PHY2464 - The Physical Basis of Music
PHY
-2464
PHY-2464
Physical Basis of Music
Presentation
Presentation 21
21
Percussion
Percussion Instruments
Instruments –– II
II
Adapted
Adapted from
from Sam
Sam Matteson’s
Matteson’s
Unit
Unit 33 Session
Session 34
34
Sam
Sam Trickey
Trickey
Mar.
Mar. 27,
27, 2005
2005
PHY-2464
PHY
PHY-2464
Pres. 21 Percussion Instruments - II
Percussion = striking
Percussion instruments divide nicely into
• those that lack a well-defined pitch
(overtones not related to harmonics of the
fundamental) Snare drums, bass drums, wood
blocks, castanets, cymbals …
• those with well-defined pitch(es) Tympani, marimba, xylophone,
carillon bells, steel drums …
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Example of percussion without pitch: A
percussionist has two nearly identical cymbals.
They have identical fundamental frequencies,
but one is 15 inches in diameter while the other
is 14 inches in diameter. What must be true
about the two?
The larger cymbal must be about 15% thicker than
the smaller one, since the frequency is proportional
to the thickness and inversely proportional to the
square of the diameter.
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Something to recall
• Piano strings exhibit inharmonicity because of the
stiffness of the wire. [Hall, Chapter 10, p. 197]
f n ≈ n f1 ⎡⎣1 + ( n 2 − 1) J ⎤⎦
J must be small!
J = 0.02 will raise 2nd mode frequency of a string by
a semitone.
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
The Percussion Instruments (*already discussed)
Strings*
Membranes
Drums
Piano*
Hammer dulcimer*
Cymbals, Gongs, Pans
PHYPHY-2464
Blocks, bells,
shells
Bars
Plates
Xylophones, chimes
Others
Pres. 21 Percussion Instruments - II
Amplitude Amplitude
Fact: Production of pitch in a percussion instrument
is an exercise in manipulating the struck-object
overtones into a (approximately) harmonic series.
fn m = xn m f01 Unpitched
f01
fn = n f1
f1
2f1
Frequency
3f1
Pitched
4f1
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Recall: The Modes of vibration of an ideal string are
harmonic.
Tension T
L
Linear density µ
• Linear density
µ= mass/length
• Tension T= force
The stiffness of the wire
increases the frequency of the
higher frequency harmonics.
fn = n /(2
/(2 L) x √(T/ µ)
₧ = 3986¢
3986¢ Log(nf1 /440) + I(₧)
n = 1, 2, 3, 4, 5, 6, 7…
7….
I(₧) = Inharmonicity
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Recall from Unit 8:
Pitch interval for two frequencies fα , fβ
Interval in cents = I (¢)
= 3986.31 log (fα /fβ)
So ₧ = 3986.31 Log(nf1 /440) + I(₧)
and I(₧) = Inharmonicity & the value of n
Give a determination of how far a given interval in
an inharmonic spectrum is from harmonic
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Inharmonicityy
Inharmonicit
Inharmonicity of Piano
40¢
40¢
20¢
20¢
-20¢
20¢
Pitch (¢
(¢)
Because of the inharmonicity of strings, the octaves are
“stretched” in a piano [Hall, pp. 196196-7]
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Tympani and Tabla
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Orchestral Percussion
Tympani
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Tympani are tuned by adjusting
the tension on the head.
Tension device
Tension pedal
Tympanum: Greek “drum”
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Reminder: Oscillation Modes of an (Ideal) Clamped
Membrane
Surface density σ
Mode: (0,1)
f0 1 = 0.7655/ d x √(S/ σ)
Surface Tension S
Mode: (1,1)
f1 1 = 1.594 f0 1
Mode: (2,1)
f2 1 = 2.136 f0 1
Fx,y: x = # radial nodes, y = # circular nodes
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Reminder: Oscillation Modes of Ideal Clamped
Membrane Fx,y: x = # radial nodes, y = # circular nodes
Mode: (0,1)
(1,1)
(2,1)
(0,2)
(3,1)
xn m / x0 1 : 1
1.594
2.136
2.296
2.653
(1,2)
(4,1)
(2,2)
(0,3)
(5,1)
2.918
3.156
3.501
3.600
3.652
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Air Loading of a Clamped Membrane
Surface density σ
Surface Tension S
The mass of air moved by
the membrane adds to the
effective surface density, thus
lowers the working frequency.
PHYPHY-2464
Air mass
Pres. 21 Percussion Instruments - II
Key Fact: Tympani kettles modify the membrane
frequencies by the interaction of the air
resonances with the surface modes.
