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. 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. Some cajóns are simple boxes and others contain snare wires or guitar wires. Different sounds can be created depending on where it is hit. 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 18 th century to work. Created by African slaves who were brought to to the coast of Perú in the 18 th 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. 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. 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 Air forced into a cavity, pressure increases After external force stops, the higher- pressure air inside will flow out After external force stops, the higher- pressure air inside will flow out Air in the port (neck) has mass Air in the port (neck) has mass Diameter of the port is related to the mass of air and volume in the chamber 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. 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) C = Speed of sound in air (1125 ft/sec) S = Surface area of the sound hole (square ft) S = Surface area of the sound hole (square ft) V = Volume of inside cavity (cubic ft) V = Volume of inside cavity (cubic ft) L = length of resonator’s neck = 1.7R L = length of resonator’s neck = 1.7R

Simplify Substitute for Substitute for “S” and “L”: “S” and “L”: Approximate pi by 3, and 2π√1.7 by 8. 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% This is accurate to three places. At four decimal places it is accurate to within 0.1%

Calculate the Volume V = *11.125*19 V = *11.125*19 V = cubic inches V = cubic inches But wait! This is not the true inside volume because there are objects inside that are taking up space. 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: Block 1: V = 1.25*1.25*9.75 = V = 1.25*1.25*9.75 = Block 2: Block 2: V = 1.25*1.25*10.63 = 16.6 V = 1.25*1.25*10.63 = 16.6 Block 3: Block 3: V = 1.25*1.25*10.94 = V = 1.25*1.25*10.94 = Block 4: Block 4: V = 1.25*1.25*10.44 = V = 1.25*1.25*10.44 = 16.31

Volume of Snare Assembly (in^3) Rest Block: Rest Block: V = 1*0.25*11 = 2.75 V = 1*0.25*11 = 2.75 Dowel Support Blocks: Dowel Support Blocks: 2.5*0.5*1.94 = *0.5*1.94 = *0.5*1.94 = *0.5*1.94 = 2.48 Dowel: Dowel: V = π*(0.63)^2*10.13 = 12.4 V = π*(0.63)^2*10.13 = 12.4 Pivot Pieces: Pivot Pieces: V = 7.63*0.5*2.38 = 9.05 V = 7.63*0.5*2.38 = 9.05 V = 6*0.5*2.13 = 6.38 V = 6*0.5*2.13 = 6.38

True Interior Volume Total volume of inside objects is cubic inches. Total volume of inside objects is cubic inches. True Interior Volume: – = cubic inches. True Interior Volume: – = cubic inches. Helmholtz formula requires this figure to be in cubic feet, so divide by 12^3 to get 1.39 cubic feet. This is our V. Helmholtz formula requires this figure to be in cubic feet, so divide 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 diameter is 4.75 inches Sound hole radius is inches Sound hole radius is inches Converted to feet, this is 0.198, so this is R. Converted to feet, this is 0.198, so this is R.

Calculate the Frequency Close to F# suspended 2 nd chord (F#sus2) Close to F# suspended 2 nd chord (F#sus2)

Real World Experiment Hit the tappa with a rubber mallet Hit the tappa with a rubber mallet Have a microphone record the sound and have a computer display sound wave Have a microphone record the sound and have a computer display sound wave Graph measures pressure versus time Graph measures pressure versus time Special thanks to the CSN Physics Dept! (Dr. Carlos Delgado & Ted Bellows) Special thanks to the CSN Physics Dept! (Dr. Carlos Delgado & Ted Bellows)

Graph of Pressure Vs. Time

Graph Analyzing The period was found by measuring trough to trough. Notice that one cycle is an “M” shape. The period was found by measuring trough to trough. Notice that one cycle is an “M” shape. Period is sec. Period is sec.

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

Thank You! Eric Hutchinson Eric Hutchinson College of Southern Nevada College of Southern Nevada REFERENCE:, Peter. "FREQUENCY AND DYNAMICS ANALYSIS OF BASS TONE OF CAJON BOX DRUM." Acoustics. Electotechnics and Applied Mechanics, n.d. Web. 16 May 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.