How Ocarinas Work (Part 2)

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Ocarina PhysicsWelcome to part 2 of this course on ocarina physics and understanding the instrument from the ground up.

In part 1 of this article, we have discovered and solved the ocarina equation, allowing us to understand how ocarinas work in detail. It was the main mathematical part of this course. Now, let us find out what we can learn from the final results.

 

Section 3 – Understanding The Ocarina

From the equations in part 1, we can understand everything about how an ocarina needs to be made. First, we can make our formula for the pitch simpler and gain another insight:

It can be shown that the speed of sound c is given by this formula

    \[c = \sqrt{p_0 \kappa / \rho}\]

With this, we can rewrite the frequency this way

(1)   \begin{equation*} f = \frac{c}{2 \pi} \sqrt{\frac{A}{V_0  l}} \end{equation*}

While the loudness of the ocarina is given by

(2)   \begin{equation*} L = \frac{v}{\omega} \end{equation*}

What can we learn from these two formulas?

 

1. If you increase the open area, the pitch gets higher

As you can see, the area A of the mouth hole is in the enumerator of (1). So if you increase the size of the open hole, the pitch gets higher.

This is why the ocarina plays higher notes when you open more finger holes – all you really do is increase the overall area that is open!

It does not matter where the holes are on the body of the ocarina; only the open area matters. What happens is that the air can now escape and oscillate through all these other holes, which changes how fast it moves and thus the pitch.

 

2. Different sizes produce ocarinas of a different voice.

In the same way, the volume V_0 of the ocarina is in the denominator. This means if you increase the size of the instrument, you divide by a larger number and hence the pitch goes down.

Large ocarinas have a low pitch, while small ocarinas have a high one. This is why sopranos are small and basses are big, with tenors somewhere in between.

Triple BassPhysically, this is because the sound waves have to fit inside the instrument – only small wavelengths (high frequency) fit in a small chamber.

Multi chamber ocarinas like the one on the right usually have 2 or 3 separate chambers of different sizes, allowing for high and low notes to be played by the same instrument.

It doesn’t matter what shape the instrument is, giving ocarina makers the freedom for very artistic designs. You can have all kinds of shapes and sizes in a variety I haven’t seen in any other instrument. It’s one of the reasons why the ocarina is special.

Compare these two gorgeous gemshorns, which differ only in size.

 

3. Different windway lengths produce ocarinas of a different voice.

pungiI have never seen anybody talk about this before, which is strange. The length of the windway is also in the denominator!
If you simply make the windway longer, you could have a soprano sized ocarina with the voice of a bass!

Such an instrument might perhaps look like a pungi, which is the flute played by a snake charmer. See the picture on the left. It’s a whole design world yet untapped for ocarinas. Why is nobody doing this? Maybe I just invented it. Yay!

 

4. Environmental conditions change the pitch

Ocarinas In SnowHave you ever noticed the ocarina changes in pitch when it’s cold or when you are at high altitudes?

This has to do with air density. When it’s cold, the air contracts and occupies a smaller volume, thus increasing in density. Denser air has a smaller speed of sound, and as you can see from our formula (2), this means the pitch goes down in cold environments. In the same way, the air expands and the pitch goes up in warm environments.

In addition to this, the body of the ocarina contracts and expands as well, but nowhere near as much as the air does. We can neglect the difference in volume.
Note: Don’t take a ceramic/clay ocarina out in cold weather, because the change in temperature (and thus change in volume) can easily crack them.

What about high altitudes? Air pressure decreases as you go higher in the atmosphere, which decreases the speed of sound and makes the ocarina sound lower. It also gets colder the higher you go, so ocarinas will always sound different on mountains than in the valley.

Overall, this poses a problem for ocarina makers. If you make a professional ocarina, you need to ensure you have standard temperature and pressure conditions in the room where you tune the instrument. Once it is made, it can’t be changed anymore. And if an ocarina goes to somebody in Australia or Alaska, or if that person lives at high altitude, it will sound wrong most of the time.

 

5. The loudness of an ocarina changes with each note

Another thing I’ve never seen anybody discuss before has to do with the loudness of the ocarina. As we have seen, it is

    \[L = \frac{v}{\omega}\]

where v is the air velocity. So the harder you blow, the louder the ocarina becomes.

As you go to higher notes and open more and more holes, air can escape the instrument easier and easier. You need to blow harder into the ocarina to build up the necessary pressure, otherwise the air will simply escape through all the holes and make a windy sound. This doesn’t only make higher notes more difficult to play, but also louder.

Although the angular frequency \omega is in the denominator of our formula and works against the increase in blow strength, the sound still gets louder overall.

Since the frequency depends on the speed of sound, all of the environmental conditions in 4. have an effect on this as well. An ocarina gets more quiet with an increase in temperature or a decrease in pressure. So on mountains or in winter, it’s not as loud as in the valley or in summer. In space, the pressure would be zero, so the sound would disappear entirely.

 

Why does playing a higher note change the loudness?

As I mentioned in point 5, ocarinas get more quiet with higher frequency, or higher note, even if you blow at the same strength. Why does this happen?

The air you blow inside the instrument has a certain amount of kinetic energy. This energy is then used to produce the sound we’re hearing.

