PH 105

1
PH 105
String Instruments
Dr. James van Howe Lecture 11
2
Name the guitarist

• Jimmy Page
• Eric Clapton
• Jimmi Hendrix
• Stevie Ray Vaughan

3
Given the recipe for a square wave below, what is
the next harmonic after the fundamental, if the
fundamental is 200 Hz.
Spectrum

• 400 Hz
• 600 Hz
• 800 Hz
• 1000 Hz

1
0.8
0.6
Amplitude
0.4
0.2
0
1
3
5
6
7
8
9
10
11
12
13
Harmonic Number
4
True or False
Bowing a violin near the bridge requires less
force
5
The Violin

• Dates back to Middle Ages, but reached peak
  development during the late 17th century in
  Italy
• Gasparo de Salo and the Amati family are credited
  with inventing the modern violin
• Antonio Stradivari was a pupil of Nicolo Amati
  and Guiseppi Guarneri
• Due to their great brilliance, Stradivari violins
  have set the standard for violin making ever
  since.
• In recent years the best Stradivari violins have
  been auctioned off for millions of dollars

6
Violin Construction
Strings G3 D4 A4 E5 (196, 294, 440, 660 Hz)
Fine tune
End Button
Finger Board
Bridge
Nut
Tail Piece
Peg Box
Scroll
Neck
f-hole
Pegs
Bass Bar
Spruce (2-4 mm thick)
f-hole
Top Plate (belly)
Corner Blocks
Ribs
Purfling
Bridge
Sound Post
Bass Bar
Sound Post
Back Plate
Curly Maple
7
Without bass bar and sound post for structural
support, fragile top plate would break
At regular tuning, taught strings push bridge
onto top plate with 100 N (44 lbs) of force
Purfling
Bridge
Bass Bar
Sound Post
Though made for support, these structures enhance
the resonances of the violin body
8
Harmonics of a Plucked a string
Right pure tones at right strength add up to make
a triangle
All harmonics having nodes at the pluck point
will be absent
Plucking at suppresses the harmonics that
are multiples of 2
9
Plucked elsewhere
Plucking at suppresses harmonics that are
multiples of 5
10
What harmonics will be suppressed if I pluck at
L/3, and the fundamental is 100 Hz?

• 100, 300, 500 Hz
• 300, 600, 900 Hz
• 300, 700, 1300 Hz
• Just 300 Hz

11
Timbre and Plucking

• The closer to the end the string is plucked the
  richer the harmonic content
• A picked string (banjo) contains more high
  harmonics than a finger plucked string (harp)
• A rapidly plucked string contains more high
  harmonics than one plucked slowly

12
Evolution of plucked string

Harmonics get out of phase reflecting back and
forth
under
Two opposite traveling pulses reflecting back and
forth
13
Bowed String
Gut or steel string
Friction from hair and rosin grabs string
Tension from string too great, overcomes
friction and slips back
Cycle repeats rapid plucking or sustained
playing
14
slipping begins
envelope
bow
sticking begins
disturbance
15
Difficulty of bowing

• If sticking and slipping doesnt happen at the
  right time, unpleasant sounds
• Counter to intuition successful sticking and
  slipping does not have to do with bowing speed,
  and does not depend on friction alone

Bowing force matters the most!
16
Difficulty of bowing too much or too little
gives an unpleasant sound
Brighter tone, requires more force and steady
experienced hand less leeway for error
Softer gentle tone, requires less force more
leeway for error
To increase the loudness of the tone
1. bow near the bridge 2. bow at higher speed
17
Timbre of bowed string
Contains both even and odd harmonics that
decrease less rapidly than the plucked string,
gives warmer tone
18
But strings are just the sound source
Vibrations from strings are transmitted to plates
by the bridge and sound post
Resonance 6000 Hz
Resonance 3000 Hz
Bridge
Bass Bar
Sound Post
19

• Main wood resonance (MWR) fundamental frequency
  of top plate (440 Hz)
• Back plate stiffer and so slightly higher
• Main air resonance (MAR) near 260-290 Hz,
  fundamental of air cavity
• MWR and MAR differ by a ration of 3/2 (perfect
  fifth)

Radiates Sound Efficiently
Low Radiation, gives feel
Low Radiation, gives feel
Radiates Sound Efficiently
Six low-frequency resonant modes of a violin
20
Video Demo Guitar Modes
21
Chladni Patterns
Method developed by Ernst Chladni in early 19th
century to investigate the modes on a surface
Sprinkle sand on the plate. The sand collects
where there are no vibrations (dead spots) or
nodes
We now have nodal lines instead of nodal points
Chladni demo
22
Hologram Interferometry uses interference of
light to show vibrational modes of plate
Old school method it to tap the plate for tap
modes.
23
Intricate interplay of resonances of the wood
material, air, geometry of plates, bass plate,
sound post, and bridge give the violin its sound

• Properties of wood the affect Modes
• Density
• Stiffness
• Internal damping
• Varnish applied (adds damping)
• Anisotropy (not uniform) wood is stiffer along
  the grain other directions spruce is great for
  this.

Go back to mass on a spring to think how you can
change frequency of a resonance
24
Bridge You can change the resonance of the
bridge by cutting out more wood or adding mass
Adding mass to the bridge does what?
Resonance 6000 Hz
Resonance 3000 Hz

• Lowers the resonant frequency
• Raises the resonant frequency

25
What makes the Strad so good?
What do you notice about the resonances
quantity, sharpness, spacing?
26
17th century Guarneri Violin
This group of resonances around 2-3 kHz similar
to singers formant found in opera singers
27
True or False
The timing of the string sticking and slipping
depends mainly on friction.
28
For Chladni Patterns sand settles where vibration
is

• Maximum positive amplitude
• Maximum negative amplitude
• No amplitude

29
Given the recipe for a square wave below, what
are the first three harmonics if the fundamental
is 200 Hz
Spectrum
1

• 200, 600, 1000 Hz
• 200, 400, 600 Hz
• 200, 400, 800 Hz

0.8
0.6
Amplitude
0.4
0.2
0
1
3
5
7
9
11
13
Harmonic Number
30
Homework

• Finish Reading Chapter 10 if havent already
• Read Chapter 11
• Review HW and Concepts so far
• Come to office hours (Capitalize on the Resources
  you pay for)