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As a (mediocre) guitarist just starting to dabble with other instruments, the first several fingering diagrams for flutes, saxophones, etc make good intuitive sense to me because you're effectively just lengthening or shortening the resonance cavity, and coming from strings it's pretty easy to grasp that short=high and long=low.

After the first several notes, however, that intuition begins to break down and I really don't understand what's going on.

Take this fingering chart, for example:

Traditional Flute Fingering Chart

D through B make total sense to me, but I don't understand how the subsequent broken-looking patterns work AT ALL. I can see how leaving the first tone hole uncovered will produce a higher pitch, but I don't understand how any open fingering after the first open fingering actually changes the note. If my original intuitions about D-B are correct, it seems like the first open hole the wave encounters should be the only one that matters...

Are you setting up two regions of resonance that interfere with one another? Can this be thought of like harmonics on a string instrument? What does a single open fingering in the middle (C# or that high E) actually DO? Is there an intuitive way to wrap my brain around this or am I just going to have to memorize diagrams?

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There are a few things that closed holes below the first open tone hole can do.

In the case of that first C, what's happening is called shading. The tone holes of this instrument aren't as wide as the bore, so not all of the air can escape out of the first open tone hole. Some of it continues and goes out of the next open hole, and the result is that the resonant length is somewhere in between. These C fingerings are taking a C# and flattening it enough to be a serviceable C. Fingerings like this are called cross fingerings. Modern flutes have tone holes almost as large as the bore, and these kinds of fingerings don't work--they will affect the pitch slightly, but not anywhere near a half step.

For the second octave D, you have the right idea with the harmonics. Just as you can place your finger at the halfway point of a guitar string in order to prevent the string from vibrating there and force a higher harmonic to sound, you can open a hole halfway along the air column of a wind instrument to cause the same thing. This is also going on with the third octave D, E, and G. For the other notes in the second octave, and the third octave F, there's no hole where it would need to be, so you have to produce the overtone yourself by blowing higher across the embouchure hole.

Whether an open hole in the middle of the fingerings acts like the first open tone hole with shading, or as a harmonic-forcing vent, is a very tricky question involving complicated physics and tiny details of instrument construction.

  • Interesting! So cross fingerings shouldn't crop up on modern instruments with large tone holes, but will probably be a feature on, say, a penny whistle or recorder? Wait... is this what the smaller holes in open-hole flutes are for? – David Perry Sep 22 '17 at 18:06
  • @DavidPerry no, they're for being able to fit your fingers over. Open holed boehm flutes (big keys with little holes) allow a player to have some of the control and freedom (e.g. half holing) given by traditional keyless flutes, but still have large boehm toneholes when the key is not pressed. These large tone holes give you volume, and better intonation. Of course, it also radically changes the sound of the instrument. I personally love wooden flutes with smaller tone holes, but the Boehm flute (with keywork to allow very large toneholes) is undoubtedly a much more "powerful" instrument. – Some_Guy Sep 22 '17 at 18:13
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    With a small tonehole, the effect of the body below the tonehole is non-neglible, whereas with a large tonehole, there almost may as well be no more flute after the tonehole. To put it another way, a small tonehole is the same as a large tonehole being nearly all whe way covered! So the pitch is still lowered by the rest of the flute. Whereas with a large open tonehole there's not that much more open it can get other than just sawing the end of the flute off – Some_Guy Sep 22 '17 at 18:22
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    @Some_Guy even "sawing the end of the flute off" is not enough to create a "mathematically perfect" open end - to do that, you would need to add a trumpet-shaped bell to the open end as well. See my answer below. – user19146 Sep 22 '17 at 18:27
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    The math is similar but not identical. For a rocket, you are modelling the steady flow of gas through the device. For a musical instrument, you are modelling the unsteady flow (i.e. "the sound waves") superimposed on that steady flow - and the steady flow component can usually be ignored, since its pressure and velocity are small (of the order of a few inches of water gauge and a few feet per second) – user19146 Sep 22 '17 at 18:49
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To add just a small amount of physics to MattPutnam's answer:

A guitar string is relatively simple to understand, because assuming that both ends of the string are fixed is a very good approximation - though it is not completely correct, since in an acoustic guitar the bridge must have some movement to transfer the vibrations from the string to the body of the instrument. That is one reason why guitars need to be "set up" to get the best intonation - the position of the frets can't be calculated "exactly" from the distance between the bridge and nut.

In high school physics courses it is often taught that the open end of a pipe corresponds to the free end of a vibrating string, and the closed end to the fixed end of a string.

That is over-simplified for the open end. When doing the simplest experiments on the resonance of a pipe, you have to apply an "end correction" by assuming the open end is actually a bit longer (in proportion to its diameter) than its measured length, but that simple "end correction formula" is itself only an approximation to the real behaviour.

To get close to the simple notion of the "open end" of a pipe, the pipe has to end in a bell or flare, like a trumpet or trombone, but that brings in different mathematical complications because the diameter of the pipe is no longer constant.

At a simple level, a flute can be considered as a pipe "open at both ends" but in fact the blowing hole affects the pitch in a similar way to the finger holes - and the size of the effect depends on the geometry of the complete system of the flute plus the player, not just on the flute!

The finger holes are not all the same diameter (as is shown in the OP's photo) and there is a tradeoff between the size of the hole and its position along the length of the pipe. It should be clear that the holes are spaced more or less evenly along the length, which is very different from the fret positions for the first octave along a guitar string.

It is easy to demonstrate the effect of the finger hole size on simple flute-like instruments (e.g. penny whistle, recorder, or even a baroque keyless flute) by playing a note while bringing a finger close to a finger-hole but not actually touching the flute.

The closer the finger, the lower the pitch. In fact this was a standard playing technique for "pitch bending" notes in baroque flute music.

Take home message: the acoustics of pipes are more complicated than they might seem at first sight - and they are much more complicated than stretched strings!

  • I hadn't even noticed or considered the spacing difference between tone holes and frets, but that makes total sense. If the resonant frequency of the cavity is determined by some function where the pressure of the standing wave is highest at the closed end and approximately atmospheric pressure at all openings, it stands to reason that the size of those openings relative to the instrument's bore would have a substantial effect on those functions - and I had noticed that some of the holes on my new flute were larger than others, but the spacing difference totally made that click a lot better! – David Perry Sep 22 '17 at 18:29
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    A more accurate acoustic model of a flute replaces each finger hole with a short tube branching at right angles to the flute, where the length of the tube is related to the hole diameter. The standard wave pressure is (approximately!) atmospheric at the "open" end of each of these tubes, but not where each side-tube branches off the main bore of the instrument. – user19146 Sep 22 '17 at 18:34
  • That makes a lot of sense - the tube length is basically modeling the back-pressure applied by constricting the airflow if I'm intuiting the right things from this? – David Perry Sep 22 '17 at 18:36
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    To add another complication, there is also a significant length of closed pipe on the opposite side of the blowing-hole from the main instrument - and that closed pipe does not have a constant diameter bore. All these details matter! – user19146 Sep 22 '17 at 18:36
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    "The tube length is basically modeling the back-pressure" - that is basically correct, but even when the finger hole is closed there is still a cavity in the side of the bore, because the instrument tube has a finite thickness, and on metal flutes (which tend to have thinner tube walls than wooden ones) there may be an external "pad" to thicken it more where the hole is bored. – user19146 Sep 22 '17 at 18:39

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