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Notes
- 1.
For a detailed physical and mathematical study of musical instruments with abundant illustrations and literature references, see the treatise by Fletcher and Rossing (1998).
- 2.
It is this energy “leak rate” through the fixed points (mainly the bridge) that is transformed into sound power in a string instrument's resonance body.
- 3.
More precisely, overtones are the higher frequency components of a complex vibration, regardless of whether their frequencies are integer multiples of the fundamental frequency or not, that is regardless of whether relation (4.2) applies.
- 4.
This does not greatly affect the pitch of the ensuing complex tone (Appendix II); but it does affect tuning of the piano in the treble and bass register (when tuning is done by octaves).
- 5.
The main objective of the pedal mechanism of the piano is based on this phenomenon: pressing the pedal lifts all dampers, and the strings are let free to vibrate by resonance. When one given note is sounded, all those strings will be induced to vibrate that belong to the series of harmonics of that note.
- 6.
There is a somewhat complicated physical reason for loudness-timbre coupling. As mentioned earlier (p. 122) the duration of the hammer-string contact influences the relative proportion of upper harmonic modes, with a longer contact leading to fewer upper modes. The duration of contact, in turn, depends on the stiffness of the felt on the hammer's head: a softer hammer stays longer in contact with the string than a harder one (for equal impact speed). But there is a remarkable fact (Hall and Askenfelt, 1988): the effective stiffness of a given hammer depends on the impact velocity with which the hammer hits the string, with a greater effective stiffness for higher impact velocities and vice versa (this is called a nonlinear behavior of stiffness). As a consequence of all this, hitting a piano key harder will not only increase the amplitude of the string oscillation (louder tone) but shorten the contact time and thus automatically increase the proportion of upper harmonics (brighter timbre).
- 7.
For a detailed discussion of the bowing of strings, see Schelleng (1973) and Cremer (1984).
- 8.
The function of the mute, when it is applied to the bridge of a string instrument, is to decrease this energy transfer for the higher frequency components, thus altering the quality of the resulting tone.
- 9.
Neither does a real string, of finite thickness, have sharp, discrete modes of oscillation.
- 10.
The position (in frequency) and the shape of this particular resonance peak is of capital importance for the quality of a string instrument (Hutchins and Fielding, 1968). See also p. 157.
- 11.
Vortices are even formed in absence of the edge, provided the slit S is small enough and the velocity v high enough (see Fletcher and Rossing, 1998). This represents the basic physics of the human whistle where slit size (lip opening) and velocity of the air stream (blowing pressure) determine the fundamental frequency.
- 12.
By the difference in dynamic pressure (not static pressure) on both sides of the reed—the same effect that keeps a flying aircraft aloft!
- 13.
Only if R is expressed in decibels.
- 14.
In real instruments, clarinet-type resonance curves (a) also display a discrepancy from harmonicity (Backus, 1974).
- 15.
Weighted with the corresponding spectral intensity values I 1, I 2, I 3, …
- 16.
It is important to understand clearly this “automatic adjustment process” for another reason: it is remarkably similar to one that has been proposed for the pattern recognition mechanism in the central pitch processor, where the set of shifting harmonics f 1, 2f 1, 3f 1, … . is called the “template” (see Secs. 2.9 and 4.8, and Appendix II).
- 17.
For a mathematical treatment of air pressure oscillations in cylindrical pipes see Fletcher and Rossing (1998).
- 18.
For almost any kind of shape of rooms and positions of the source therein, there may be regions practically inaccessible to sound waves emitted from S (blind spots), or regions into which sound waves are focused (e.g., the focal points in elliptic enclosures).
- 19.
It is assumed here that I m represents the diffuse, omnidirectional sound energy flow.
- 20.
Italian organs in the Baroque have preserved the basic timbre control through the inclusion of many mutation (upper harmonic) stops; much later, the sounds of the first electronic Hammond organs were based entirely on the possibility of a separate intensity control of individual, electronically generated, harmonics.
- 21.
We strongly recommend that the reader review Sect. 2.9.
- 22.
It should be pointed out that what is shown in Fig. 2.25(b) are the mathematical predictions for a cochlear model which does not include a sharpening mechanism mediated by the outer hair cell motility (Sect. 3.6).
- 23.
The fact that the seventh harmonic is a dissonance has been worrying musicians for a long time. This worry is unfounded though: the seventh harmonic is extremely difficult to be singled out, even in constantly sounding tones from musical instruments. This is now recognized in the fact that some large modern organs do have a 1 1/7' mutation stop sounding the seventh harmonic of the written note, which gives a very particular timbre when used judiciously with other stops, but does not disturb in any way the “smoothness” of the sound.
- 24.
Although there are about 15 critical bands in the musically relevant frequency range the intensities of which ought to be specified in order to determine the spectrum, a study of vowel identification (Klein et al., 1970) indicates that only four independent intensity parameters (each one a specific linear combination of the intensities in all critical bands) are sufficient to specify a complex tone within the “timbre resolution capability” of the auditory system.
- 25.
Although this is a book on sound, it easier to visualize in a picture the information routes of the visual system. The principal lower-stage auditory areas of the cortex are hidden behind cortical folds.
- 26.
We have a similar situation with present-day electronic computers. The main limitation to their speed is given quite simply by the spatial distance between computing units!
- 27.
Research in music processing has a lower priority and receives less funding than speech!
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Roederer, J.G. (2008). Generation of Musical Sounds, Complex Tones, and the Perception of Timbre. In: The Physics and Psychophysics of Music. Springer, New York, NY. https://doi.org/10.1007/978-0-387-09474-8_4
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