The Unbounded Organ — Architecture & Theory

0. The Governing Invariance

A standard acoustic instrument defines an operating range and excludes everything outside it. The Unbounded Organ fixes a single 1D physical waveguide and evaluates it continuously across ten orders of magnitude in pipe length, from 8.575 mm (the 20 kHz ceiling of human hearing) to 40,075 km (the equatorial circumference of Earth).

The mathematical model does not branch or modify its rules. As length $L$ scales, the output resolves successively as a musical note, a sequence of discrete echoes, and an ultra-slow drift.

The Invariant Recursion: At every length, the waveguide simulation evaluates one feedback loop: $$y[n] = x[n] + g \cdot y[n - D]$$ where $D = \frac{2L}{c} f_s$ is the round-trip sample delay, $x[n]$ is the excitation, and $g$ is the boundary reflection factor.

1. Acoustic Physics (The Continuous Medium)

1.1 The 1D Wave Equation and Pipe Resonances

Sound propagation in an ideal 1D column of air is governed by the second-order wave equation:

$$\frac{\partial^2 p}{\partial t^2} = c^2 \frac{\partial^2 p}{\partial x^2}$$

where $p(x, t)$ is acoustic pressure and $c$ is the adiabatic speed of sound ($c \approx 343\text{ m s}^{-1}$ in air at 20 °C).

By d’Alembert’s solution, the acoustic disturbance resolves into forward- and backward-traveling waves: $p(x, t) = p^+(t - x/c) + p^-(t + x/c)$. Boundary reflections determine the resonant frequencies:

Open Pipe (Open-Open): Both ends open to atmospheric pressure ($p = 0$). Reflection inverts phase ($r_1 = -1, r_2 = -1$). The net round-trip reflection is $(-1) \times (-1) = +1$. Reinforcement occurs at round-trip time $\tau = 2L/c$, producing all integer harmonics: $$f_n = n \frac{c}{2L}, \quad n = 1, 2, 3, 4, \dots$$
Stopped Pipe (Open-Closed): One end open ($r_1 = -1$), one end rigidly closed ($r_2 = +1$). The net round-trip reflection is $(+1) \times (-1) = -1$. The wave traverses the tube twice ($\tau = 4L/c$) to return in-phase, producing only odd harmonics: $$f_n = (2n - 1) \frac{c}{4L}, \quad n = 1, 2, 3, 4, \dots$$

1.2 Stratified Atmosphere & Acoustic Cutoff Frequency

In an atmosphere under surface gravity $g$, density $\rho(z)$ decreases with altitude according to the density scale height $H$:

$$H = \frac{c^2}{\gamma g}$$

where $\gamma = 1.4$ is the ratio of specific heats. For Earth, $H \approx 8.57\text{ km}$.

Exponential stratification imposes a lower frequency bound on vertical acoustic propagation—the vertical acoustic cutoff frequency $\omega_a$:

$$\omega_a = \frac{\gamma g}{2c} \implies f_a = \frac{\gamma g}{4\pi c}$$

On Earth, $f_a \approx 3.185\text{ mHz}$ (a period of $5.23\text{ minutes}$).

The Propagation Threshold ($L_{\text{crit}}$): Below $f_a$, vertical acoustic waves cannot propagate; vertical wavenumber $k_z$ is purely imaginary ($k_z^2 < 0$), making disturbances evanescent.

A pipe longer than $L_{\text{crit}} = 2\pi H \approx 53.84\text{ km}$ (open) or $\pi H \approx 26.92\text{ km}$ (stopped) has its fundamental frequency below $f_a$. Its fundamental component does not propagate through the atmosphere.

1.3 Physical Failure Boundaries

Short-End Failure (End-Correction Error): Real organ pipes radiate from open ends, adding acoustic mass that increases effective acoustic length by $\delta \approx 0.61 a$ (where $a$ is the bore radius). Fractional error is $\delta / L$. At $L = 8.575\text{ mm}$ with a $1\text{ mm}$ bore, the error is $12\%$. Below roughly $1\text{ cm}$, ideal 1D column arithmetic diverges from physical pipe behavior.
Long-End Failure (The Vacuum Boundary): Above the Kármán line ($\sim 100\text{ km}$), atmosphere ceases to function as a continuous acoustic medium. For an Earth-radius pipe ($6,371\text{ km}$), over $98.4\%$ of the column sits in vacuum.

