Tuning
φ-tuned C — why 266.67 Hz
The trilogy anchors every chord and frequency claim on a single fundamental: C = 266.67 Hz. This page explains what that number is and why it's not the C your piano tunes to.
The standard Cs
Several different pitches all get called "C" depending on the tuning system in use:
- 12-TET, A=440 → C4 = 261.626 Hz. The note your piano tunes to. The global concert standard since the mid-20th century.
- 12-TET, A=432 → C4 = 256.87 Hz. The alternative concert pitch some early-music traditions and contemporary harmonic-series advocates prefer.
- Just intonation, C=264 → C = 264 Hz. The pythagorean baseline; chosen so the major triad above it lands at the exact 4:5:6 ratios (264 · 330 · 396).
The trilogy picks a fourth value, distinct from all three:
φ-tuned C = 266.67 Hz
About 33 cents (a third of a semitone) above the standard 12-TET C, and about 2.67 Hz above the just-intonation C=264. Audible difference, not a typo.
Why 266.67
The number is chosen so that C × φ ≈ A = 432 Hz:
266.67 Hz × φ ≈ 431.36 Hz ≈ 432
That single relationship is the architecture. From this C, every other note in the trilogy's chord work follows by powers of φ:
| C × φ⁻¹ ≈ | 164.81 Hz | root of the Webb-triangle chord (E, one φ-interval below) |
| C × √φ ≈ | 339.20 Hz | middle note of Sable's chord (E at the geometric midpoint) |
| C × φ ≈ | 431.36 Hz | top of Sable's chord (G♯) · within 0.15% of A=432 |
| C × φ² ≈ | 698.13 Hz | two φ-intervals up · near F5 in standard 440-tuning |
The choice is structural, not aesthetic. 266.67 is the C that anchors a φ-progression of frequencies whose first multiple lands on (essentially) the 432-Hz A.
The Hallelujah coincidence
The +33-cent offset from standard C is almost exactly the geometric midpoint between the pitches Jeff Buckley actually sang in his Hallelujah recordings — he was unconsciously navigating an interval the equal-temperament keyboard can't write. The trilogy treats this as a coincidence and a wink: an artist of his caliber locating, by ear alone, the φ-tuned region the math says is there.
Where to hear it
- Tunings page — button t4 plays the bare 266.67 Hz tone, just-intonation chord built on it, and adjacent comparisons.
- Reading & References — the Webb-triangle widget plays the chord on this C transposed down one φ-interval (E·G♯·C at 164.81·209.64·266.67).
- José & Alex's chord — the augmented triad C/E/G♯ stacked at 5:4 thirds on this C (16:20:25).
- Sable's chord — the same three pitch-classes stacked at 1:√φ:φ ratios on this C.
- The Webb-triangle explainer — how the angles and frequencies share the same φ at different powers.
The bare frequency is one tone; the chord work above it is where the trilogy's argument actually lives. Hold the chord against the 5:4 major triad on the same C to feel the difference: same notes, different irrationality.