Modes of air vibration
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Modes of Oscillation of Tympani
Strike point
Mode: (0,1)
(1,1)
(2,1)
(0,2)
(3,1)
fn m/f01 : 1
1.594
2.136
2.296
2.653
(1,2)
(4,1)
(2,2)
(0,3)
(5,1)
2.918
3.156
3.501
3.600
3.652
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Amplitude
Key Fact: Tympani achieve pitch by (1) suppression of
radial modes; (2) modification of other mode
frequencies by both air loading and the effect of
the kettle ; (3) attenuation of the lowest mode.
f0
(0,1)
2f0
3f0
4f0
5f0
6f0
(1,1) (2,1)
(3,1)
(4,1) (0,3)
(3,2)
(3,2)
(0,3)
(0,2)
(1,2)
(2,2)
(2,2)
(5,1)
(0,2)
(1,2)
Frequency
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Steelpans (steel drums): MidMid-20th century metalophone
innovation. Scribed areas on the end plate of a 55 gallon
drum are tensioned to tuned notes by hammering &
heating
Note antianticlockwise
circle of 5ths
on outer two
rings
Figure from Rossing,
Rossing, Moore, and Wheeler “The Science of Sound” 3rd Ed.
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Steelpans (steel drums): Typical playable ranges
Figure from Rossing,
Rossing, Moore, and Wheeler “The Science of Sound” 3rd Ed.
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Steelpans (steel drums): Modes of vibration
Figure from Rossing,
Rossing, Moore, and Wheeler “The Science of Sound” 3rd Ed.
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Steelpans (steel drums): Spectra G3 (196 Hz), C4# (277 Hz)
Figure from Rossing,
Rossing, Moore, and Wheeler “The Science of Sound” 3rd Ed.
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Traditional metalophones: Glockenspiels,
Xylophones, Marimbas, & Vibraphones
Xylo:
Xylo: wood
Phone: sound
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Metalophones:
Glockenspiels, Xylophones and Marimbas
Bar
h = thickness
w = width
L = Length
Density ρ = mass/volume
Young’
Young’s Modulus Y = Force/elongation
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Metalophones:
Glockenspiels, Xylophones and Marimbas
Longitudinal Waves in a Bar
vL = √Y/ ρ
node
AntiAnti-node
AntiAnti-node
Longitudinal
Wave Velocity
fn = n/2L√Y/ ρ like an open pipe
Density ρ = mass/volume
Young’
Young’s Modulus Y= Stress/Elongation
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Reminder: Bending Wave in a Plate
vbend
h: thickness
ρ: density
Y: Young’s Modulus
• Density
ρ= mass/volume
• Young’
Young’s Modulus
Y= stress/elongation
=stiffness
• vL = √Y/(.91 ρ)
vbend = √[1.8 f h vL ]
fnm = 0.0459 h vL( ynm /d)2
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Bending Modes in Bars:
End Clamped
f1= 0.1782 fo
f2= 1.116 fo
f3=3.125 fo
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Bending Modes in Bars:
Free Ends
f1= 1.133 fo
0.224 L
f2= 3.125 fo
f3=6.125 fo
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Glockenspiel, Orchestra Bells:
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Orchestral Chimes
Free Ends
End Plug
f1= 1.133 fo
f2= 3.125 fo
f3=6.125 fo
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Marimba
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Mode Frequencies in Undercut Bar:
Undercut Bar in Xylophone, Marimba and Vibraphone
Xylophone
f1/f1 = 1.00
f2 /f1 = 3.00
f3 /f1 =6.1
λ/4
Marimba/Vibes
f1 /f1 = 1.00
f2 /f1 = 4.00
f3 /f1 =6.5
Vibraphone
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
What are the differences among a Xylophone,
Marimba, and Vibraphone?
•
•
•
•
The depth of the undercut: a marimba is
undercut more than a xylophone.
The first harmonic of a xylophone is 3x the
fundamental; for a marimba and “vibe” 1st
harmonic is 4x the fundamental.
The xylophone sounds “brighter” and the
marimba more “mellow.”
Vibes have a tremolo mechanism.
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Carillons, Chimes, handbells
Carillon by definition is 23 or more bells played from a
keyboard (less than 23 is a Chime).
Chime).
Strike Note:
determined by
Octave, Twelth,
Twelth, &
Upper Octave
(2:3:4 frequency
ratio), not the Prime
nor Hum
Handbells:
Handbells: pp 169169-70 Hall
Figure from Rossing,
Rossing, Moore, and Wheeler “The Science of Sound” 3rd Ed.
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 21 Percussion Instruments - II
Summary:
•
Piano strings exhibit inharmonicity because of
wire stiffness.
•
Some percussion instruments have pitch.
•
Pitch results from a harmonic series of
overtones.
•
Tympani, Tabla, & Steelpans are pitched drums
•
Orchestra Chimes, Glockenspiel, Xylophone,
Marimba and Vibraphone have intonation.
•
Marimba are undercut more than xylophones.
•
Carillon, chimes, and handbells are tuned,
deformed plates.