Chapter 8
Real Hammers and Plectra
Simulating Broad Plectra
Simulating Complex Shapes
• Use a collection of narrow plectra to
simulate the broad one in any pattern.
• Initial vibrational pattern can be deduced by
taking the individual patterns of the narrow
plectra and combining.
Example Case
½
¼
Fundamental Mode Excitations
• For the fundamental mode the central plectrum is
at the antinode (maximum excitation).
o Amplitude a1 if it acts alone
• The plectrum at the ¼ point excites the
fundamental 0.707 as much.
o Amplitude b1 = 0.707 a1 if it acts alone
• When both act together, the displacement is
constant between the plucking points and equal to
a1 + (⅔)b1
Note: I’m not quite sure where the ⅔ comes from, but clearly the second
position should get less weight in predicting the excitation of this mode
Net Mode 1 Excitation
• Mode 1 is excited to an amplitude of
a1 + (⅔)[(0.707)a1] = 1.47a1
Changing the Method of Plucking
a1 - (⅔)(0.707)a1 = 0.53a1
Notice the minus sign – the second plectra
gives a negative contribution for this mode
Another Example
Mode 7 Oscillation (3½ wavelengths)
p’s mark positive maxima
q’s mark negative maxima
Both the magenta and red plectra grab the
string at p positions
Constructive Use of Plectra
• Two plectra at p’s should give twice the initial amplitude
as one at a p position.
• The following situation should also produce
twice the initial amplitude as one plectrum.
To see this ask yourself which way the string
would move when the plectra are removed.
Notice that both plectra reinforce the mode
Destructive Use of Plectra
• If only the left plectrum were there, then after
release all p’s move down and all q’s move up.
• If only the right plectrum were there, then after
release all q’s move down and all p’s move up.
• With both plectra in place, mode 7 oscillations are
destroyed.
Generalize
(Four Points)
1. The result of using several plectra is given by
combining the results of each plectrum acting
alone (call this the narrow plectra substitution
model).
2. We must take proper account of the sign of the
individual results. Some plectrum positions may
cancel others.
Continued
3. For two plectra close together (less than ⅓ rd the
spacing of mode maxima) and pulling in the
same direction, we get nearly the same effect as
pulling one back twice as far.
4. Two plectra pulling in the same direction
separated by exactly ½ wavelength will cancel
out that mode (the destructive example two
slides ago). If the separation is approximately ½
wavelength, that mode is only weakly excited.
Application to Wide Plectra
W
• Recall that a wide plectrum (width W) acts the
same as two narrow plectra (separted by W).
Finger Picking a Guitar
• Use a thumb and forefinger on a guitar string
(separation W about 2 cm)
o modes are weakly excited whose wavelength is around
4 cm (W then would be ½ l and point 4 applies).
• Since the length of the string is 60 cm (L), modes
that fit this condition are
n = L/W = 60 cm/4 cm = 15
(that is the 15th harmonic would have a wavelength of 4 cm)
Results of model
• Mode 15 is absent and near 15 are weak.
• Point three tells us that modes of wavelength
greater than three times W (>6 cm) are excited by
a large amount and are accurately predicted by this
narrow plectra substitution model
n = L/(3W) = 60 cm/6 cm = 10
• Modes up to 10 are predicted well
• Modes between 10 and 15 are not predicted
Using a Pick
• Pick width 0.2 cm
• Harmonics around mode 150 are missing
and simple theory carries us to mode 100
o Since the amplitudes fall off with n (use of a
pick is like striking the string), these are all the
usable modes
Hammer Width for Struck Strings
• Force is not uniformly distributed
• Imagine the hammer hitting a wax block
• The indentation gives an accurate view of
the forces at work
Restoring Force
• The simple model uses a linear restoring
force F = -kx (Hooke’s Law)
• When a steady force is applied to the felt of
a piano hammer, the felt becomes stiffer
with use.
o Larger force must be applied to produce the
same compression. F = Kxp
o p is the non-linearity coefficient
Comparing forces
Force (F)
F = kx
F = Kx^p
Compression (x)
Effective Nonlinearity Exponent
• Hammers taken from pianos
o p = 2.2 – 3.5
• Unused hammers
o p = 1.5 – 2.8
• Preferred range of values is 2 - 3
o p < 1 hammer is too linear and loud notes are simply
amplified soft notes
o p > 3 there is too much contrast - fortissimos are too
harsh and pianissimos are too bland
Hammer Forces
Spatial Variation of the Force
Force
Fmax
Wh
Distance Along String
½ Fmax
Observations
• For Wh < ¼ wavelength (l) vibrational
modes are excited just like a narrow
plectrum.
• For Wh  ½ l excitations are about half as
strong as a narrow plectrum
• For Wh > l that mode receives very little
excitation.
Impact Duration
Temporal Variation of the Force
Force
Fmax
Th
Time
½ Fmax
Observations
• When Th < P/4 (¼ Period), vibrational
modes are excited that are the same as an
instantaneous hammer strike (impulse).
• For Th  P/2 are excited at about half the
strength as an instantaneous hammer strike.
• For Th > P that mode received little
excitation.
Hard and Soft Hammers
(expected linear response)
Soft Hammer
Hard Hammer
Hard Blows
Soft Blows
• Soft hammers apply the force over a
longer period of time.
• Soft and hard blows last the same time
Real Piano Hammers
• Solid curves
show results
• Dotted curves
show attempts
to fit data to
various models
Soft and Hard Blows
(real piano)
Hard blow results from
higher impact speed
String Stiffness
• So far we have considered flexible strings
• Stiffness makes the string shape rounded
o Higher frequency modes are blocked
Summary
• Major assumptions of the method:
o Excitation of a mode depends on the relative
amplitude of the mode at the point of striking or
plucking.
o Extended plectra or hammers may be treated by
adding the result of several narrow plectra or
hammers arranged to give the same initial
shape.
Continued
• More assumptions:
o Modes whose wavelength are comparable to the
width of the plectrum or hammer are not
excited. The same holds if the time interval of
the hammer strike is comparable to the period
of the mode.
o Plucked string modes decrease in initial
amplitude by 1/n2. Struck string modes
decrease in initial amplitude by 1/n.
Examples
• Pluck guitar string 6 as normal and then damp at
the mid-point.
o The odd-numbered modes have antinodes at the mid-
point and are damped.
o Only the even-numbered modes remain.
o The normal 329.6 Hz tone (E4) is replaced by 2*329.6
Hz = 659.2 Hz or E5.
• If the damper is applied at the 1/3rd point, then
modes 3, 6, 9, 12 survive, giving a tone close to
B5.