Case study: clarinet

val:Reed is a cylindrical bore, stopped by the mouthpiece at one end and open at the other, driven by a pressure-controlled valve. It is the same machinery as the flute with the opposite boundary condition, and almost everything that distinguishes the two instruments follows from that one difference. The full Turtle is at examples/clarinet.ttl.

What a stopped bore does

A wave travelling down the tube reflects at the open end with its sign inverted, and at the closed end without. One round trip therefore inverts once, so the tube supports only those frequencies that fit an odd number of quarter wavelengths into it.

Three familiar facts come straight out of that:

The even harmonics are missing. Measured at D4, the third harmonic comes out at 0.33 of the fundamental and the fifth at 0.19, while the second, fourth and sixth are all under 0.005. That hollow, woody tone is not a filter setting; it is the geometry.
A clarinet sounds an octave below a flute of the same length, because a quarter wavelength fits where the flute needs a half.
It overblows to the twelfth, not the octave: the next mode up is three times the fundamental, not twice.

tests/dsp/ClarinetTest.cpp measures the first of those directly, and compares it against val:Flute at the same pitch: 0.005 against 0.144, a factor of thirty, from nothing but the boundary condition.

The reed

The reed is a valve that the pressure across it closes. The bore's returning pressure works against the player's, and the reed's opening is very nearly a straight line in that difference until it slams against the lay and shuts. Clipped like that, it is the whole nonlinearity of the instrument, and where the cycle spends its time against the limit is what decides the tone.

That gives a reed a band of mouth pressure it works over, and the two ends of the band are always a factor of two apart whatever the reed is like. Below the band nothing sounds. Above it the reed is held shut and nothing sounds either, which is what happens when a beginner blows too hard. The pressure control is mapped onto that band, so most of its travel plays while it still passes through zero, and an unblown instrument is silent.

stiffness moves the band rather than the tone:

Reed	At very little air	Blown hard
Soft (`stiffness` 0)	speaks	chokes
Hard (`stiffness` 1)	will not start	takes everything it is given

That is what a player feels when they change reeds, and it is measured in testStiffnessChangesWhatItTakesToPlay.

Tuning

A waveguide plays at a pitch set by its total loop delay, and the loss filter in the loop is part of that delay. Its phase delay is known in closed form and is taken out of the delay line exactly, which is what stops the damping control pulling the instrument flat as it is turned up: measured across the whole range, the pitch moves by six cents while the third harmonic falls from 0.33 to 0.19.

What is left over measures as a straight line in frequency, about 0.079 cents per hertz, and is taken out by a calibration fitted to measurements at eight pitches. Rendering MIDI notes through the engine:

Note	Sounded	Error
D3 146.83	146.79	−0.5 cents
D4 293.66	294.48	+4.8 cents
D5 587.33	585.37	−5.8 cents

The upper register comes out purer than the lower one, which is also true of the instrument: at D5 the third harmonic is 0.18 where at D3 it is 0.34.

The circuit

MidiPitch ─────────────────────► Reed.frequency
Envelope ──► blowScale ────────► Reed.pressure
Envelope ──────────────────────► VCA.cv

Reed ──► OnePole (bell) ──► VCA ──► Output

The envelope drives blowing pressure as well as the amplifier, because pressure is not a volume control: the Blow parameter sets how hard the note is played and the envelope's attack is the time the player takes to get there. The one-pole after the reed stands for the bell, which radiates the high partials more readily than the low ones.