| Copyright© by Joe Wolfe. | ||
When we were playing two recorders, we heard a low buzzing sound, sometimes making a chord with our notes. What is that?
It is called the Tartini tone. It is most often noticeable when the two notes being played are sustained, about equally loud, not too low in pitch (say treble clef and above) and moderately loud. I've put a section about it on a page called Interference beats and Tartini tones where you can hear sound files made with pure sine waves that show the effect quite well. If you listen for it, you can use it to tune your chords: the tuning you will arrive at is Just Intonation, and not Equal Temperament. In sustained chords, the former usually sounds better.
The effect may be due to nonlinear effects (heterodyning) in the ear, or perhaps due to higher processing in the auditory cortex, or (more likely, I think) both of these.
The strong Tartini tone that we sometimes hear is usually generated in the ear. In the case of the two recorders (above), there is no vibration in the air at the difference frequency. However, when two notes are played simultaneously on the same instrument -- as played here on a violin by John McLennan -- it is possible to produce a difference tone via small non-linearities in the instrument. This is explained in more detail in Interference beats and Tartini tones.
What are heterodyne and comination tones? How do they relate to beats and difference tones? And where do they come from?
Nonlinear systems, in response to two signals with frequencies f and g, produce heterodyne or combination signals, with frequencies mf +/- ng, where m and n are integers. To me, the two terms have the same meaning, although heterodyne is used more frequently in radio and TV.
One of the heterodyne terms, that with frequency f-g, is called the difference tone in acoustics and the Tartini tone in music. (I'm choosing f > g.) If (f-g) is rather smaller than f or g, then the phenomenon is called interference beats or just beats: the combined signal is an oscillation with frequency (f+g)/2, amplitude modulated by the beat frequency (f-g). When the beat frequency is less than a dozen Hz or so, it can be clearly heard as a variation in amplitude, and is commonly used to tune instruments. (Beats at mf-ng are also thus used.) See Interference beats and Tartini tones for examples and sound files.
Where do such terms come from?
Non-linear systems inherently produce them. I shan't go through the algebra here
(though it's not difficult) but one can expand a non linear response using a
Taylor series. The series includes higher order terms, which become more
important as the signals get larger. Simple trigonometric identities substituted
in the product terms give the heterodyne terms.
In electronics, the nonlinear response of diodes was traditionally used for modulation and demodulation (producing heterodyne terms).
In music, nonlinearities occur in several places. The basilar membrane in the inner ear is nonlinear, and it is plausible that Tartini tones are generated here. I have read, however, that Tartini tones can be heard when one note is input to one ear and another to the other, via headphones. I've not experienced this myself but, if it happens, it suggests that they are produced in the neural processing. (Neural processing is highly nonlinear.)
They may also be produced in a nonlinear source. For instance, when they are produced by double stopping on a string instrument, one would expect the two notes to interact through the hair of the bow and through the bridge. The bow-string interaction is highly nonlinear, so this can produce heterodyne terms. This is shown in the example given above. When wind players produce multiphonics, they also produce heterodyne signals. To take one particular example that we have studied in detail: when a didjeridu player vocalises at a pitch different from that of the drone he is playing, strong heterodyne terms are produced.
Finally, it is possible to produce heterodyne terms in a microphone. No microphone is perfectly linear and so sufficiently strong signals could, in principle, produced heterodyne terms in the microphone itself.
Tartini tones sound unusual because one doesn't have a sense of direction associated with them. One can imagine that they are being produced 'inside one's own head', which may in fact be true. However, it's worth noting that, even in a slightly reverberant space, the same may be said of a pure tone. Listen to the sound files on Interference beats and Tartini tones and see if you agree.
What is the frequency, and how does it relate to pitch? Where do harmonics fit in? Does the frequency of the lowest harmonic determine the pitch?
The frequency f is the number of vibrations per unit time. For example, when a tuning fork sounds the note A4, its tines vibrate 440 times per second. Its frequency f is 440 cycles per second, which is usually written as 440 Hertz or 440 Hz. (The unit of frequency is named for Heinrich Hertz, a pioneer of electromagnetic radiation.) The pitch of a musical note is determined by the frequency of its sound wave.
Pitch differences depend on the ratio of frequencies. If the frequency is doubled, the pitch rises by an octave, independent of starting frequency. If it increases by a factor of 3/2, the pitch rises by a fifth. (It follows that pitch is proportional to the logarithm of the frequency.)
The duration of each cycle of a vibration is called its period, T. If there are f cycles in one second, then each cycle must last 1/f seconds. In other words, the frequency is the reciprocal of period, or f = 1/T. So, for the note A4 at 440 Hz, the period is 2.27 milliseconds.
To relate pitch to frequency and back, see notes, frequencies and MIDI numbers. We give an introduction to pitch, loudness and timbre in Quantifying Sound.
