September 25 , 2007
modified January 14, 2010
Attack, Decay, Sustain
Rasmus Storjohan and his brother Kristoffen have indicated a few improvements that might be made in the animations. Rasmus in particular has gone to a bit of work to explain some of the math involved and has granted permission to put the files on this website. A sincere Thank You.
Basically, the problem involves what is known as "attack", "decay" and "sustain" in a scientific sense. (It is also a part of musicianship but the word "attack" was used by my guitar teachers more often to express angle, touch, pressure, etc. of the hand, fingers, plectrum, etc. during the act of plucking the string - hardly anything was more carefully tended to - so there might be some confusion between the scientist and the artist on this point).
Here we are talking about the change in the shape of the motion of a plucked string (and its consequent sound) after the string has been released and as the higher partials lose energy very much faster than the lower harmonics (this "transient" - generally stated to be the "twang" of a guitar seems to be measured in milliseconds yet a distinctive and important feature of an instrument's sound.
The pdf file from Rasmus:
(modified January 14, 2010)
The file upon which much of the math is based is
at:
V. Howle and L. N. Trefethen,
Eigenvalues
and Musical Instruments
J. Comp. Appl. Math, Vol. 135, No. 1, pp. 23-40,
October 1, 2001.
http://www.math.ttu.edu/~vhowle/publications/ewJCAM.pdf
http://www.math.ttu.edu/~vhowle/publications/pubs.html
or_______________
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.132.1922&rep=rep1&type=pdf
or______________
http://www.comlab.ox.ac.uk/people/nick.trefethen/publication/publication.html
http://www.comlab.ox.ac.uk/people/nick.trefethen/publication/PDF/2001_93.pdf
Some quick research under the keyword "Attack" is shown below:
http://www.ee.washington.edu/research/sahr/pages/physics.html
(ror - I particularly like figure 7 at the very bottom for a quick idea of the effect of decay.)
Observe Fig. 1. This shows a string that is fixed at both ends (a boundary-value problem). As there is no applied force on the string, the net force f(x,t) must be zero. Notice that f(x,t) is a function of both the transverse location x and time t. We desire to derive a model for the displacement y(x,t) of the string. If the mass density &vepsilon; is distributed uniformly along the x-axis, and the tension T is constant along x, then if the maximum displacement y is small « length l, the resulting oscillation will be identical in space and time. Put another way, the resulting 'wave' (the shape of the string) can be viewed as a constant shape that 'moves' along x-axis.
Link Not Working August, 2006
Some student projects - mostly pdf files
http://wug.physics.uiuc.edu/courses/phys398emi/Student_Projects/Fall00/TSchmitter/fourieranalysis.pdf
________
9. Stephen Treharne investigated the sustain of an electric guitar, and measured the decay time constants associated with the fundamental and the 2nd harmonic for the open strings of an electric guitar.
http://wug.physics.uiuc.edu/courses/phys398emi/Student_Projects/Fall00/STreharne/STreharne_P398EMI_Final_Report.pdf
...Since the overall waveform is theoretically nothing more than
the
superposition (summation) of its harmonics, the decreasing amplitude of
each harmonic can be tracked by applying Fourier analysis to the waveform.
Measuring the decrease in the amplitudes of the harmonics over time is the only
precise way of objectively measuring the...
http://www.jhu.edu/~signals/phasorlecture2/indexphasorlect2.htm
Various forms of the Fourier series description for periodic signals are based on alternate ways of writing a cosine signal.
http://ccrma.stanford.edu/overview/pastmodeling.html#SECTION00092700000000000000
Synthesis of Transients in Classical Guitar Sounds (April 2000)
Cem Duruoz
...Synthesis of acoustic musical instrument
sounds using computers has been a fundamental problem in acoustics. It is well
known that, the transients heard right before, during and right after the attack
portion of an instrumental sound are the elements which give the instrument most
of its individual character. Therefore, in a synthesis model, it is crucial to
implement them carefully, in order to obtain sounds similar to those produced by
acoustic instruments...
http://hyperphysics.phy-astr.gsu.edu/Hbase/sound/timbre.html
The plucking action gives it a sudden attack characterized by a rapid rise to its peak amplitude. The decay is long and gradual by comparison. The ear is sensitive to these attack and decay rates and may be able to use them to identify the instrument producing the sound.
http://www.physics.purdue.edu/~giordano/tmp/book_strings/book_strings.html
Nick Giordano
1997
Some results obtained with the algorithm (3) are shown in Fig. 1. Here the initial string profile (shown at the top of the figure) was chosen to be triangular, with the string at rest, as would be appropriate for a plucked guitar string. The kink associated with this pluck is seen to split into two separate kinks, one propagating to the left and one to the right, which reflect from the ends of the string. In our simulation we kept the ends of the string fixed, so the reflections are inverted.
http://www.geofex.com/Article_Folders/noteproc/noteproc.htm
The Guitar Effects Oriented Web Page
Note Processing for Guitar Effects
Copyright 2001 R.G.
Keen.
