Mathematics, Music, and
the Guitar
Martin Flashman
Visiting Professor of Mathematics
Occidental College
April 21,2006
Something Old, Something New,
Something Borrowed, and …
The Blues
Bird Studio
Program
• Mathematics, Music, and the Guitar
– General Guitar Overview
– The Problem of Scales
• Pythagorean / Ptolemaic Proportional Scales
• Even (Well) Tempered Scales
– Fretting and Scales on the Guitar
– Some Guitar Intonation Problems
• Where and how to play a note.
• The Bridge and the Saddle.
The Guitar Parts
• Head
– Nut
• Neck
• Body
– Bridge and Saddle
The Head
The strings pass over the
nut and attach to tuning
heads, which allow the
player to increase or
decrease the tension on
the strings to tune them.
In almost all tuning heads,
a tuning knob turns a
worm gear that turns a
string post.
Between the neck and the
head is a piece called the
nut, which is grooved to
accept the strings
The Neck
• The face of the neck,
containing the frets, is
called the fingerboard.
The frets are metal
pieces cut into the
fingerboard at specific
intervals. By pressing a
string down onto a fret,
you change the length of
the string and therefore
the tone it produces when
it vibrates
The body
The body of most acoustic guitars
has a "waist," or a narrowing.
This narrowing happens to
make it easy to rest the guitar
on your knee.
The most important piece of the
body is the soundboard. This
is the wooden piece mounted
on the front of the guitar's
body, and its job is to make the
guitar's sound loud enough for
us to hear.
The two widenings are called
bouts. The upper bout is
where the neck connects, and
the lower bout is where the
bridge attaches.
In the soundboard is a large hole
called the sound hole.
The Bridge
Attached to the soundboard is a piece called the
bridge, which acts as the anchor for one end of
the six strings. The bridge has a thin, hard piece
embedded in it called the saddle, which is the
part that the strings rest against.
Building Scales
• Choose one tone:
– A: frequency = 440 cycles/sec (Hertz)
• Double the frequency
– A2: frequency = 2* 440 = 880 (Octave)
• Triple the primal tone frequency then
divide by 2
– E: frequency = 3*440/2 = 1320/2 = 660
• Divide A2 frequency by 3 then double.
– D: frequency = 2*880/3 = 4/3* 440 = 586.666
MORE SCALE TONES
• A=440 D = 586.66 E = 660 A2=880
• Continue multiply by 3/2, 4/3…
• Multiply A by 9/4 then divide by 2
– B: 440*9/4=990… 990/2 = 495
• Multiply A by 16/9
– G#: 440*16/9 = 782.22
– Pythagorean Pentatonic Scale:ABDEG#A
(Play This)
The round of Perfect Fifth’s
• FCGDAEB F#C#G#D#A# FCGDAEB
• This gives a total of 12 distinct “chromatic”
tones.
• The intervals between these tones in the
same octave are roughly the same ratio.
• HOWEVER: The scales are not the
same if you start with a different tonic.
A Pythagorean Scale based on 3:2
“Pythagorean
Scale”
Frequency ratio
F to 1 (1<F<2)
String “Fret”
ratio
Factor to obtain
next ratio
Do
1:1=1
1
9/8
Re
3/2:2/3=9/4
=9/8
8/9
256/243
Mi
16/9:3/2
=32/27
27/32
9/8
Fa
Perfect Fourth
2:3/2=4:3
=4/3
3/4
9/8
Sol
Perfect Fifth
3:1=3:2
=3/2
2/3
9/8
La
9/8:2/3
=27/16
16/27
256/243
Ti
4/3:3/2=8/9
=16/9
9/16
9/8
Do
2:1 = 2
1/2
Pythagorean A Major Scale
“Pythagorean
Scale”
Frequency
ratio
F to 1 (1<F<2)
String “Fret”
ratio
Factor to obtain
next ratio
A
1= 440
1
9/8
B
9/8
8/9
256/243
C#
32/27
27/32
9/8
D
Perfect Fourth
4/3
3/4
9/8
E
Perfect Fifth
3/2
2/3
9/8
F#
27/16
16/27
256/243
G#
16/9
9/16
9/8
A
2=880
1/2
Just Intonation Scale (Ptolemy)
Based on triad 4:5:6
“Ptolemaic
Scale”
Frequency
ratio
F to 1 (1<F<2)
String “Fret”