Musical tones usually comprise vibrations that are periodic. Such tones may be considered as the sum of pure tones from the harmonic series. So a note with frequency f usually contains also components with frequencies 2f, 3f, 4f etc. These components are called harmonics and the component with frequency f, the highest common factor, is called the fundamental. This is discussed in more detail in What is a sound spectrum?
We usually hear a pitch corresponding to the fundamental. However, we hear that pitch even if the fundamental is absent -- called the missing fundamental. For periodic tones, our sense of pitch is determined by the spacing of harmoncs in the region of several hundred Hz. So, for example, a small loudspeaker is very inefficient at 40 Hz. Consequently, when the sound of a bass playing its lowest note (E1 at 40 Hz) is played on such a speaker, the radiated sound has strong high harmonics, but almost no fundamental. Nevertheless, the pitch we hear is E1. The radiated sound might include the tenth, eleventh, twelfth etc harmonics at 400, 440, 480 Hz etc, and the spacing between these is 40 Hz.
What is an undertone in flute playing and what causes it?
Sometimes when a flutist plays a high note, one hears a faint note at a lower pitch. Normally there is no exact harmonic relation between the two. This is what I call an undertone. (This is not to be confused with Tartini tones or other combination tones, in which two notes, often from different instruments, interact.)
The undertone is a special case of a multiphonic, usually produced accidentally. Let's take a simple example. Play the note D6, but play it with a relatively large opening between the lips. Then, still with large lip opening, reduce the air speed gradually. When the jet is slow enough, the note will drop down to C5 with a rather breathy tone. However, on the way you will pass through the multiphonic C5&D6. By adjusting the jet speed, you can vary the proportions of the two notes: when they are about equally loud, you have a standard multiphonic. (If you have never played multiphonics before, this is an easy one with which to start, although it's not very interesting. See multiphonic fingerings for more.)
Let's now see why this works by consulting the database on flute acoustics. The fingering for D6 is a bit like that for G4, except that you raise your left index finger and open one of the small holes on the flute. The harmonics of G4 are G4, G5, D6 etc, and all can be played on the G4 fingering. When you open the index finger key, you create a register hole that both weakens substantially and mistunes the second resonance, the one that supports G5, so that G5 becomes unplayable. This is the object of a register hole: you don't want your D6 dropping down to G5 when you decrescendo. The lowest resonance is also very much changed: it is weakened a little, and its frequency is raised. You can think of this lowest resonance as a cross fingering for C5.
To understand this, open a new window for G4 and another for D6. The impedance curves tell us (approximately) which notes the flute can play for any fingering. The flute will usually play a note where it has a (sufficiently strong) resonance, and the resonances are the deep minima in the curves of impedance. The first three of these for G4 are at about 400, 800 and 1,200 Hz, and they play G4, G5 and G6. They also support the first three harmonics of G4, so these frequency components are consequently strong in the sound spectrum of the note played. The first three minima for D6 show a reasonably strong minimum for C5 (near 520 Hz), a shallow (unplayable) minima above that (near 900 Hz), then the strong minimum for D6 at 1.2 kHz. There are no minima to support harmonics of C5, which is why this (and other crossfingered notes) sounds darker and weaker than a normal fingering. (In fact, this is the normal fingering for C5 on a baroque flute and explains why that note is darker than its neighbours.) If you found this paragraph heavy going but are still keen to understand it all, you might want to read our Introduction to flute acoustics. If you would like to see this particular multiphonic discussed in technical detail, download this this paper.
In the sound spectrum for D6, you will see the fundamental of D6 (near 1.2 kHz), which corresponds with its resonance in the impedance graph, but no trace of undertone: the impedance curve shows a resonance near 520 Hz (C5) but there is no signal in the sound spectrum at this frequency. No surprise: the recording is of a professional flutist. However, you can now see how a beginning flutist, with a wide lip opening and not much breath control, might play D6 with an undertone: s/he is playing a multiphonic with a weak lower note.
In the fourth octave, it is much easier to produce an undertone, because the required jet speed is very high, and so it's easy to blow too slowly! For instance, if you look at the sound file for the note F7 (open a new window for F7), there are a few things to observe. First, there is more breath sound because the required jet speed is much higher. Second, the resonances are much weaker (this is because there is more 'friction' loss between the air and the flute tube at high frequency). You will also observe in the sound spectrum that, as well as the strong harmonics at F7 and its harmonic F8 (about 2.8 and 5.6 kHz), there is a component a little above 2 kHz (near C7). Personally, I cannot hear this pitch but, if I could, that would be an undertone. When less accomplished flutists play F7 with this fingering, they are more likely to produce noticeable undertones.
Finally, we should note that Geoff recorded these sound files before we had built the virtual flute (TVF). If we asked him to record F7 today, he would use one of the fingerings that TVF recommends, some of which are substantially easier to play and so yield less breath sound and less chance of an undertone.