Mathematics Music and the Guitar Martin Flashman Visiting

							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