Math Matters The mathbehind...AGuitar Luthier, timbre, rule of18, chromatic scale, octave, frets, length, scale Pythagorean principle Technical termsused: Reference: warmer tones withmoremake sustain. Harder woods inanacoustic guitar’s for soundboard make brighter notes withlesssustain. woodsspruce) (like Softer sizes,tar body etc. unlimited.The variety ofsoundsemittingfrom guitarsisvirtually The face ofaguitaristhesoundboard. An instrument’s timbre results from theuniquetones created lengths, scale by varying tone woods, pickups, electronic gui- Interesting facts: Stewart-MacDonald website. the secondnoteinchromaticscale(thefirstbeingopenstring).Adigitalfretcalculatorisavailableon 17.817 is1.4312173”(36.482011mm),roundeddownto1.431”(36.48mm).Thistheplacementforfirstfret; For example,atypicalguitarscalelength(FenderTelecaster)is25.5inches(648mm).The25.5”scalelength,dividedby referred toasthe“ruleof18.” is thedistancefromfrontedgeofnuttofirstfret.Forseveralcenturies,thisnumberwasrounded18and preferences, gaugeofthestringsandlengthneck.Divideanyscale’sopenstringby17.817,result There areseveraldifferentscalelengthsonvariousguitars;determinedbytheinstrument’sdesigner,player’spersonal determined thecalculationofdistancebetweentonesinitiallyuncoveredbyPythagoras. 12th fretistheoctaveofthatstring.Aformula,developedbyVincenzoGalileiin16thCentury,mathematically Pleasant totheWesternear,guitarneckisdesignedona12-pitchchromaticscalebasedhisprinciple.The controlled bytheplacementoffretsonneck. octave. Thelengthofthestring,tension,anditsgauge(thickness)determinepitch.Onaguitar,stringis if thelengthofthisstringweretobecutinhalf,pitchmadewouldequaltonebutattwicefrequency;an Stretched tightlybetweentwopoints,astringwhenplucked,makesspecifictonalpitch.Pythagorasdiscoveredthat called luthiers. The guitarproducesverysophisticatedmusicbyproducingdifferentsoundfrequencies.Guitarsdesigners/buildersare How itworks: octaves, and notes infifths eventually resultedtwelve-pitch inthe chromatic guitarscale. Before theGreek Philosopher, Pythagoras (570-475BC), instruments musical were limited. ofcreating Hisprinciple musical Uses andapplications: Softer woods(likespruce)makewarmertoneswithmoresustain. a guitaristhesoundboard.Harderwoodsinanacousticguitar’s soundboardmakeforbrighternoteswithlesssustain. electronic pickups,guitarbodysizes,etc.Thevarietyofsounds emittingfromguitarsisvirtuallyunlimited.Thefaceof Interesting Fact:Aninstrument’stimbreresultsfromtheunique tonescreatedbyvaryingscalelengths,tonewoods, warmer tone.Checkoutdifferentbrandsandmodelstoseeforyourself. neck. Asthestringgetsshorter,distancebetweenfretsshrinksproportionately.Theshorterscalelengthcreatesa Calculating theformuladownremainingscalelengthonelocatespositionsoffretson Math,%20Music,%20&%20Guitar.pdf Mathematics, Music,andthe Guitar, David Hornbeck, http://jwilson.coe.uga.edu/EMAT6450/Class%20Projects/Hornbeck/ *http://www.guitarbuilding.orgNational STEMGuitarBuildingProject: Apply It. Submitted by Tom Morrissey, Professor Community College 2016 Island, ofRhode October ofArt Formula for Calculating Fret Spacing: * Society forIndustrial andAppliedMathematics Dn =[(L–Dn-1)÷17.817]+Dn-1 www.siam.org/careers/matters.php www.siam.org .