ratio and
complement
Factor to obtain
next ratio
Do
4:4=1
1
0
9/8
Re
3/2:2/3=9/4
=9/8
8/9
1/9
10/9
Mi
5:4=5/4
4/5
1/5
16/15
Fa
Perfect Fourth
2:3/2=4:3
=4/3
3/4
1/4
9/8
Sol
Perfect Fifth
6:4=3:2
=3/2
2/3
1/3
10/9
La
2*5/6=10/6=5/3
3/5
9/8
Ti
3/2*5/4=15/8
8/15
16/15
Do
2:1 = 2
1/2
1/2
A major Scale with Just Intonation(Ptolemy)
“Ptolemaic
Scale”
Frequency ratio
F to 1 (1<F<2)
String “Fret”
ratio and
complement
Factor to
obtain next
ratio
A
1=440
1
9/8
B
9/8
8/9
1/9
C#
Major Third
5/4
4/5
1/5
16/15
D
4/3
Perfect Fourth
3/4
1/4
9/8
E
Perfect Fifth
3/2
2/3
1/3
10/9
F#
Major Sixth
5/3
3/5
9/8
G#
15/8
8/15
16/15
C Octave
2=880
1/2
0
1/2
Even Tempered Scale
Based on Equal “step” R1.05946
String “Fret”
ratio
Factor to obtain
next ratio
“Even Tempered
Scale”
Frequency
ratio
F to 1 (1<F<2)
Do
1
Re
R2
Mi
R4
Fa
Perfect Fourth
R5 1.335
0.793701 R
0.749154 R
Sol
Perfect Fifth
R7 1.498
0.66742 R
La
R9
Ti
R11
Do
R12= 2
1 R
0.890899 R
0.561231 R
0.529732 R
0.5
A Major Even Tempered Scale
Based on Equal “step” R1.05946
String “Fret”
ratio
Factor to obtain
next ratio
“Even Tempered
Scale” A
Frequency
ratio
F to 1 (1<F<2)
A = 440
1 = 440
B = 493.88
R2
C# = 554.37
R4
D = 587.33
R5 1.335
0.793701 R
0.749154 R
E = 659.26
R7 1.498
0.66742 R
F# = 739.99
R9
G# = 830.61
R11
A = 880
R12= 2 = 880
1 R
0.890899 R
0.561231 R
0.529732 R
0.5
Comparison
Just vs Even Tempered
Just F ratio
Just Fret Ratio
1
1
9/8
ET F ratio
ET Fret Ratio
1
1
8/9
1.122462
0.890899
5/4
4/5
1.259921
0.793701
4/3
3/4
1.33484
0.749154
3/2
2/3
1.498307
0.66742
5/3
3/5
1.781797
0.561231
15/8
8/15
1.887749
0.529732
2
1/2
2
0.5
Scales, Frets, and logarithms
Frequency
Fret
“cent”
1
1
0
1.059463
0.943874
100
1.122462
0.890899
200
1.189207
0.840896
300
1.259921
0.793701
400
1.33484
0.749154
500
1.414214
0.707107
600
1.498307
0.66742
700
1.587401
0.629961
800
1.681793
0.594604
900
1.781797
0.561231
1000
1.887749
0.529732
1100
2
0.5
1200
2.118926
0.471937
1300
2.244924
0.445449
1400
2.378414
0.420448
1500
2.519842
0.39685
1600
2.66968
0.374577
1700
Frets and scales
Frequency
(1st string)
Fret position
from saddle on
Martin 0-16NY
329.6
25
Note
Fret
E
open
F
1
349.2
23.597
F#
2
370.0
22.272
G
3
392.0
21.022
G#
4
415.3
19.843
A
5
440.0
18.729
A#
6
466.1
17.678
B
7
493.8
16.685
C
8
523.2
15.749
C#
9
554.3
14.865
D
10
587.3
14.031
D#
11
622.2
13.243
E
12
659.2
12.5
Some Guitar Intonation Issues
• Where and how to play a note.
– At the fret.
– Vibrato and Bending.
– String qualities- multiple positions.
• The Bridge and the Saddle.
– Varying string length proportions from bridge
to nut.
– Added tension: “sharper” on higher frets.
10 Minute Intermission

Music Program
Selections from
• Something “Old”
Ain’t She Sweet
Java Jive
Teddy Bears’ Picnic
Sunshine / Railroad
This Land
Johnny B Goode
• Something “New”
Tomorrow I’ll be gone
Whisper It in My Ear
I Wanna’ Be with You
The Rain Song
I gotta’ woman
• Something “Borrowed”
Lulu’s Back in Town
S’Wonderful
Good Luvin’
Be Friends with you
Don’t think Twice
The Story of Love
• The Blues
Down and Out
Jesse Fuller Medley
The Dink Song
You got me …
Trouble in Mind
Thanks
The End!

Refreshments Outside
PleaseNo food in Bird Studio
C Major Ptolymaic Scale
•
•
•
•
•
•
•
•
264 Hz - C, do (multiply by 9/8 to get:)
297 Hz - D, re (multiply by 10/9 to get [5/4]:)
330 Hz - E, me (multiply by 16/15 to get [4/3]:)
352 Hz - F, fa (multiply by 9/8 to get [3/2]:)
396 Hz - G, so (multiply by 10/9 to get [5/3]:)
440 Hz - A, la (multiply by 9/8 to get [15/8]:)
495 Hz - B, ti (multiply by 16/15 to get [2]:)
528 Hz - C, do