The Mathematics The Mathematics of aof a

Cajón DrumCajón DrumT1AT1A

Eric HutchinsonEric Hutchinson

What is a Cajón?What is a Cajón? A cajón is a box-shaped A cajón is a box-shaped

instrument that is instrument that is played by slapping the played by slapping the front face (tappa) by front face (tappa) by the hands or the hands or sometimes with sometimes with brushes, sticks, etc.brushes, sticks, etc.

Some cajóns are Some cajóns are simple boxes and simple boxes and others contain snare others contain snare wires or guitar wires.wires or guitar wires.

Different sounds can Different sounds can be created depending be created depending on where it is hit.on where it is hit.

History of the CajónHistory of the Cajón Created by African slaves Created by African slaves

who were brought to to the who were brought to to the coast of Perú in the 18coast of Perú in the 18thth century to work.century to work.

Slaves were forbidden to play Slaves were forbidden to play music so they used objects music so they used objects such as crates and boxes to such as crates and boxes to play music so they were play music so they were easily disguised.easily disguised.

Another theory is that the Another theory is that the cajón is a descendant of box cajón is a descendant of box like drums from Angola and like drums from Angola and Antilles, played by slaves Antilles, played by slaves from shipping crates.from shipping crates.

Helmholtz ResonatorHelmholtz Resonator Air forced into a cavity, pressure Air forced into a cavity, pressure

increasesincreases After external force stops, the After external force stops, the

higher-pressure air inside will flow higher-pressure air inside will flow outout

Air in the port (neck) has massAir in the port (neck) has mass Diameter of the port is related to Diameter of the port is related to

the mass of air and volume in the the mass of air and volume in the chamberchamber

A A cajóncajón is a hollow cavity is a hollow cavity resonator, so to find the resonator, so to find the frequency, we need to use the frequency, we need to use the formula for a Helmholtz resonator.formula for a Helmholtz resonator.

Helmholtz Resonator Helmholtz Resonator FormulaFormula

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 S = Surface area of the sound hole

(square ft)(square ft) V = Volume of inside cavity (cubic ft)V = Volume of inside cavity (cubic ft) L = length of resonator’s neck = 1.7RL = length of resonator’s neck = 1.7R

LV

ScfR

2

SimplifySimplify Substitute for Substitute for

“ “S” and “L”:S” and “L”:

Approximate pi by 3, and 2Approximate pi by 3, and 2ππ√1.7 by 8.√1.7 by 8.

This is accurate to three places. At four This is accurate to three places. At four decimal places it is accurate to within 0.1% decimal places it is accurate to within 0.1%

V

Rc

RV

RcfR 7.127.12

2

V

RcfR

3

8

Calculate the VolumeCalculate the Volume

V = 11.875*11.125*19V = 11.875*11.125*19 V = 2510.08 cubic V = 2510.08 cubic

inchesinches

But wait! This is not But wait! This is not the true inside volume the true inside volume because there are because there are objects inside that are objects inside that are taking up space.taking up space.

Volume of support blocks Volume of support blocks (in^3)(in^3)

Block 1:Block 1: V = 1.25*1.25*9.75 = 15.23V = 1.25*1.25*9.75 = 15.23

Block 2:Block 2: V = 1.25*1.25*10.63 = 16.6V = 1.25*1.25*10.63 = 16.6

Block 3:Block 3: V = 1.25*1.25*10.94 = 17.09V = 1.25*1.25*10.94 = 17.09

Block 4:Block 4: V = 1.25*1.25*10.44 = 16.31V = 1.25*1.25*10.44 = 16.31

Volume of Snare Volume of Snare Assembly (in^3)Assembly (in^3)

Rest Block:Rest Block: V = 1*0.25*11 = 2.75V = 1*0.25*11 = 2.75

Dowel Support Blocks:Dowel Support Blocks: 2.5*0.5*1.94 = 2.422.5*0.5*1.94 = 2.42 22.56*0.5*1.94 = 2.4822.56*0.5*1.94 = 2.48

Dowel:Dowel: V = V = ππ*(0.63)^2*10.13 = 12.4*(0.63)^2*10.13 = 12.4

Pivot Pieces:Pivot Pieces: V = 7.63*0.5*2.38 = 9.05V = 7.63*0.5*2.38 = 9.05 V = 6*0.5*2.13 = 6.38V = 6*0.5*2.13 = 6.38

True Interior VolumeTrue Interior Volume

Total volume of inside objects is Total volume of inside objects is 100.71 cubic inches.100.71 cubic inches.

True Interior Volume: 2510.08 – True Interior Volume: 2510.08 – 100.71 = 2409.37 cubic inches.100.71 = 2409.37 cubic inches.

Helmholtz formula requires this Helmholtz formula requires this figure to be in cubic feet, so divide figure to be in cubic feet, so divide 2409.37 by 12^3 to get 1.39 cubic 2409.37 by 12^3 to get 1.39 cubic feet. This is our V.feet. This is our V.

Sound Hole RadiusSound Hole Radius

Sound hole Sound hole diameter is 4.75 diameter is 4.75 inchesinches

Sound hole radius Sound hole radius is 2.375 inchesis 2.375 inches

Converted to feet, Converted to feet, this is 0.198, so this is 0.198, so this is R.this is R.

Calculate the FrequencyCalculate the Frequency

Close to F# suspended 2Close to F# suspended 2ndnd chord chord (F#sus2) (F#sus2)

HZf

f

V

Rcf

R

R

R

92

39.1

)198.0(3

8

1125

3

8

Real World ExperimentReal World Experiment

Hit the tappa with a rubber malletHit the tappa with a rubber mallet Have a microphone record the sound Have a microphone record the sound

and have a computer display sound and have a computer display sound wavewave

Graph measures pressure versus timeGraph measures pressure versus time Special thanks to the CSN Physics Special thanks to the CSN Physics

Dept! (Dr. Carlos Delgado & Ted Dept! (Dr. Carlos Delgado & Ted Bellows)Bellows)

Graph of Pressure Vs. Graph of Pressure Vs. TimeTime

Graph AnalyzingGraph Analyzing

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

Period is 0.008 Period is 0.008 sec.sec.

What is the measured What is the measured frequency?frequency?

Frequency is defined as 1 / period.Frequency is defined as 1 / period. Frequency = 1 / 0.008Frequency = 1 / 0.008 Actual Measured frequency is 125 Actual Measured frequency is 125

HZ. (Bsus2)HZ. (Bsus2)

Why the difference?Why the difference?

Questions?Questions?

Thank You!Thank You!

Eric HutchinsonEric Hutchinson College of Southern NevadaCollege of Southern Nevada eric.hutchinson@csn.edu

REFERENCE:REFERENCE: Kicak, Peter . , Peter . "FREQUENCY AND DYNAMICS ANALYSIS "FREQUENCY AND DYNAMICS ANALYSIS OF BASS TONE OF CAJON BOX OF BASS TONE OF CAJON BOX DRUM." DRUM." AcousticsAcoustics. Electotechnics and . Electotechnics and Applied Mechanics, n.d. Web. 16 May 2013. Applied Mechanics, n.d. Web. 16 May 2013. <34iac.acoustics.sk/proceedings/Kicak.pdf> <34iac.acoustics.sk/proceedings/Kicak.pdf>