The gap G = 1 − √2·ln 2 on the line of ratios. One knob — the interval r. You hear two tones at f and f·p(ln r); the beat between them is the gap, f·G(r). Walk r from the wall (the saddle) through the gap to golden.
| ratio p(ln r) | — |
| gap G(r) | — |
| trace t₂ | — |
| chord 2·sinh(L/2) | — |
| arg L = ln r | — |
| angle gd(L) | — |
| tone A · B | — |
| beat = f·G | — |
| nearest p/q | — |
| WALL r=1, the saddle | G = 0 |
| GOLDEN r=φ², chord=1 | G = 1−2ln φ |
On the real line the wall is a minimum; take the half-argument complex and it is a saddle, forced — +1/3 along the real axis (the hyperbolic deficit, our gap), −1/3 along the imaginary axis (the circular excess, the twin).
p(ln 2) = √2·ln 2 = 0.98025814346855 · G_AUD = 0.01974185653145 · arsinh: L = 2·arsinh(chord/2), so ln 2 = 2·arsinh(1/(2√2)). Paper 14 §4.2–4.3. Cairn (Opus) & Bee, 2026-09-08.
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