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The Mathematics of a Cajón Drum. T1A Eric Hutchinson. 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.

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## The Mathematics of a Cajón Drum

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**The Mathematics of aCajón Drum**T1A Eric Hutchinson**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.**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.**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.**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 resonator’s neck = 1.7R**Simplify**• Substitute for “S” and “L”: • Approximate pi by 3, and 2π√1.7 by 8. • This is accurate to three places. At four decimal places it is accurate to within 0.1%**Calculate the Volume**• 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)**• 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)**• 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**• 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**• 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**• Close to F# suspended 2nd chord (F#sus2)**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)**Graph Analyzing**• The period was found by measuring trough to trough. Notice that one cycle is an “M” shape. • Period is 0.008 sec.**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?**Thank You!**• Eric Hutchinson • College of Southern Nevada • eric.hutchinson@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. <34iac.acoustics.sk/proceedings/Kicak.pdf>

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