The Mathematics of Stringed Instrument Construction…

And All That JazzMilos Podmanik

Chandler-Gilbert Community CollegeChandler, AZ

Background

• Started with a small Craftsman table saw about 10 years ago doing small home-carpentry projects (standard home repairs)• Eventually began working on home cabinetry and custom furniture

projects• Had leftover birch and thought, “Hey, let’s build a banjo!”

Why Instrument-Building?

• Luthier – someone who makes/repairs stringed instruments (lutes, violins, guitars, banjos, etc.)• Because you love woodworking• Because you love mathematics and sciences• Because you…• Have too much time on your hands• Are mildly insane• Or both of the above

How Does This Talk Relate to Math Pedagogy?• Real problems are hardly about• Maximizing the area contained within four fences for Farmer John• Figuring out how long a ladder is that rests 10 feet above the ground against a

wall

• Real problems are about• Having a deep enough understanding of tools to use them creatively• Persevering in problem solving when the solution isn’t quite feasible (try

getting an angle of 2.489 degrees on your next project!)• Coming up with an appropriate model and understanding its practical

limitations

How Does This Talk Relate to Math Pedagogy?• Precision is merely a pipe dream that mathematics makes us believe is

real. • The tasks we face daily may involve some learnt mathematics, but

those tools need to be understood deeply to be appropriately utilized.• We need to understand solutions, connect them to reality, modify

solutions to fit reality, and mathematically validate the effect of such changes• Reality: Students don’t use it because they don’t understand it

So You Want to Build a Guitar?

• Calculations are important up-front. This determines many important features of the guitar (scale-length, body size, neck angle, etc.)• Limitations of those calculations are necessary in order to realize

where things may not pan out as initially believed• We don’t try to just “give it a shot”, especially when you are using

Mahogany that runs $12.99 per board foot (when about 6 board feet are needed) and other exotic woods that run upwards of $30.00 per board foot. • If you do, you will have pricey wall-hangings!

What Type of Mathematical Thinking Is Needed to Problem-Solve?

How Does a Guitar Work?

• A “string”, which is often wound metal (or nylon, for classical guitars) is tensioned between two points, often referred to as the bridge/saddle and the nut

How Does a Guitar Work?

• The distance between the nut and the bridge is called the scale length. • The Gibson SG has a scale length of 24.75” (which is actually 24.6”…

Ugh! Note how the bridge is actually slanted… not a mistake!)

How Does a Guitar Work?

• When plucked, the string vibrates between the bridge and nut creating sound waves. • “Beautiful” electric guitars are really just over glorified boards with

tensioned metal between two endpoints!

What is Sound?

• “In physics, sound is a vibration that propagates as a typically audible mechanical wave of pressure and displacement, through a medium such as air or water. In physiology and psychology, sound is the reception of such waves and their perception by the brain.” – Wikipedia

We Need to Quantify Sound!

• Sound is measured in hertz, which replaces the unit .• Let be the displacement of a string from its natural rest (equilibrium)

when under tension between nut and bridge.• Let be the time since the string was plucked, measured in seconds

, where is the frequency, measured in Hz, is a reasonable model, assuming no damping of the string’s displacement (in a vacuum).

We Need to Quantify Sound!

• The note referred to as Middle C (typically a piano term), is a C-note that is quite literally in the middle of the keyboard, and one that has a frequency of about 261 Hz. This is sometimes referred to as C4, since it is the 4th octave of C, relative to the minimum C note perceived by the human ear (C0 is about 16 Hz and the human ear can perceive as low as about 12 Hz)

Overview of the Electric Guitar Build

• Acquire/design plans• Create templates• Rough-cut body and neck• Truss rod channeling and install• Shape body and neck• Route electronics cavities• Fretboard cutting/slotting• Inlays • Pre-finish assembly• Finishing• Assembly• Set-up (nut filing/truss rod adjustments/pickups/action/etc.)

Problem 1: Get Lumber

• MATHEMATICAL COMPETENCY – Unit Analysis and Proportional Reasoning• To account for waste and logistics, the body “blank” will need to be about 15” wide

by 24” long. The Gibson SG has a body thickness of about 1 9/32”, so 8/4 lumber will be required (and then planed to thickness)

• One board-foot of Genuine Mahogany is about $12.99 per board-foot• One board-foot is a measure of volume equivalent to a piece of lumber that is 12”

wide by 12” long and 1” thick, or 144• bf for body = • Cost of body = 5 bf * = $64.95 for body• This is just the cost of the body… it’s apparent why big mistakes must be avoided!• The neck will cost about $15 and the fretboard about another $10-$15

Problem 1: Get Lumber

• PROBLEM WITH THE SOLUTION: Lumber supply houses often require that a board that is cut for purchase leave at least 4 linear feet of waste… otherwise you must purchase the whole board, which means a greater expense.• New problem: Optimize the use of a board (perhaps plan for a second

guitar so that you don’t end up with a small block of wood that will sit in the heat of your Phoenix garage - 117F, anyone?)• To consider: kerf of the blade (the thickness of the blade) and

whether or not the desired stock is of “roughly” equal width from length’s end-to-end

Problem 2: Neck Angle• MATH COMPETENCY: Geometric Problem-Solving• The neck will be glued into the body, but it requires a backward angle

because a Tune-O-Matic bridge will be used.• Purpose: Counter the tension of the strings on the neck, have good

playing “action” (string height) and make playing more ergonomic (arguably)

Problem 2: Neck Angle

Okay… now what?

