The Mathematics of a Caj

1
The Mathematics of aCajón Drum

• T1A
• Eric Hutchinson

2
What is a Cajón?

• 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.

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History of the Cajón

• 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.

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Helmholtz Resonator

• Air forced into a cavity, pressure increases
• After external force stops, the higher-pressure
  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.

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Helmholtz Resonator Formula

• 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 resonators neck 1.7R

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Simplify

• Substitute for
• S and L
• Approximate pi by 3, and 2pv1.7 by 8.
• This is accurate to three places. At four
  decimal places it is accurate to within 0.1

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Calculate the Volume

• V 11.87511.12519
• V 2510.08 cubic inches
• But wait! This is not the true inside volume
  because there are objects inside that are taking
  up space.

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Volume of support blocks (in3)

• Block 1
• V 1.251.259.75 15.23
• Block 2
• V 1.251.2510.63 16.6
• Block 3
• V 1.251.2510.94 17.09
• Block 4
• V 1.251.2510.44 16.31

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Volume of Snare Assembly (in3)

• Rest Block
• V 10.2511 2.75
• Dowel Support Blocks
• 2.50.51.94 2.42
• 22.560.51.94 2.48
• Dowel
• V p(0.63)210.13 12.4
• Pivot Pieces
• V 7.630.52.38 9.05
• V 60.52.13 6.38

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True Interior Volume

• 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 123 to get 1.39
  cubic feet. This is our V.

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Sound Hole Radius

• 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.

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Calculate the Frequency

• Close to F suspended 2nd chord (Fsus2)

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Real World Experiment

• 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)

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Graph of Pressure Vs. Time
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Graph Analyzing

• The period was found by measuring trough to
  trough. Notice that one cycle is an M shape.
• Period is 0.008 sec.

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What is the measured frequency?

• Frequency is defined as 1 / period.
• Frequency 1 / 0.008
• Actual Measured frequency is 125 HZ. (Bsus2)
• Why the difference?
• Questions?

17
Thank You!

• Eric Hutchinson
• College of Southern Nevada
• eric.hutchinson_at_csn.edu
• REFERENCE Kicak, Peter . "FREQUENCY AND
  DYNAMICS ANALYSIS OF BASS TONE OF CAJON BOX
  DRUM." Acoustics. Electotechnics and Applied
  Mechanics, n.d. Web. 16 May 2013.
  lt34iac.acoustics.sk/proceedings/Kicak.pdfgt