Vessel flute


A vessel flute is a type of flute with a body which acts as a Helmholtz resonator. The body is vessel-shaped, not tube- or cone-shaped.
Most flutes have cylindrical or conical bore. Vessel flutes have more spherical hollow bodies.
The air in the body of a vessel flute resonates as one, with air moving alternately in and out of the vessel, and the pressure inside the vessel increasing and decreasing. This is unlike the resonance of a tube or cone of air, where air moves back and forth along the tube, with pressure increasing in part of the tube while it decreases in another.
Blowing across the opening of empty bottle produces a basic edge-blown vessel flute. Multi-note vessel flutes include the ocarina.
A Helmholtz resonator is unusually selective in amplifying only one frequency. Most resonators also amplify more overtones. As a result, vessel flutes have a distinctive overtoneless sound.

Types

Fipple vessel flutes

These flutes have a fipple to direct the air at an edge.
A referee's whistle is technically a fipple vessel flute, although it only plays one note.

Edge-blown vessel flutes

These flutes are edge-blown. They have no fipple and rely on the player's mouth to direct the air at an edge.
The shepherd's whistle is an unusual vessel flute; the fipple consists of two consecutive holes, and the player's mouth acts as a tunable vessel resonator. A nose whistle also uses the mouth as a resonating cavity, and can therefore vary its pitch.

Acoustics

Sound production

Sound is generated by oscillations in an airstream passing an edge, just as in other flutes. The airstream alternates quickly between the inner and outer side of the edge.
Some vessel flutes have a fipple to direct the air onto the labium edge, like a recorder. Others rely on the player's lips to direct the air against the edge, like a concert flute. Fippleless flutes are called edge-blown flutes.
The pitch of a vessel flute is affected by how hard the player blows. Breath force can change the pitch by three semitones. This is why vessel flutes generally have no tuning mechanism, and why it is hard to learn to play a vessel flute in tune.
Fingering holes and fingers that are too close to the labium disrupt the oscillation of the airstream and hurt the tone.

Amplification

At first the sound is a broad-spectrum "noise", but those frequencies that match the resonant frequency of the resonating chamber are selectively amplified. The resonant frequency is the pitch of the note that is heard. Vessel flutes use the air in a vessel for amplification; the vessel acts as a Helmholtz resonator.

Pitch and fingering

The resonant frequency of a vessel flute is given by this formula:
From this, one can see that smaller instruments are higher-pitched. It also means that, in theory, opening a specific hole on an instrument always raises the pitch by the same amount. It doesn't matter how many other holes are open; opening the hole always increases the total area of the open holes by the same amount.
A vessel flute with two fingering holes of the same size can sound three notes. An vessel flute with two fingering holes of different sizes can sound four notes. The number of notes increases with the number of holes:
Number of holes012345678910
Number of notes12481632641282565121024
Powers of two20212223242526272829210

In theory, if the smallest hole were just big enough to raise the pitch by a semitone, and each successive hole was twice as big as the last, then a vessel flute could play a scale of 1024 fully-chromatic notes. Fingering would be equivalent to counting in finger binary.
In practice, the pitch of a vessel flute is also affected by how hard the player blows. If more holes are open, it is necessary to blow harder, which raises the pitch. The high notes tend to go sharp; the low notes, flat. To compensate, fingering charts soon diverge from the plain binary progression.
The exact shape of the vessel is also not critical, as long as the cavity resonates as a Helmholtz resonator. This is why vessel flutes come in a variety of shapes.

Overtones

The resonator in the ocarina can create overtones, but because of the common "egg" shape, these overtones are many octaves above the keynote scale. In similar instruments with a narrow cone shape, like the Gemshorn or Tonette, some partial overtones are available. Overblowing to get a range of higher pitched notes is possible on the ocarina, but not widely done, because the resulting notes are not "clean" enough.

Multiple resonant chambers

Some ocarinas are double- or triple-chambered, often with the chambers tuned an octave or a tenth apart. This allows the player to play chords, but it also allows an increased range.

Physics [|simplifications]

A less-simplified formula for the resonant frequency of a Helmholtz resonator is:
Where f is the resonant frequency, v is the speed of sound, A is the total area of openings in the vessel, and V is the volume of air enclosed in the vessel, and π is a geometric constant.
The pitch of a Helmholtz resonator is also affected by how far the air has to go to get in or out of the resonator; in other words, the thickness of the material the holes are cut in.
Variations in the speed of sound
The speed of sound, assumed to be constant above, is in fact somewhat variable.
The speed of sound in air varies with temperature, meaning that a vessel flute's pitch will change in hot or cold air. However, varying the playing airspeed can change the pitch by three semitones. This is enough to cancel the expected pitch effects of any probable temperature change.
Air pressure variations do not affect pitch.
Humidity has a comparatively small effect on the speed of sound. Going from zero to 100% relative humidity should change the frequency by less than a two-degree-Celsius change in room temperature. As the player's breath has ~100% relative humidity, the humidity can't vary that much anyway.