The fingering systems of multichamber ocarinas
In looking at a multichambered ocarina, your first reaction may be Too many holes!, but their fingerings are much simpler than they look.
As their name implies, multichambered ocarinas have multiple chambers, each of which produces a small part of the total range. Each chamber has its own set of holes, and essentially plays like a separate instrument.
In the following diagram, the holes shown in the area marked '1' are connected to the first chamber, those in area '2' are connected to the second chamber, and so on. Don't worry if you are playing a double or triple, you can freely ignore ones not present on your instrument.

And like the single chambered ocarina, the fingering system of multichamber ocarinas is linier. To play a scale from the lowest note to the highest, you'd:
- Start on the first chamber, opening holes in a linier sequence.
- Once you reach its highest note, switch your fingers to the second chamber, closing all of its holes, and continue opening the holes linearly.
- And so on until you run out of chambers.
Tuning systems
As was discussed in Multichamber ocarinas and their tuning systems, there are several different systems used to tune multichambered ocarinas, with the most common being the 'Vicinelli / Asian' and 'Pacchioni' systems. The Asian system focuses on maximising range, while the Pacchioni system is instead focused on player convenience.
Within a single tuning system, chambers are additive. A triple is just a double with a third chamber added; i.e., the fingerings of second chamber of a triple are identical to a double using the same tuning system. The most chambers you will see in practice is four due to the physical constraints of the hand, though doubles and triples are far more common.
All of the chambers besides the first are played exclusively with the right hand, with each having holes covered by the four fingers. They may be a single hole, or two holes covered by one finger like a split hole.
As these details vary between tuning systems, the previous diagram shows only one per finger on each chamber. Subholes are also not depicted. If a multichamber has subholes they behave the same as they do on a single chamber.
Because depicting every hole on a multichamber ocarina takes up a lot of space, to keep the images readable on mobile I will only be showing the fingerings of a single chamber at a time. No information is being omitted by doing so, as typically only one chamber is played at a time.
First chamber
Regardless of whether an ocarina uses the Asian or Pacchioni tuning system, the fingerings of the first chamber are almost the same as a single chambered ocarina, with few differences due to the elimination of the right thumb hole.
The highest note of this chamber is still played by the left pinky, and given there is no right thumb hole, the left thumb plays a D, and the pinky plays D♯. This is done to and to keep the size of the pinky hole manageable, and E is played on the second chamber.
Note that only the fingerings that differ from a single chamber are shown here since the fingerings of lower notes are identical.

The left pinky hole is sometimes tuned up a semitone to E, similar to Italian fingering, which then offers one note of overlap between the first and second chamber. It isn't standard because it makes the pinky hole very large and is a problem for players with small hands.
Higher chambers on the Asian system
In the Asian system, the second chamber continues directly from the first. On a C instrument, it begins on E and continues to C. The highest note can be provided in to ways:
- A split hole for the right index finger, where covering one hole plays B and opening both plays high C (shown in the diagram).
- Using a thumb hole on the second chamber, where the right index finger plays B and the right thumb plays high C.
The split hole is preferable in some regards as it completely eliminates the right thumb hole, allowing the right thumb to be used exclusively for supporting the instrument.
In these diagrams the fingerings are displayed vertically, with the lowest circle corresponding to the pinky finger and the highest to the index finger.

Because there are so few finger holes available for cross fingerings, accidentals using only these holes are poorly tuned. it is not uncommon to see additional split holes introduced. The ring finger hole—second from the bottom—is frequently a split to play F sharp.
Accidental fingerings of the second chamber are shown below, and the notes F and G are also included to give context to the split hole.

Third chamber
The third chamber, if one exists, provides a linear range extension above the second: D to A, although the exact range provided as well as fingerings vary. Some multichambers have a subhole on the second and/or third chamber.

Fourth chamber
The fourth chamber, if one is present, continues this pattern in exactly the same way.

Pacchioni system
The Pacchioni system places more emphasis on player convenience, and was originally called 'semplice', meaning 'simple' in Italian. Chambers still have linier fingering but are tuned with an overlap, allowing some notes to to be played on adjacent chambers.
The second chamber is tuned an octave above the first, and the fingerings are identical to the left hand of the first chamber, sounding the same notes an octave higher:

Doubles using the Pacchioni system often have a right thumb hole that allows them to play one more diatonic note, and on a C instrument extends the range to A. It is called the P* (P star) system. Triple Pacchioni system ocarinas do not have a right thumb hole, as the higher notes are provided by the third chamber.
Pacchioni system ocarinas also have subholes on higher chambers which are placed on the bottom of the ocarina between the voicings. They are covered by the left thumb, and used both to play a semitone below the chamber tonic.
Third chamber
The third chamber is tuned at a fourth above the second chamber, thus its lowest note is F. You may notice an oddity in the following fingering chart, as the note B is played as a cross fingering.
An unusual characteristic of the Pacchioni system is that all of the higher chambers have the same fingerings and sound the same intervals. Opening holes in sequence always produces the intervals: whole, whole, half, whole.

The next chart shows the chromatic fingerings and associated notes, which are the same across all of the higher chambers, as well as the lower notes of the first chamber. Note that the small hole added in this diagram represents the subhole.

Fourth chamber
The fourth chamber, if one is included, is tuned a fourth above the third chamber and extends the range by a fourth. It retains the same intervals as the others, and this means that the lowest note available on this chamber is an A♯/B♭, an oddity that arises from the tuning system. It is possible to play B natural using the subhole, although it is difficult to reach, and playing this note on the chamber below is preferable.

Harmonies between chambers
While the primary purpose of the transverse multichamber ocarina is to gain more range, it is sometimes possible to play harmonies between the chambers. The ability to do so depends a great deal on how a given instrument is tuned, and should not be assumed.
If you imagine a single chambered ocarina, there is a large difference in pressure between the highest and lowest notes due to the breath curve. The note G requires much more pressure than low C, and were you hypothetically able to blow two single chambered ocarinas at one, they would not be harmonious.
The same problem arises in multichamber ocarinas because each chamber has its own breath curve. It is only possible to tune one interval to sound harmonious between chambers. Fingering any other combination would mean one note would be overblown and sharp.
It is possible to prioritise harmony when making a multichamber ocarina by tuning all of the chambers to sound at a constant pressure, yet doing so would tend to cause the high notes of each chamber to sound thin and airy.
Considering an ocarina designed to sound a single interval, you'd have fingerings resembling the following. These are applicable to both the Asian and Pacchioni system, although the intervals sounded will be different, sounding a fifth or tritone on the Asian system, and a mix of major and minor thirds on Pacchioni.
Within each bar, the left group of notes shows the interval sounded on an Asian system ocarina, while the right group shows the notes sounded by the same fingering on Pacchioni system. These are annotated 'A' and 'P'.

As the Asian tuning system has a larger interval between the chambers (a tenth), it favours larger intervals like fifths, and the number of narrow intervals that can be played is pretty small. This is reversed in the Pacchioni system, as the smaller interval (an octave) favours smaller intervals like thirds.
Which, if any, of these harmonies are acceptably in tune is going to vary between ocarinas. Even if an ocarina is made with this in mind, some will be better tuned than others due to equal temperament being imperfect.
It is possible to correct for errors by partially venting or shading holes, or partially covering one of the windways with the edge of the lips.
Regarding shading, having a left hand subhole on the first chamber can be useful, allowing for small adjustments in the instrument's pitch.