Problem 2: Neck Angle

Problem 2: Neck Angle

145 cm

7 cm

tan𝑥= 7145

→𝑥=tan− 1( 7145 )≈2.76 °

𝑥

Problem 2: Neck Angle

• Problem with the Solution: I can really only be confident that I get the angle precise to the nearest degree. Do I truncate or round down? What are the consequences?• By the way… a “good” craftsman makes do with what he has on-

hand… so no buying more equipment just for one job• Can I be off by one degree?• What is the consequence of avoiding a neck angle entirely?

Problem 3: Routing the neck pocket

• MATH COMPETENCY: Geometric reasoning in 3D• We have the neck angle… but how do we get the neck angled?

Problem 3: Routing the neck pocket

• Easy! Build a template and use a bearing-guided straight router bit• PROBLEM: The router bit take out material parallel to the surface

Pocket is angled

What do we do now?

Problem 3: Routing the neck pocket

• Build a “ramp” for the router so that it is cutting parallel to an angled surface!• But, what is the angle of this ramp?• PROBLEM with the solution: Uh, oh! My table saw has a minimum

angle of relative to the fence!Ramp

Problem 4: Slotting the fretboard

• MATH COMPETENCY: Exponential functions and proportional reasoning• Pythagoras experimented with music and had a preference for “equal

temperament”• Now known as the Western Chromatic Scale• From notes to (or whatever note) is a doubling of the frequency of

the sound wave• Can be thought of as a set of 12 notes between the same note of

adjacent octaves (e.g. is double the frequency of )

Problem 4: Slotting the fretboard

• MATH COMPETENCY: Exponential functions and proportional reasoning• Pythagoras experimented with music and had a preference for “equal

temperament”• Now known as the Western Chromatic Scale• From notes to (or whatever note) is a doubling of the frequency of

the sound wave• Can be thought of as a set of 12 notes to the same note of adjacent

octaves (e.g. is double the frequency of )

Fretting: Or How Pythagoras Heard the World

• C C# D D# E F F# G G# A A# B C

• It is purported that Pythagoras believed this chromatic scale would sound “good” if notes had common ratios• Known as equal temperament, or, more commonly, the Western Chromatic

Scale• That is, if is a common ratio between notes, or frequencies, then and so

Fretting: Or How Pythagoras Heard the World

• C C# D D# E F F# G G# A A# B C

• Can we place frets (thing bars made out of nickel) arbitrarily?• Of course! Just not if you want your guitar to play most forms of modern

music (some instruments are fretted to be in the harmonic minor scale)• NOTE: when a string is pressed against the fretboard, the scale length is

shortened, thereby increasing the frequency of vibrations and producing higher sounds• The length of the string is inversely related to the frequency of sound waves

Fretting: Or How Pythagoras Heard the World

• C C# D D# E F F# G G# A A# B C

• To increase the frequency by a factor of , we must reduce the length of the string to of the original length. Mathematically, if is the ith fret distance from the nut and is the scale length, then:

Fretting: Or How Pythagoras Heard the World

• C C# D D# E F F# G G# A A# B C

• In general, the distance of the ith fret relative to the fret for a chromatic scale, should be:

Fretting: Or How Pythagoras Heard the World

• PROBLEM:

Problem 5: Optimal Electronics

• Admittedly, somewhat elusive to the presenter • A standard circuit with volume acts according to Ohm’s Law• Add a capacitor after the resistor, and suddenly you have a low-pass

circuit (filter high frequencies and pass low frequencies to the amplifier). This is what guitarists refer to as the function of the “tone knob”

Electronics

One More Problem: Getting the banjo neck angle and radius

One More Problem: Getting the banjo neck angle and radius

One More Problem: Getting the banjo neck angle and radius

So You Want to Build GuitarS

• MATH COMPETENCY: Generalize and Form Conceptual Abstractions• Lumber, neck pockets, fretting, oh, my!• Generalizability is the type of thinking we want students to embrace• Can I build one jig that can rout neck pockets for a variety of angles?• Can I build a spreadsheet in Excel that will compute lumber volume

required, lumber volume to be purchased, and total cost?• Can I construct a fret-spacing calculator for general scale lengths of

inches with total frets? (there are a lot of “wannabe” Eddie Van Halens out there)

• If the reduction in time to complete a stringed instrument was to be the same, it has taken approximately 47.5% less time to build each new instrument.

You, Too, Can Build Guitars… In Your Classroom!• STEM Guitar Institute• http://www.guitarbuilding.org/• All resources can be purchased as kits for students (they don’t build

from scratch… but they build guitars and learn math and science!)

Closing Thoughts

• Our focus should be on getting students to make conceptual sense of mathematics, not on getting them to differentiate

• A deep level of understanding is evident when you can make a hammer hammer… pull nails, and build a guitar!

Thank you for your time!

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