A NOTE BEFORE YOU LISTEN.
What is exact here are the relations: the ratios 9/7 and 5:6:7:9, the doublings, the constants in the panel below — theorem-level, computed live in this page’s source. What is chosen is the anchor: A3 = 220 Hz and the octave spans are rendering decisions — anchor the root anywhere else and every relation survives unchanged. Nothing in the framework hums at 220 Hz. What is observed — and labeled so — is the numerical closeness of the corridor’s width to the blues bend, and the ceiling beyond which no measured system has been found: the silence renders “nothing observed there,” not “nothing there.” What that quiet contains is an open question, and your questions about it are the right response. And what you feel while listening is yours: beauty is welcome here, and it is not evidence — the two can share a room without sharing a column. This page exists so you can listen, with the labels left on.
I. The Corridor
Click and drag across the corridor. Floor (1/φ ≈ 0.618) to ceiling (KAUD ≈ 0.980).
The tone maps your position. The last 2% — between ceiling and unity — is silence.
1/φKAUD
0.618 — Floor0.980 — Ceiling1.000
—
II. The Binary Tower
64 steps of n × G. Click any step to hear its tone. Landmarks glow:
step 18 (√2/4, Shell 3), 32 (√φ/2, k=5 tower target — sits ~0.67% from 32·G),
35 (ln 2, Landauer crossing — last sub-Landauer step), 36 (1/√2, Geometric-damping threshold),
37 (1/(2 ln 2), Shannon bound), 50 (KAUD, Ceiling), 64 (√φ, Pivot).
The corridor floor 1/φ ≈ 0.618 is a separate reference that sits ~2.2% below 32·G.
Click a step to hear it.
III. The Blues Interval
log₂(9/7) = 0.36257. Corridor width (KAUD − 1/φ) = 0.36222. Difference: 0.42 cents.
A musician bending from the flat-5 to the flat-7 traverses the corridor.
The half-diminished seventh chord 5:6:7:9 contains the corridor.
The interval 9/7 between the 7th harmonic and the 9th is the bridge between floor and ceiling.
IV. The Gap
A tone rises from floor toward unity. It reaches KAUD and stops.
The remaining 2% is the breathing room. The space where the real thing happens.
KAUD1
V. How the Sound Is Made
Nothing is hidden: every frequency on this page is computed from the constants below, in the page’s own source. Three mappings, each one line.
Why doublings? Because the ear is binary-native.
An octave — the most consonant interval in music — is exactly a doubling of frequency: A3 = 220 Hz, A4 = 440 Hz, A5 = 880 Hz. The ear hears equal ratios as equal steps, which is why every mapping here uses 2^x: equal moves along the corridor or the tower become equal musical intervals. A framework built on doublings and a sense built on doublings need no translator.
I. Corridor → pitch: two octaves from floor to unity.
Your position v between the floor (1/φ) and unity is rescaled to t = (v − 1/φ)/(1 − 1/φ), then mapped as freq = 220 · 2^(2t) — A3 at the floor, rising smoothly through two octaves toward 880 Hz at unity. But the tone is cut at v > KAUD: the last 2% is rendered as silence, because that is what the gap is — the part of the climb that never sounds.
II. Tower steps → pitch: the fractional part sings.
Step n carries the value n × G. Its tone is freq = 110 · 2^(3 · frac(n·G)) — the fractional part of n·G spread over three octaves from A2. This is a display choice, honestly labelled: the pitch encodes where the step sits inside its current unit interval, so you hear the staircase’s internal rhythm rather than its absolute height. The info line below the steps always shows the exact value, and for landmark steps, the named constant and the deviation — a step names the nearby constant; it does not equal it.
III. The blue notes: why 7/5, 9/5, and the 9/7 bend.
The root is A3 = 220 Hz. The flat-5 is 220 · 7/5 = 308 Hz and the flat-7 is 220 · 9/5 = 396 Hz — the septimal (harmonic-seventh family) tunings that blues bends gravitate toward. Bending from flat-5 to flat-7 therefore traverses the ratio (9/5)/(7/5) = 9/7. Measured the way the ear measures — in octaves — that bend spans log₂(9/7) = 0.3625701. The corridor’s width, KAUD − 1/φ = 0.3622242, is numerically 0.42 cents away (a cent is 1/1200 of an octave; trained ears distinguish ~5 cents, so the two are audibly identical). Tier note: this proximity is an Observation — a numerical coincidence between an octave-measure and a dimensionless width, reported as numbers, with no mechanism claimed. The chord button plays 5:6:7:9 — the half-diminished seventh — in which the 9/7 interval sits between its two upper voices.