2. Psychoacoustics (Perception & Mode Density)

2.1 The Auditory Integration Window

Human auditory processing integrates periodic pulses within a window of roughly $50\text{ ms}$ ($20\text{ Hz}$):

Repetition rates $> 20\text{ Hz}$ ($L < 8.575\text{ m}$): Pulses fuse into a continuous pitch percept.
Repetition rates $< 20\text{ Hz}$ ($L > 8.575\text{ m}$): Pulses arrive slower than the integration window, resolving as rhythmic impulses and discrete reflections.

2.2 Mode Density

In an open pipe, frequency spacing between adjacent harmonic modes is constant: $\Delta f = f_1 = c / (2L)$.

Length ($L$)	Fundamental ($f_1$)	Harmonics below 20 kHz	Perceptual Regime
2.44 m (8 ft)	70.33 Hz	284 modes	Note Distinct pitch
8.575 m	20.00 Hz	1,000 modes	Note Hearing threshold
53.84 km ($L_{\text{crit}}$)	3.185 mHz	6,278,000 modes	Echo Diffuse reflection
6,371 km ($R_{\oplus}$)	26.92 µHz	743,000,000 modes	Drift Continuous drift

At 53.84 kilometres, over 6 million harmonic modes fall within the audible spectrum. When excited by broadband energy, thousands of modes overlap inside every critical band of the cochlea. Pitch sensation collapses into dense reverberant texture before the atmospheric acoustic cutoff is reached.

3. Digital Medium & DSP (The Discrete Medium)

3.1 Digital Waveguides vs. Additive Synthesis

Additive synthesis creates sound by summing individual sinusoidal oscillators: $s(t) = \sum_{k=1}^N A_k \sin(2\pi k f_1 t)$. For a $53.84\text{ km}$ pipe, computing 6 million oscillators simultaneously is computationally intractable.

A digital waveguide (Smith, 1992) discretizes 1D wave propagation into a circular delay line. Reading and writing to the delay line takes $O(1)$ operations per sample, independent of length. Resonant modes emerge mechanically from feedback loop interference.

3.2 Fractional Delay & Interpolator Dispersion

Continuous delay $\tau = 2L/c$ rarely falls on an integer number of discrete sample periods:

$$D = \tau \cdot f_s = M + d, \quad M = \lfloor D \rfloor, \quad d \in [0, 1)$$

Non-integer delay $d$ is realized through an interpolation filter (linear FIR, 4-point cubic Lagrange FIR, or 1st-order Thiran allpass IIR).

Numerical Dispersion: An ideal delay line maintains constant phase delay $\tau_p(\omega) = d$. Real interpolators have frequency-dependent phase response $\theta(\omega)$, altering the effective loop length at higher frequencies: $$\tau_p(\omega) = -\frac{\theta(\omega)}{\omega} \implies f_n^{\text{realized}} = n f_1 \left(\frac{D}{\tau_p(\omega_n)}\right)$$ This introduces mode inharmonicity ($\Delta f_n / n f_1$). The Unbounded Organ measures this phase dispersion live and reports it alongside the physical pipe end-correction.

3.3 Digital Medium as a Second Atmosphere

Just as air is specified by $(\gamma, g, c)$, the digital apparatus is defined by sample rate $f_s$ and bit depth:

Nyquist Limit ($f_s / 2$): Modes above $f_s / 2$ cannot exist in discrete buffers.
Memory State Size: State buffer size is $S = \left(\frac{2L}{c} f_s\right) \times 8\text{ bytes}$. At $48\text{ kHz}$, a 6,371 km pipe requires $7.1\text{ GB}$ of RAM. Lowering $f_s$ to $10\text{ Hz}$ reduces state size to $1.5\text{ MB}$, making planetary columns runnable without altering the algorithm.
WAV RIFF 32-bit Container Limit: Standard WAV headers encode file length in 32-bit unsigned integers, capping files at $4\text{ GiB}$.

4. The Five Limits Matrix

At both extremes of the length scale, the physical atmospheric model and the digital representation encounter distinct, independent boundaries:

Axis Extreme	Physical Medium Boundary	Digital Medium Boundary
Short Pipes ($L \le 8.575\text{ mm}$)	End-correction error grows as $\delta / L$ ($12\%$ error at $8.575\text{ mm}$ for a $1\text{ mm}$ bore). 1D ideal column mechanics break down.	Delay buffer shrinks below 2 samples. Resonant frequency approaches Nyquist limit ($f_s / 2$).
Long Pipes ($L \ge 53.84\text{ km}$)	Fundamental falls below the atmospheric acoustic cutoff $f_a$ ($3.185\text{ mHz}$). Above $100\text{ km}$, air density vanishes into vacuum.	Delay buffer exceeds browser memory headroom. File output reaches the 32-bit WAV limit ($4\text{ GiB}$).