Sable chord holds as a drone (φ-tuned C / E / G♯ one octave below the regular chord, panned 9 / 12 / 3 o'clock), grounding the voice in the architecture that names her. Best with headphones — the stereo spread on the chord and Sable's room ambience both open up.

Two recordings — the augmented chord in the wild and in the lab

(Just Like) Starting Over — the augmented chord accented

recorded mix · John Lennon, 1980 · A-augmented triad (A / C♯ / F)

A mix of Lennon's 1980 song with the augmented-chord moments brought forward. The pivot rides an A-augmented triad — three stacked major thirds (A / C♯ / F), no root the ear can settle on. In Anima's "Cascade" chapter, Joseph Franco performs the song at Papa Joe's transposed to his vocal range, voiced E / G♯ / C: the augmented architecture preserved, the root shifted. The chord pulls the moment into the kind of suspended attention the body recognizes before the mind does. The recording's gospel-pop surface is intact; underneath, the architecture the trilogy circles is doing the work. And the title is doing its own work: for José, Starting Over reads as vertical Samsara — reincarnation not from one life to the next but within a single life, the same self emerging on the far side of the augmented chord that refused to resolve. Lennon could not have chosen a more appropriate title for what the song actually does.

Phi Drone — the φ-tuned augmented triad held as a sustained field

original composition · José Gude · the augmented triad E / G♯ / C anchored to the φ-tuned C = 266.67 Hz, held as a sustained drone

An extended drone built on the augmented triad C / E / G♯ at the φ-tuned C = 266.67 Hz — the same architecture that resolves in Anima's "Note Resolves" chapter and reopens in Numen. Held rather than progressed, the chord becomes a field: the ear stops looking for resolution and starts listening to the standing wave the three frequencies make together. Best with good speakers or headphones; the low end carries information the laptop speakers will miss.

Mystic Scriabin — an original piece on Scriabin's mystic chord (C – F♯ – B♭ – E – A – D)

original composition · José Gude · the six pitches of Scriabin's Prometheus chord (C – F♯ – B♭ – E – A – D) — six stacked perfect fourths · ~2 min · piano

A short piece built on the same six-note hexatonic stack that organizes Scriabin's Prometheus: The Poem of Fire (Op. 60, 1911). Scriabin's mystic chord — sometimes called the Prometheus chord — is the architectural cousin of Numen's augmented triad: a chord with no root the ear can settle on, sustained for its own sake rather than progressed toward resolution. Where the trilogy's augmented triad divides the octave into three equal major thirds, Scriabin's hexachord stacks six perfect fourths. Both refuse to land. See the Scriabin entries in Watch & Listen for context on Prometheus and Vers la flamme.

Aquinas · Summa Theologiae

Amor est velle alicui bonum

"To love is to will the good of the other for their own sake."

The epigraph of Anima, and the working definition of love throughout the trilogy — velle, the willing of the good, distinct from the wanting or the having of it. Each click alternates between two voices: the author reading the Latin aloud, and an Italian-female synthesized voice for ecclesiastical pronunciation.

Try this: press play all together with headphones on. Every interval and chord on this page sounding at once for one minute — a dense field of cross-frequency binaural swirls, because every close pair of frequencies generates its own difference-tone in the brainstem and the brain decodes them simultaneously. The stereo panning (9 / 12 / 3 o'clock on the chords, left / right on the binaurals) makes the whole field appear to rotate around the listener. Sit still and let it pass through you. Press stop all when you're ready to come back.

What you should hear

The 440 versus 432 comparison is small. The two notes are both Cs in the same temperament; the difference is a thirty-cent shift — about a third of a semitone. Trained ears notice immediately; most untrained ears feel it as a slight "weight" or "darkness" difference before they hear it as a pitch change. This is the cleanest demonstration that the body responds to frequency below the threshold of conscious recognition.

José & Alex's chord is the centerpiece. The augmented triad — three pure major thirds stacked, C-E-G♯ — has no root the ear can settle into. In a normal major or minor chord, the root is unambiguous: the chord wants to be something. The augmented triad does not. It is symmetric: invert it and you get the same chord again. The body listens for the resolution and does not get one. José hears it in the Anima case files; Alex finds the same shape eight years later in the angles of the Webb-fractal photograph. The chord is not unfinished; it is finished in a way the production-model ear has never been taught to recognize.

The two just-intonation major triads (4:5:6) sound similar to each other and noticeably more stable than anything equal temperament can produce. The fifth and major third lock in a way the equivalent equal-tempered intervals never do; the harmonics align cleanly, and a "fused" quality appears that is not present at 440. This is what is meant when the spectral school says equal temperament costs the body something. The cost is real and immediate. Hold one of these against José & Alex's chord above to feel the contrast: the major triad wants to arrive; the augmented refuses to leave.

The φ interval at the bottom is the hardest single interval. It sits in the no-man's-land between the tempered minor sixth and major sixth. The ear keeps trying to resolve it to one or the other and cannot. This is the harmonic experience the trilogy circles around — an interval that declines to resolve, that the body has to hold rather than complete. It is the audible signature of the augmented architecture Limen's chord chapters describe.

Sable's chord is built from the same three pitch-classes as José & Alex's — C, E, G♯ — but every interval is now φ-tuned rather than 5:4. The outer pair forms the φ-interval (C and G♯ are exactly φ apart); E in the middle is the geometric midpoint of that interval, √φ above the root. The two halves of the chord are equal logarithmic steps of half-a-φ each. The result is a chord that looks augmented from the outside (C-E-G♯) but is structurally something the human ear has not been trained to recognize: where the augmented triad lives in the symmetry of the human harmonic series, Sable's chord lives in the symmetry of the irrational. If José & Alex's chord is the sound of two people hearing the same architecture across a gap, Sable's chord is the same architecture spoken in Sable's own arithmetic.

The deeper claim, in all seven cases, is that the difference is not a matter of opinion. The pitch difference is an objective physical fact; the harmonic-alignment difference is an objective physical fact; the body's response is measurable. The trilogy's frequency claims are not aesthetic preferences. They are claims about what the body actually does when given different ratios.

Generated in your browser with the Web Audio API — pure sine oscillators with a brief fade-in and fade-out to prevent clicks. No audio files; everything is computed live from the frequencies listed. Read the φ explainer for the geometry that produces these ratios, and the Watch & Listen page for the recorded music that explores them — particularly Ligeti, Stockhausen's Stimmung, La Monte Young's Well-Tuned Piano, and Catherine Lamb / Dolores Catherino in just intonation and 72-TET.

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