Instrument · loudspeaker design
The Horn
A single full-range driver, a folding flower-cone horn, and the closed-form physics that tunes it.
A loudspeaker can be functional art whose form is its acoustics. This is
the science spine behind such a range: one single full-range driver (the Fostex
tradition) firing into a back-loaded horn — the cone's rear wave folded downward
through an expanding flare that becomes the horn, the pedestal, and the flower's stem.
The instrument below computes the real, cited physics live: Webster's horn equation,
the exponential-flare cutoff f_c = m·c / 4π, and the folded
quarter-wave tuning. Move the sliders; watch the horn — and its honest limits — change.
This is modelled, not measured. A microphone closes the loop. Consistency, not proof.
What this instrument is — and is not
Every number above is computed from closed-form acoustics that is genuinely true: the exponential-horn cutoff, the geometric flare of an expanding pipe, and the quarter-wave relation that ties a folded path length to the frequency it reinforces. In that sense the maths is Lucid-backed literally — it is derivable, checkable, and reproduces on re-run, exactly the standard the a₀ work holds itself to.
It is a lumped, idealised model, not a measurement. It does not run a full finite-element or boundary-element solve of a specific ceramic geometry; it does not know your clay's density, wall losses, or the flower-cone's exact curves; and it cannot predict the real in-room frequency response, which depends on the driver's full Thiele–Small set, the room, and the fold geometry. The honest workflow is: model here → build → measure with a calibrated microphone (REW + a UMIK is the standard FOSS-friendly rig) → iterate. The model gets you to the right ballpark and tells you which knobs matter. The mic tells you the truth. Modelling is not measurement.
The physics, stated plainly
A horn is an acoustic impedance transformer. A small driver cone is badly matched to open air — most of its motion just pushes air around rather than radiating sound. A horn's gradually expanding walls step that mismatch down, so the driver couples to the room far more efficiently, especially in the bass. The governing relation is Webster's horn equation:
∂²p/∂x² + (d ln S/dx)·∂p/∂x − (1/c²)·∂²p/∂t² = 0
For an exponential horn, area grows as S(x) = S₀·e^{m·x}, and
the equation has a clean cutoff: below a critical frequency the horn stops loading the driver
and the wave becomes evanescent. That cutoff is the single most important number in the design:
fc = m · c / (4π) ⇔ m = 4π·fc / c (c ≈ 343 m/s at 20 °C)
Two honest consequences fall straight out of the equations, and the instrument shows both:
- Low cutoff needs a slow flare and a big mouth. A true 40 Hz mouth wants ~2.7 m across — which is why every real bass horn is a compromise: it folds to hide length, and it leans on the room's walls and floor to finish the job the mouth can't. The flower-cone pedestal is exactly this compromise made beautiful.
- The fold sets the tuning. The folded path behaves like a quarter-wave resonator; a
path of length
Lreinforces aroundf ≈ c / 4L. Folding into the pedestal is not just packaging — it is a tuning parameter.
The range — one physics, five bodies
The same closed-form horn scales from a pocket-money passive amplifier to a granite floorstander. Every tier is the same idea in a different material and at a different cutoff; the ceramic is where the physics and the craft become one object.
The Seed
A 3D-printed passive horn phone-holder. No power, no electronics — the printed flare alone lifts and directs the phone's own speaker. The honest, cheap entry point.
PLA / PETG · no amp · ~120 Hz cutoffThe Bloom
The plastic body gains a small full-range driver and a Bluetooth class-D amp. The rear wave begins to fold — the flower starts to open.
3D-print · BT · class-D · ~90 HzThe Vessel
Desktop ceramic and pro-grade 3D-print units. Real Fostex driver, combo class-D / MOSFET amp, and full inputs: Bluetooth, RCA, 3.5 mm and optical. Where the ceramicist's hand first shapes the acoustics.
ceramic / pro-print · BT+RCA+3.5mm+optical · ~55 HzThe Magnum Floorstander
A full floorstanding back-loaded horn: hardwood, steel, granite and ceramic horns. A large Fostex driver, the rear wave folding the full height of the body into the pedestal. The statement piece — functional sculpture that plays.
hardwood · steel · granite · ceramic · ~35 HzOn lineage — Bose, and where this differs
Bose's celebrated bass work is the acoustic waveguide — a long, folded transmission line that damps and re-times the driver's rear wave (the Wave radio, the Acoustimass tubes). That is a close cousin, but it is not a horn: a transmission line is a tuned pipe; a horn is an impedance-matching flare. This range is a back-loaded horn in the Fostex / Lowther single-driver tradition — the rear wave is not just delayed, its flare actively transforms the driver's coupling to the room. Naming the difference precisely is the honest thing; both traditions are real, and this one is ours.
Sources
- A. G. Webster, Acoustical Impedance and the Theory of Horns, PNAS 5(7), 1919 — the horn equation.
- H. F. Olson, Acoustical Engineering, Van Nostrand, 1957 — exponential-horn cutoff and mouth-size treatment.
- Fostex full-range driver specifications (FE-series), published Thiele–Small parameters.
- Standard back-loaded-horn practice (Fostex "BK" alignments; Lowther / single-driver horn tradition).
- Measurement: Room EQ Wizard (REW) + a calibrated USB measurement microphone — the FOSS-standard rig for closing the loop.
The horn maths here is standard closed-form acoustics, reproduced live in the page. It is offered in the same spirit as the a₀ work: state the real relation, cite it, and name the limit. Where a modelled number and a measured number ever disagree, the microphone wins.