...The very first signal we get, starting from nil, is a loud transient. This might correspond to the pick attack, when a guitar pick pulls the string massively further than it will vibrate normally. That initial loudness attack decays very quickly to the sustaining level of the note, where the note volume decays at a slower rate.
http://en.wikipedia.org/wiki/Talk:Eigenvalue,_eigenvector_and_eigenspace
(ror-editing comments - these came up on the search - the main article itself is difficult enough but these comments among the editors go quite a bit beyond that - not for me- go to "Miscellany>In the Past>Eigenvalues" to see my own animation which I felt a bit too far beyond my understanding to include with the general animations)
If one considers the transformation of the rope as time passes, its eigenvectors (or eigenfunctions if one assumes the rope is a continuous medium) are its standing waves -- the things that, mediated by the surrounding air, humans can experience as the twang of a bow string or the plink of a guitar . The standing waves correspond to particular oscillations of the rope such that the shape of the rope is scaled by a factor (the eigenvalue) as the time evolves.
http://www.novaxguitars.com/info/technical.html
Transcript of Ralph Novak’s Lecture on Scale Length and
Tone
to the 1995 G.A.L. Convention
...A program that analyses the harmonic content of a tone over a specified time period captures a sample and "transforms" the sound into a spectral analysis of the harmonic components and their relative intensity. The graphic output is called a Fourier transform, and is not a waveform such as you might see with an oscilloscope. This is not attack, sustain, and decay. It is a graphic analysis of the harmonic content of a tone with the "spikes" representing the most intense frequencies. Attack, sustain, and decay do figure in some aspects of scale length tone, but we're focusing on harmonic content.
... it is also generally accepted that harmonics with an intensity of less than about 20% of the magnitude of the harmonic of greatest intensity contribute insignificantly to the tone.
...Another "law" applying to the formation of harmonics of the vibrating string is that the first six partials are generally agreed to be harmonious. We can add the eighth, tenth, twelfth, fifteenth, and sixteenth without impairing the consonance of the tone. Due to natural string division, the seventh, eleventh, thirteenth, and fourteenth are discordant. They are mathematical multiples, but they do not belong to the musical scale; therefore they impair the consonance of the tone.
...Something that has me particularly excited is the "clang tone" concept. The clang tone is the result of the elasticity of the vibrating string. It is the tone of the string stretching and relaxing as it performs its transverse vibrations, and is referred to as longitudinal vibration.
...A taut string actually vibrates in three modes: transverse, which we are most concerned with; longitudinal, which is usually the concern of piano builders because the struck string has greater potential for excitation of higher harmonics; and torsional, or twisting of the string, which has no musical value, only mathematical value.
(ror - The following has some practical considerations for rock guitarists (as well as tuning "purists" in general I should think)).
http://www.endino.com/archive/tuningnightmares.html
By Jack Endino
www.endino.com
The intent is to enlighten and entertain. Musicians do not respond well to technical papers.
Several things to note:
1) The harmonics that are "powers of
two" (2nd, 4th, 8th) are all octaves of the lowest, or "fundamental" note. To a
tuner, they are the SAME note.
2) All the other harmonics represent
DIFFERENT notes. It's the unique combination of fundamental plus these various
harmonics that give any instrument it's particular character or timbre.
3)
How you pluck the string, where you pluck the string, and where the pickup is
located under the string, determines the blend of "fundamental vs. harmonics"
that you hear. Pluck it near the bridge, and you get a twangy sound with lots of
high tones. Pluck it near the middle, and you get mostly a deeper, more "pure"
tone. Pluck it hard, and that initial burst of energy will cause more high
harmonics. Put a pickup near the end of the string (at the bridge), and it will
pick up more of those high harmonics; put it closer to the middle, and the
fundamental tone will come through louder.
An excellent pdf file
http://liam.musique.umontreal.ca/LIAM_Publications/Traube_C_WASPAA01.pdf
EXTRACTING THE FINGERING AND THE PLUCKING POINTS
ON A GUITAR
STRING FROM A RECORDING
Caroline Traube and Julius O. Smith III
This paper presents a signal processing technique for extracting the plucking point on a guitar string from an acoustically recorded signal. It also includes an original method for detecting the fingering point, based on the plucking point information.
3.1. Plucking an ideal string
The plucking excitation
initiates wave components traveling independently in opposite directions. The
resultant motion consists of two bends, one moving clockwise and the other
counterclockwise around a parallelogram. Ideally, the output from the string
(force
at the bridge) will lack those harmonics that have a node at the
plucking point. Figure 2 illustrates a plucking position at 1=5th of the length
from one end: the spectrum will lack the harmonics that are multiples of
5.
3.2. Plucking a real string
A real plucking differs from an
ideal plucking in the following ways. The finger or plectrum exciting the string
has a non-zero touching width, which adds more lowpass filtering to the
excitation. A real excitation is not an event that can be modeled with linear
and time-invariant operations. In fact, the finger may grab the string for a
short time, while causing nonlinear or linear, but timevarying interactions.
Also, the modes of the string vibration are
in general nonlinearly coupled so
that a mode with zero initial energy will begin to vibrate, gaining energy from
other modes [11]. Finally, in the case of an acoustic guitar, the resonating
body of the instrument filters the output wave of the string, according to the
modes that have been excited (which depend on the plucking angle and plucking
style). The forces parallel and perpendicular to the bridge excite different
linear combinations of resonances, resulting in tones that have different decay
rates [10].
http://www.leeds.ac.uk/music/studio/rproj_swss/tuning/htmlpap0.htm
Robert Asmussen
1997
...With a burst of well-aimed air, a trumpeter plays a single note, setting into motion a subtle and complex cascade of events. During the rise and fall of the note's amplitude known as attack and decay, the individual harmonics will change dynamically in amplitude according to numerous factors. These factors include the frequency of the fundamental; the amount of tubing currently used; the ratio of cylindrical to exponential tubing; the acoustics of the concert hall; and the real-time input of the performer through the use of muscles in the face, lips, tongue, throat, chest, abdomen and hands. Needless to say, such a complex sound is impossible to analyze in complete detail, especially when other criteria such as combination tones are considered. It is testament to the complexity of sound that a single note from a single instrument cannot be completely understood.