# What is the math behind how keys/holes on woodwind instruments affect the pitch?

To make this question more specific, I'll provide an example of what I mean.

Let's say you have an instrument that's just a tube you blow into, and it has one hole exactly halfway down the tube (I know that's not an "instrument" but I'm speaking theoretically). If one blows into the instrument without covering the hole, let's say it produces a B flat 4. If one were to cover the hole in the center and blow, how would that affect the sound? Would it change the Bb's octave from 4 to 3 or 5 since the hole is exactly halfway down the tube?

https://en.wikipedia.org/wiki/Music_and_mathematics

In this article under "Frequency and Harmony" there is a chart with frequencies and stuff, but I don't understand it, and how it would apply to a wind instruments. I understand the "Common Name" column and the "Example name Hz", but the other ones confuse me greatly. I'm not sure if this would help me understand what I asked more, but it seems like it.

Hopefully this makes some sort of sense.

• Just to be clear: this theory applies to the woodwinds, where a reed or a "pocket" (in the case of the flute) is the source of vibrations. Brass instruments, where the source (lips) is essentially a square wave, sound awful if you try to change the pitch via holes in the tube. – Carl Witthoft Nov 20 '15 at 13:12
• @Carl Wtthoft look at the question. Notice that it says "woodwind" – Sam Nov 20 '15 at 15:26
• Yes, Sam, but there will be lots of people reading this question and it's worth cautioning that the answers do no apply universally. – Carl Witthoft Nov 20 '15 at 16:30
• @CarlWitthoft It's completely fine. I shouldn't have to specify every single little thing about this question. As long as it's understandable. Chris Erwin was kind enough to answer my thoroughly, proving that he understood it. This question is more so about sound waves and instruments, not about how "good" a brass instrument's sound is. – Sam Nov 20 '15 at 20:17

Covering the hole would bring the note down to Bb 3. With the hole covered (or no hole at all) you would get a standing wave with a wavelength of twice the length of the instrument as the fundamental pitch.

The reason for this is because the instrument is a tube capable of holding pressure in the middle, but not capable of holding pressure at either end (let's assume a flute with open ends). When you visualize a wave, remember that the wavelength is from 0, up to the highest peak, back to zero, down to the lowest peak, and back to zero again. Line two of those zeros up at either end of the tube and you now have half the wave length inside the instrument, and this is the longest wave that fits in the instrument.

When you open the hole at exactly the center, think about a wave that has 0 at either end and 0 at the hole. The longest wave capable of this has a wavelength of 1L, where L is the length of the instrument.

Basically, whatever combination of holes is open, the only notes that will resonate are ones where the wavelengths work out so the pressure nodes line up at the holes.

Now let's go back to having all the holes closed. Don't forget that you can blow harder on a flute (and more importantly, brass instruments) to raise the frequency. The only notes that will resonate are those that have nodes (0 pressure) at either end of the instrument. Since Bb3 is the fundamental pitch, you could potentially play Bb4, F5, Bb5, D6, F6, Ab6, and so on.

That's where we get into that chart on the wikipedia page. It shows A2 having a frequency of 110Hz. To get A3, we double that (220). To get A4, we need to double THAT, right? so 440. But what about 3x 110 (330)? That's where you get E4.

The "ratio within octave" means the ratio between the note given and the A below it (since that chart is based around A). So, in regards to frequency, E4 is 3x A2, but 1.5x A3. C#5 is 5x A2 (the fundamental) but 1.25x the closest A, A4. There are 1200 cents per octave (100 per note), so that column is just giving how many out of 1200 increments it is above the A below it.

That's all confusing, but think about tuning a piano. You start by tuning A440. Now you play that A440 with the A below it. If it's out of tune, the frequency won't be exactly double. If it was, the waves would line up and it would sound great. When the waves don't quite line up, you get 'beating' where the waves cycle between adding their sound together and slightly canceling each other out. You tweak the A3 until the beats stop, meaning it's at exactly 220.

You can then tune the E to that. Since E4 is 1.5x A3, the waves also line up when they're in tune. When they're out of tune, you hear the beats, and you can adjust until they line up. That's why that 1.5x is important.

Which brings us to the circle of 5ths. One you tune that E4 to A3, you can tune all the Es. And then you can tune the Bs because they're 1.5x the nearest E. Then the F#s, and so on. 12 notes later and you're back at A and you've tuned the piano. I won't get into temperament though.

Here's a great article on flute accoustics that has helpful images: https://newt.phys.unsw.edu.au/jw/fluteacoustics.html

• Thank you so much! It makes sense now, but I cannot open the article because my computer, and my cell phone also, block it. – Sam Nov 19 '15 at 13:53
• Not sure why you can't open the article. Maybe the https version is causing issues? Try this link: newt.phys.unsw.edu.au/jw/fluteacoustics.html – Chris Erwin Nov 19 '15 at 19:23
• Thank you! However, I want to ask you one more thing. From what the article says, the pitch depends on the frequency of a wave from the end of the instrument to the first open hole (I'm probably wrong). What if you were to have a sound wave in between two open holes? What "pitch" would play? Sorry if this doesn't make sense. – Sam Nov 20 '15 at 20:21
• Totally missed your comment and you'll probably never see this, but the short answer is that, if you have two holes, and two open ends, the instrument will play the note where each end and the two holes line up on pressure nodes (0 pressure). It's not just the distance between the two holes, it's the least common multiple of the distance between the end and the first hole, the first hole and the second hole, and the second hole and the other end. – Chris Erwin Aug 10 '16 at 20:52