A sound wave carries more energy with higher frequency or higher amplitude (loudness). So if you play a higher note, more energy is used up to produce the higher frequency, which means less energy is left over for the loudness. Opening a hole forces the system to produce a higher frequency, so it also forces it to be more quiet. In other words, the reason is conservation of energy.

Ocarina Physics Experiment

I have done a few sound experiments and recorded this sample, where I play two notes after each other to demonstrate how the amplitude changes with pitch. It was difficult to blow at the same strength – ocarinas are very delicate on blow strength.
Physics works :)

 

A Word About Hole Sizes

Ocarina HolesDepending on the notes you want an ocarina to play, you have to make the hole sizes so that it works out.

If you open no finger hole, it plays the fundamental note. Then you open one hole and add its area to the hole of the mouth piece, thus causing the ocarina to play a higher note.

The hole sizes are different, because the area A is under a square root – if you double the open area, you don’t get a pitch that’s twice as high as before.

Also, you want the ocarina to play more notes than it has holes, so you get to different notes by hole combinations. In order to make it all work out, no two holes can ever have the same area.

 

A Final Note

If you have gone through all of this with me, I commend you!

I truly think understanding your instrument is a key to really appreciating and working with it. A true master should always do this, especially when the explanation is readily available, like I have done it for you here.

May the music be with you,

Allen

 

Ocarina Physics

13 Responses to “How Ocarinas Work (Part 2)”

  1. Dennis Silverman says:

    I think in sec. 4 you meant to say that the speed of sound increases with higher density, therefore increasing the frequency. Thanks for explaining the physics of the ocarina.
    Dennis Silverman
    Physics and Astronomy
    UC Irvine

  2. Maria Scutari says:

    The length of the windway doesn’t affect the ocarina’s frequency. If you take some plastic tubing or something to virtually lengthen the windway and try blowing into it, you will find that the frequency remains the same. So, I think that the final formula above is wrong.

    • Allen says:

      No, the sound is still produced at the first hole, where the air hits the labium (edge). For your experiment, you’d need to move the labium all the way up to the beginning of the tube :)

  3. Matthew says:

    Thanks so much! I recently got into the ocarina and was wondering exactly how it worked. It took me a while to find this explanation, but it was worth it. I appreciate all the math and work that went into it.

  4. Lemon says:

    Would you give a rundown on what each variable in your equation stands for? I got lost in the math component where you combined the two equations and don’t know what some of the variables stand for now.

    • Allen says:

      p0 = air pressure
      A = area of all open holes together
      c = speed of sound
      f = frequency of the sound produced by the ocarina
      V0 = volume of the ocarina interior
      l = length of the ocarina mouth piece (bottle neck)
      rho = air density
      k = 1.402 (for dry air)

  5. Lemon says:

    Thanks, another quick question. I swapped it around to solve for A (and filling in f with a chromatic -> Hz chart). And I’m getting huge A values (19cm for 55Hz) using this data:
    V0 * l * (2 * f * pi)^2 = A

    V0 = 3.9m^3 (13x6x5cm)
    c = 340.29 m/s
    l = 3cm (0.03m)
    f = variable (from 55 Hz -> 3520 Hz [A1 -> A7])

    For example I am getting 0.12066m^2 for 55Hz. I feel like this is wrong as that is 12cm^2 (or a 19cm radius circle).

    Do you happen to know what I am doing wrong? I feel like it has to do with units, but I’ve been messing around with it for a few hours and am no longer confident that I’ll recognize the answer when I run across it.

    Thank you so much for your help.

    – Lemon

  6. Ben says:

    Thanks so much for this!

    I have a question:
    There are ocarinas with a (relatively) constant breath slope, such as those made by Hind and Mountain Ocarinas. These don’t require you to blow harder at the higher notes, with the tradeoff that the high notes aren’t really louder than the lower ones (this is actually nice, if less expressive, because sometimes you want to play a high note softly). According to the formulas you derived, it seems like such a thing shouldn’t be possible. So how is it done?

  7. Pierre P says:

    Thank you very much for all these explanations. I am making a project with my son who really likes playing ocarina. I would like to make an ocarina on my CNC. Yesterday I was designing the instrument with Solid Works ( parametric design software) and was looking for all this information you kindly published. Would you have an excel spreadsheet that will drive the equations and values to make the instrument sound right. I have build and designed aircraft all my life, so I’m no music specialist but I do understand the need of making it right. So if you can help with the spreadsheet this will help me a lot in making this project. I plan to make the 12 hole ocarina out of Iroko wood. With the cnc I can be very tight on volumes and values. I was looking to make a tenor range instrument.

    I really did enjoy your article!

  8. Nina says:

    THANKS SO MUCH,

  9. Name says:

    I want to verify I understood you right: When you talk about “mouth hole”, do you mean the sound producing hole called “window” and not the end of the airway inside of the mouth piece?

  10. Jordan says:

    Hey I’ve done some reasearch on how one goes to a higher range on the ocarina. I can go to the the higher pitches of my recorder by simply uncovering 1/4of the bottom hole of it. Surely there would be some way to switch to the higher notes on my ocarina. If you know, please tell me.

    Thanks, and happy new year :)

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