Flute Acoustics – Part 3
Flute Acoustics – Part 3
How the air jet and pipe work together
To sum up the preceding sections: the bore of the flute has several resonances, which are approximately in the ratios of the harmonics, 1:2:3:4 etc, but successively more approximate with increasing frequency–we’ll see why below under frequency response. The air jet has its own natural frequency that depends on the speed and length of the jet. To oversimplify somewhat, the flute normally plays at the strongest bore resonance that is near the natural frequency of the jet. (We shall see below how register holes are used to weaken the lower resonance or resonances and thus make one of the higher resonances the strongest.)
When the flute is playing, the jet is oscillating at one particular frequency. But, especially if the vibration is large, as it is when playing loudly, it generates harmonics (see What is a sound spectrum?). For low notes, the first several harmonics are supported by standing waves. However for high notes, the resonances of the flute are no longer harmonic, so only a small number of harmonics—only one in the third and fourth octave are supported by resonances of the bore. Played loudly however, harmonics of the vibration are present in the spectra, as you can see by looking at the spectra for any note.
Opening tone holes
If you open the tone holes, starting from the far end, you make the pressure node move closer up the pipe – it’s rather like making the pipe shorter. On the Boehm flute, each opened tone hole raises the pitch by a semitone. After you open 4 holes on a C foot flute, as shown below, you have the fingering for E4, which is shown below. (Open a new window for E4).
For the moment, we can say an open tone hole is almost like a ‘short circuit’ to the outside air, so the first open tone hole acts approximately as though the flute were ‘sawn off’ near the location of the tone hole. We shall return to qualify these assumptions below when we discuss register holes and cross fingerings. (For the technically minded, we could continue the electrical analogy by saying that the open tone hole is actually more like a low value inductance, and so it behaves more like a short circuit at low frequencies than at high. We return to this point when discussing cut-off frequencies below.)
Holes can also serve as register holes. For instance, if you play C4 and then lift your left thumb, you are opening a hole halfway down the instrument. This makes the fundamental and the odd harmonics impossible, but hardly affects the even harmonics, which have a node there. So the flute ‘jumps up’ to C5 (2f1), and will also play C6, G6 etc. Here the register hole makes the played note (at least) one octave higher, because it is halfway along the working length of the flute and so permits the second harmonic of the fundamental C4. The example shown is not a standard fingering, but a register hole at half the length is used for the standard fingerings for D5 and others.
When the desired wavelength is short (i.e. for high notes) one can open a register hole at a different fraction of the length. For example, the fingering for D6 uses a register hole at approximately one third of the working length for G4, and so facilitates the third harmonic of G4 (and thus produces a note a twelfth higher than G4). The fingering for G6 also uses the working length for G4, but has a register hole about one quarter of the way along, and so facilitates the fourth harmonic.
One of the alternative fingerings for D#6 uses the working length for D#4 but has two register holes, at one quarter and one half the wavelength. Notes in the third octaves of all flutes rely heavily on using tone holes as register holes. Specific examples are explained on the pages for these notes. (See Flute Acoustics and choose a note above D#6.)
Acoustic impedance of the flute
The way in which the jet flows into and out of the flute depends upon the acoustic impedance at the embouchure hole, which is why we measure this quantity. The acoustic impedance is the ratio of the sound pressure to the oscillating air flow. (See Acoustic impedance for more detail.) If the impedance is low, air flows in and out readily and a loud sound can be produced. In fact, the resonances, which are the frequencies for which the acoustic impedance is very small, are so important that they ‘capture’ the behaviour of the air jet, and so the flute will play only at a frequency very close to a resonance. We discuss the acoustic impedance below, under Frequency response of the flute. There is further explanation on What is acoustic impedance and why is it important? and also a discussion of the impedance for C4.
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