The modal voice and the breath bow

A voice is a chain of cells

A Suzu voice is up to sixteen magic-circle cells, each tuned to a ratio of the note — the placed partials — each with a decay it declares, and coupled to its neighbours by a spring in phase space. Every sample does three things, in this order: the coupling kick, computed from where every cell was before the sample began; each cell’s own rotation; the declared damping. Each stage is symplectic, so the chain as a whole neither makes nor loses energy on its own: a bell with every decay set to zero, struck once, rings for ten minutes within two thousandths of a decibel.

Partials, harmonics, overtones. Partials are whatever frequencies a voice contains; harmonics are the integer-ratio ones; overtones are the partials above the fundamental. Here partials arise in exactly three ways: placed by a preset’s ratios, created by the phase-space shears (the cells chapter), and split by coupling.

The modal voice, live: each bar a partial's energy placed at its frequency; a strike sets them ringing and each fades at its own rate. Breathe, and the bow feeds each partial toward its target and no further — the servo strip shows the lattice's energy converging. Raise the coupling and the partials exchange energy but stay in tune; the engine refuses a coupling it cannot compensate, saying why.open the lab page ↗

The presets

preset	ratios	strike weights	what it is	fitted to (VCSL)
harmonic string	$k\sqrt{1 + Bk^2}$	$1/k$	a string; $B$ its stiffness (0 for nylon)	Concert Harp, acoustic guitars
stiff bar	$(2k + 1)^2/9$ — 1, 2.78, 5.44, 9 …	$1/k$	a free bar	Glockenspiel, marimba, vibraphone
bell	0.5, 1, 1.2, 1.5, 2, 2.5, 2.667, 3 …	as cast, the prime loudest	hum, prime, tierce, quint, nominal …	Tubular Bells 1 & 2, hand bells
glass	1, 2.32, 4.25, 6.63 …	$1/k^2$	a struck glass	Crotales, glass
plucked string	$k\sqrt{1 + Bk^2}$	$\sin(k\pi p)/k^2$	Karplus–Strong in modal form; $p$ the pluck position	Dan Tranh (zither), Concert Harp

The decays follow the string’s law: the fundamental’s T60 is the patch’s, and each partial above dies faster by a rate that grows with its ratio squared — highs first. The plucked string’s weights are the triangular pluck’s spectrum in closed form: pluck near the bridge and the strike brightens; pluck at the middle and the even partials are not fed at all.

Coupling, and why it had to be compensated

The spring between neighbours is a shared potential, so energy migrates between partials — the physical origin of a soundboard’s richness — and the swirl dimension (poly pressure) adds to it: pull down and hear the overtones stir. But a spring stiffens each mode it touches. The first lattice played its fundamental 149 cents sharp at the default coupling with the swirl down: the coupling knob was a pitch knob before it was a timbre knob.

So the coupled chain is solved at patch load — its normal modes found, and each cell’s own stiffness adjusted until every normal mode lands on its ratio — and the voice runs on that, tabled over the coupling’s range and corrected for the pitch it is playing. The partials stay where the preset put them to a tenth of a cent from C2 to C8 at any coupling the gate admits, and what remains of the coupling is the exchange of energy. Splitting survives where it belongs: two cells at the same pitch split into a close beating pair, exactly the joint map’s normal modes.

Two cells at the same pitch, coupled, split into a close pair and the sum beats at their difference. The measured beat sits on the analytic line to a hundredth of a percent from κ = 0.0125 to 0.5.

The load gate

A patch is refused at load, with a message, on either of two counts: the coupling would take a partial’s whole spring (the compensation cannot place it), or the chain’s stiffness would reach the sampling bound somewhere in the playable range — every semitone from A0 to a C8 bent up four octaves, the modes past the output’s Nyquist muted as the voice mutes them. For every shipped preset the first count binds first; the second is the chain’s ceiling, and its red control lives on two cells just under that ceiling: five percent over the bound and the harness blows up at once, five percent under and it rings.

The breath bow

A cell’s damping can be signed: below a target orbit the cell is anti-damped — the bow grips — and above it damped — the bow slips. The composite is a limit cycle by construction, never free growth: from silence and from twice the target the orbit settles alike, within a few time constants, and with no breath there is no bow — the declared decays alone, and silence is a gated property.

Breath (CC 2, or CC 11) sets the target; the bow’s onset sets its time constant — slow blooms, fast speaks at once — and a bow position says which partials it feeds. One consequence of the servo’s honesty: it can only inject energy as fast as its onset allows, so a partial whose own decay is faster than that cannot be held — it rings from the strike and dies. At the defaults the first three partials of the harmonic string sing; shorten the onset and more join.

This is the wind player’s reason to be here: the instruments that breathe finally have a voice that holds. On the tablets the Bowed string patch is this voice under the brass layouts’ valves and slide.

The harmonic string with breath 0.63 held through every strike: a flat singing line across the keyboard, the second partial level with the first at bow position 0.3.

The profiles

The harmonic string: partials at 1, 2, 3 … with weights $1/k$, the highs decaying first. Above C7 the upper partials cross the 24 kHz ceiling and are muted — fewer partials, a little less held energy, and the level line bends by a decibel.

The bell: the hum an octave under the prime, the tierce, the quint, the nominal and the cluster above. The peak line’s sawtooth is the measurement’s — a 0.4 s window catches the prime–tierce beat at a different phase on each note; the RMS underneath is flat to 0.8 dB. The bar, the glass and the plucked string are on the profiles page.