# Mathematics Music and the Guitar Martin Flashman Visiting by nikeborome

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```									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     Frequency ratio   String “Fret”   Factor to obtain
Scale”           F to 1 (1<F<2)    ratio           next ratio
Do               1:1=1             1               9/8
Re               3/2:2/3=9/4       8/9             256/243
=9/8
Mi               16/9:3/2          27/32           9/8
=32/27
Fa               2:3/2=4:3         3/4             9/8
Perfect Fourth   =4/3
Sol              3:1=3:2           2/3             9/8
Perfect Fifth    =3/2
La               9/8:2/3           16/27           256/243
=27/16
Ti               4/3:3/2=8/9       9/16            9/8
=16/9
Do               2:1 = 2           1/2
Pythagorean A Major Scale
“Pythagorean     Frequency        String “Fret”   Factor to obtain
Scale”           ratio            ratio           next ratio
F to 1 (1<F<2)
A                1= 440           1               9/8
B                9/8              8/9             256/243
C#               32/27            27/32           9/8
D                4/3              3/4             9/8
Perfect Fourth
E                3/2              2/3             9/8
Perfect Fifth
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       Frequency        String “Fret”   Factor to obtain
Scale”           ratio            ratio and       next ratio
F to 1 (1<F<2)   complement

Do               4:4=1            1        0      9/8
Re               3/2:2/3=9/4      8/9     1/9     10/9
=9/8
Mi               5:4=5/4          4/5     1/5     16/15
Fa               2:3/2=4:3        3/4     1/4     9/8
Perfect Fourth   =4/3
Sol              6:4=3:2          2/3      1/3    10/9
Perfect Fifth    =3/2
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      Frequency ratio   String “Fret”   Factor to
Scale”          F to 1 (1<F<2)    ratio and       obtain next
complement      ratio
A               1=440             1        0      9/8
B               9/8               8/9     1/9
C#              5/4               4/5     1/5     16/15
Major Third
D              4/3                3/4     1/4     9/8
Perfect Fourth
E               3/2               2/3      1/3    10/9
Perfect Fifth
F#              5/3               3/5             9/8
Major Sixth
G#              15/8              8/15            16/15
C Octave        2=880             1/2      1/2
Even Tempered Scale
Based on Equal “step” R1.05946
Frequency        String “Fret”   Factor to obtain
“Even Tempered   ratio            ratio           next ratio
Scale”           F to 1 (1<F<2)
Do               1                            1   R

Re               R2                    0.890899   R

Mi               R4                    0.793701   R
Fa               R5 1.335             0.749154   R
Perfect Fourth
Sol              R7 1.498               0.66742 R
Perfect Fifth
La               R9                    0.561231 R
Ti               R11                   0.529732 R
Do               R12= 2                     0.5
A Major Even Tempered Scale
Based on Equal “step” R1.05946
Frequency        String “Fret”         Factor to obtain
“Even Tempered   ratio            ratio                 next ratio
Scale” A         F to 1 (1<F<2)
A = 440          1 = 440                      1         R

B = 493.88       R2                    0.890899         R

C# = 554.37      R4                    0.793701         R

D = 587.33       R5 1.335             0.749154         R

E = 659.26       R7 1.498               0.66742 R
F# = 739.99      R9                    0.561231 R
G# = 830.61      R11                   0.529732 R
A = 880          R12= 2 = 880                     0.5
Comparison
Just vs Even Tempered
Just F ratio     Just Fret Ratio   ET F ratio   ET Fret Ratio
1                1                         1               1
9/8              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
Fret position
Frequency
Note   Fret                     from saddle on
(1st string)
Martin 0-16NY

E      open       329.6              25
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”         • Something “Borrowed”
Ain’t She Sweet           Lulu’s Back in Town
Java Jive                 S’Wonderful
Teddy Bears’ Picnic       Good Luvin’
Sunshine / Railroad       Be Friends with you
This Land                 Don’t think Twice
Johnny B Goode            The Story of Love
• Something “New”         • The Blues
Tomorrow I’ll be gone     Down and Out
Whisper It in My Ear      Jesse Fuller Medley
I Wanna’ Be with You      The Dink Song
The Rain Song             You got me …
I gotta’ woman            Trouble in Mind
Thanks
The End!


Refreshments Outside
Please-
No 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

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