Transcript Mathematics of Stringed Instrument Construction* And All

The Mathematics of Stringed
Instrument Construction… And
All That Jazz
Milos Podmanik
Chandler-Gilbert Community College
Chandler, 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
1
.
sec.
• Let 𝑦 be the displacement of a string from its natural rest
(equilibrium) when under tension between nut and bridge.
• Let 𝑡 be the time since the string was plucked, measured in seconds
𝑦 𝑡 = sin 𝑓 ∙ 2𝜋 ∙ 𝑡
, where 𝑓 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)
2𝜋
1
1
𝑝𝑒𝑟𝑖𝑜𝑑 =
=
=
seconds
1
2𝜋 ∙ 𝑓 261
261
seconds
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
3
12” wide by 12” long and 1” thick, or 144 in
720 in3
• bf for body = 144 in3 = 5 bf
1 bf
$12.99
1 bf
• 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 - 117°F, 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
tan 𝑥 =
𝑥
7
7
→ 𝑥 = tan−1
≈ 2.76°
145
145
145 cm
7 cm
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 onhand… 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 30° 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 𝐶𝑛 to 𝐶𝑛+1 (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. C1 is double the frequency of 𝐶0 )
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 𝐶𝑛 to 𝐶𝑛+1 (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. C1 is double the frequency of 𝐶0 )
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 𝑟 is a common ratio between notes, or frequencies, then 𝑟12 = 2
12
and so 𝑟 = 2
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#
12
B
C
• To increase the
frequency by a factor of 2, we must reduce the length of the
12
string to 1 − 2 of the original length. Mathematically, if 𝐹𝑖 is the ith fret distance
from the nut and 𝐿 is the scale length, then:
12
𝐹1 = 1 − 2 ∙ 𝐿
12
𝐹2 = 1 − 2 ∙ 𝐹1
𝐹2
1−
12
2
= 1−
12
2 ∙ 𝐿 ⇒ 𝐹2 = 1 −
12
2
2
∙𝐿
Fretting: Or How Pythagoras Heard the World
•C
C#
D
D#
E
F
F#
G
G#
A
A#
B
• In general, the distance of the ith fret relative to the fret for a chromatic
scale, should be:
𝐹𝑖 = 1 −
12
2
𝑖+1
∙𝐿
C
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”
• 𝑓𝑐𝑢𝑡𝑜𝑓𝑓 =
1
2𝜋𝑅𝐶
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 𝐿
inches with 𝑛 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
−𝑥 ln
2
−𝑒
𝑥 +2
𝑥2 +3𝑥
cot
sec
csc
𝑒
3𝑥+𝑒𝑥
sin 3
• 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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