by Chris Clayton
(Chris was a student at M.I.T. when he wrote this paper for an acoustics course)
There exists in the field of acoustics a specific phenomenon known as the Wolf Tone. The Wolf Tone exists due to imperfections in the construction of musical instruments that leads to "intonation difficulties." These imperfections work with resonant properties to create a most unpleasant sound on even the finest instruments. Wolf Tones have been known to occur on numerous instruments, but are most prevalent on the stringed instruments: violin, viola, and violoncello.
To a musician, a Wolf Tone is "a wildly fluctuating and uncontrollable tone" that "deviates in tuning or loudness" from a given note on a major scale . This is the result of an eccentricity of resonance which either enhances or damps the note in question. A Wolf Tone is essentially a potent or sharply defined resonance frequency that is just slightly sharp or flat in relation to said note. The Wolf Tone sounds like a strange mixture of beats because of the periodic diminution and augmentation of the odd harmonics of a string. When the string is bowed, its oscilloscope reading would be a neat, even sawtooth wave.
Under Wolf Tone conditions, the oscilloscope would display a considerably less elegant wave form which oscillates periodically between the initial sawtooth and a sawtooth of a frequency twice that of the initial sawtooth. The interstitial spaces between these stages see the introduction of the odd harmonics (See Figure 1). The Wolf Tone is often found at the major sixth or seventh above the open G on a cello, or an octave above the open G on a poorly made violin.
In order to understand Wolf Tones, one needs to have a basic understanding of coupled oscillators. An excellent example of a coupled oscillator is apparent in the motion of two pendulums attached by an elastic cord. If the pendulums are set in motion at roughly the same frequency, they will eventually fall out of step with each other. The result will be a gradual and perpetual transfer of kinetic energy between the two pendulums through the medium of the cord.
As more and more energy is exchanged, one of the pendulums will appear to stop moving until its counterpart passes some energy back to it. "If the influence of each on the other is small, the coupling is said to be 'loose,' and the system is unchanged. . . if the two systems interact considerably, the coupling is said to be 'tight.' "
An interesting feature to note is that this coupled relationship between two resonant frequencies has the ability to actually split the fundamental of the driven oscillator into two frequencies, one slightly above and the other slightly below the fundamental. Essentially, where each system would have one resonant peak, the coupled system has two which compete for dominance.
The Wolf Tone is the audible product of the coupled oscillation of the string and body of a stringed instrument. As the string is bowed, energy passes through the bridge into the body of the instrument. There are two resonances involved here, one belonging to the string and one to the body of the instrument. In order for this coupled oscillation process to occur, it is necessary for the two resonances to be very near equal in frequency (a situation primarily dictated by the main wood resonance frequency), thus bringing Helmholtz oscillation into the problem.
Two types of Helmholtz oscillation are involved, one being the slip-stick process of bowing the instrument and the other being the bottle/ocarina effect of the main body of the instrument. The Wolf Tone disrupts the normal bowing pattern of the instrument and excites the air resonance within the body of the instrument. When this occurs, the bridge of the instrument will yield too much, because "the impedance of the bridge is no longer [sufficiently greater] than the string's for the string to resonate well. Its resonance is drained away too quickly."
Impedance here is a measure of the vibrational resistance of an object, or the tendency of that object to remain still in the face of a driving force. As a result of all this, the musician loses control of the instrument. If the two resonances are not very "active," they will behave in the manner of the coupled oscillator described above. This shifting of energy results in a warbling sound due to the beat frequency between the two resonant frequencies. If both resonances are active, then "it can become impossible to put in enough energy with the bow to sustain both," causing the production of a "growly" tone as the resonances take turns "shutting down." If one of the resonances is inaudible, it may accumulate energy and then release it in the form of an audible mode, which "can bleat like a sheep." If the resonances are on octaves, the pitch will jump up and down from one to the other.
Wolf Tones can come and go with changes in humidity. Sound travels faster in moist air, so the resonant frequencies of the air go up. The wood resonances drop as the wood gets heavier and less stiff. A Wolf Tone becomes more entrenched the more it is played, but it can sometimes vanish if the instrument is left to sit for long periods of time. This idleness allows the wood to stiffen back up again, and the instrument will require "playing in" before the resonant conditions will be back to normal.
Decades of experience with Wolf Tones have provided string musicians with several options for removing the Wolf Tone. The easiest of these is to replace the culprit string with a string of lighter gauge. This makes it possible to play notes of the offending frequency at different positions on the neck of the instrument, reducing the chance of touching off the Wolf Tone. A lighter string also has less mass, and will have different resonant frequencies from the wood of the body.
A second solution involves bowing with greater pressure on the string, a practice that can prevent the string from running away with the body. Increasing the bowing pressure allows the higher modes of the string to be excited, thus regulating the detuned fundamental.
Third, one could purchase what is called a "Wolf Mute," and mount it on the short section of the string located below the bridge of the instrument, effectively damping the offending mode of resonance by forcing it to sound in that section of the string only. A Wolf Mute functions by adding to the mass of the string anchor in the vicinity of the offending string. This has the effect of raising the impedance of the bridge back to an acceptable level, and also serves to stiffen the sound board so that it cannot resonate as dramatically.
As a fourth option, a German company called Gewa "has made a marvelous damper which is glued to the inside of the cello top [plate] just below the bass F-hole that can rid the cello of the Wolf completely." Or, more economically, the instrument can be gently squeezed laterally while playing. This compressive method dampens the vibration of the wood of the body, and alters very slightly the shape and volume of the air cavity inside the body (which alters the resonant frequencies!). This is most easily accomplished on a cello, which is squeezed by the cellist's knees.
Most luthiers follow the following rules of thumb when constructing a cello:
1) Avoid too thin a top plate, particularly in the corners on the bass side
of the instrument.
Any of these methods, except for increased bowing pressure and changing strings, will decrease the loudness, or sonority, of the instrument just slightly. Fortunately for string players and luthiers, "the very strength of the troublesome wolf-note resonance in the [sound box] also makes it an extremely narrow one." Effectively, the width of the Wolf Tone resonance is slightly less than a semitone. That basically means that a little bit of selective tuning will actually be able to hide the Wolf Tone from normally played notes.
The Wolf Tone has been an interesting subject to study. It is not only due to the resonances of a string, but also requires the wood of the body to resonate at a frequency near that of the string. The Wolf Tone operates on the fundamental physical concept of the coupled oscillator. This basic principle is the underlying model for the periodic nature of the Wolf Tone. Visually, the oscillation cycles from a normal sawtooth pattern to a random pattern to a sawtooth of twice the frequency of the original, and back again.
The Wolf Tone is a periodic oscillation between frequencies which is caused by the coupled periodic oscillation of two systems. Despite its intriguing nature, however, most of the efforts directed towards learning about the Wolf Tone are devoted to its eradication. I believe that empirical tinkering with an intuitive grasp of acoustics has managed to all but eliminate the Wolf Tone in modern instruments of significant quality.
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