The Mathematics of a
Cajón Drum
T1A
Eric Hutchinson
What is a Cajón?
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A cajón is a box-shaped
instrument that is played by
slapping the front face
(tappa) by the hands or
sometimes with brushes,
sticks, etc.
Some cajóns are simple
boxes and others contain
snare wires or guitar wires.
Different sounds can be
created depending on where
it is hit.
History of the Cajón
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Created by African slaves who were
brought to to the coast of Perú in the
18th century to work.
Slaves were forbidden to play music
so they used objects such as crates
and boxes to play music so they were
easily disguised.
Another theory is that the cajón is a
descendant of box like drums from
Angola and Antilles, played by slaves
from shipping crates.
Helmholtz Resonator
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Air forced into a cavity, pressure
increases
After external force stops, the higherpressure air inside will flow out
Air in the port (neck) has mass
Diameter of the port is related to the
mass of air and volume in the chamber
A cajón is a hollow cavity resonator, so to
find the frequency, we need to use the
formula for a Helmholtz resonator.
Helmholtz Resonator Formula
c
fR 
2
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S
V L
C = Speed of sound in air (1125 ft/sec)
S = Surface area of the sound hole (square ft)
V = Volume of inside cavity (cubic ft)
L = length of resonator’s neck = 1.7R
Simplify
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Substitute for
“S” and “L”:
c
R 2
c
R
fR 

2 V 1.7 R 2 1.7V
Approximate pi by 3, and 2π√1.7 by 8.
c 3R
fR 
8 V
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This is accurate to three places. At four decimal places it is
accurate to within 0.1%
Calculate the Volume
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V = 11.875*11.125*19
V = 2510.08 cubic inches
But wait! This is not the true
inside volume because there
are objects inside that are
taking up space.
Volume of support blocks (in^3)
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Block 1:
V = 1.25*1.25*9.75 = 15.23
Block 2:
V = 1.25*1.25*10.63 = 16.6
Block 3:
V = 1.25*1.25*10.94 = 17.09
Block 4:
V = 1.25*1.25*10.44 = 16.31
Volume of Snare Assembly (in^3)
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Rest Block:
V = 1*0.25*11 = 2.75
Dowel Support Blocks:
2.5*0.5*1.94 = 2.42
22.56*0.5*1.94 = 2.48
Dowel:
V = π*(0.63)^2*10.13 = 12.4
Pivot Pieces:
V = 7.63*0.5*2.38 = 9.05
V = 6*0.5*2.13 = 6.38
True Interior Volume
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Total volume of inside objects is 100.71 cubic
inches.
True Interior Volume: 2510.08 – 100.71 =
2409.37 cubic inches.
Helmholtz formula requires this figure to be in
cubic feet, so divide 2409.37 by 12^3 to get 1.39
cubic feet. This is our V.
Sound Hole Radius
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Sound hole diameter is
4.75 inches
Sound hole radius is
2.375 inches
Converted to feet, this is
0.198, so this is R.
Calculate the Frequency
c 3R
fR 
8 V
1125 3(0.198)
fR 
8
1.39
f R  92 HZ
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Close to F# suspended 2nd chord (F#sus2)
Real World Experiment
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Hit the tappa with a rubber mallet
Have a microphone record the sound and have a
computer display sound wave
Graph measures pressure versus time
Special thanks to the CSN Physics Dept!
(Dr. Carlos Delgado & Ted Bellows)
Graph of Pressure Vs. Time
Graph Analyzing
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The period was found by
measuring trough to
trough. Notice that one
cycle is an “M” shape.
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Period is 0.008 sec.
What is the measured frequency?
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Frequency is defined as 1 / period.
Frequency = 1 / 0.008
Actual Measured frequency is 125 HZ. (Bsus2)
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Why the difference?
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Questions?
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Thank You!
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Eric Hutchinson
College of Southern Nevada
eric.hutchinson@csn.edu
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REFERENCE: Kicak, Peter . "FREQUENCY AND
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DYNAMICS ANALYSIS OF BASS TONE OF
CAJON BOX DRUM." Acoustics. Electotechnics and
Applied Mechanics, n.d. Web. 16 May 2013.
<34iac.acoustics.sk/proceedings/Kicak.pdf>