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Physics, Mathematics, Music
and Perhaps a Bit of Biology
Steven A. Jones
This Presentation Draws From
1.
2.
3.
4.
General Engineering Background
One year graduate sequence in acoustics
Twenty nine years of playing guitar
19 Years of Research in Medical Ultrasound
Stuff I Didn’t Know
1. Where the notes are on the guitar (vs piano).
2. Importance of rhythm
1. Triples
2. 5-tuples
3. Other African animals
3.
4.
5.
6.
7.
What is interpretation?
Concept of Voices
Cerebellar Function
How keys work
How chords work
More Stuff I Didn’t Know
8. Harmonic Analysis
9. What tone is
10. Rubato
11. Dynamics
12. Major/Minor 3rd/5th etc.
13. Major/Minor Scales
Vibration of a String
The wave equation
2
2 y

y
2
c
2
t
x 2
T
c

• Behavior in time is the same as behavior in space
• Wave Speed depends on tension (T) and string density
per unit length (  )
t1
y
x
t2
General Solution to the
Equations
y( x, t )  f (ct  x)  g ct  x 
Meaning: The string shape can propagate along the
string in the forward and/or reverse direction.
Initially Stationary String
If the string is not moving initially, must have:
v 0 
y ( x, t )
 0  cf ( x )  cg x   0
t t 0
f ( x)   g  x 
-x
0
+x
Forward and reverse
waves are inverted
copies (except for
constant).
Boundary Conditions
Boundary Conditions
Initial Conditions
y (0, t )  0
y ( L, t )  0
y ( x,0)  f ( x )
y ( x,0)
 v( x )
t
Plucked String
Struck String
Boundary Conditions Constrain
Allowable Frequencies
Assume simple harmonic motion:
y( x, t )  A1ei t kx   A2 ei t kx 
y(0, t )  0  A1eit  A2 eit  0  A2   A1
 y x, t   Aeit e ikx  e ikx   2iA sin kxe it
Re   2 sin kx Ar sin t   Ai cost 
2nc
y L, t   0   
; n  1, 2, 3, ..., 
L
Harmonics of a String
L
  ; n  1, 2, 3, ..., 
n
String Shapes/Vibration
Modes
– 1st Harmonic is a Sine Wave
– 2nd Harmonic is 2x the frequency of the 1st
– Since the middle of the string doesn’t move
for 2nd harmonic, can touch it there & still get
vibration.
– 3rd harmonic has two nodes (at 1/3 and 2/3rds
the string length)
– The “harmonics” give pure tones.
– Can do harmonics with fretted strings.
Color
Fourier Interpretation:
Tone depends on the relative loudness &
phases of each harmonic.
The Frequencies (Musician’s
Terminology)
•
•
•
•
•
•
•
C, C#, D, D#, E, F, F#, G, G#, A, A#, B
Do
Re Mi Fa
Sol
La
Ti
Major 3rd (C to E)
Major 5th (C to G)
Minor 3rd (C to D#)
Major/Minor 7th (C to A# / B)
Barbershop Quartet (C, E, G, Bb)
The Frequencies (Musician’s
Terminology)
C, C#, D, D#, E, F, F#, G, G#, A, A#, B
Do
Re Mi Fa
Sol
La
Ti
Why are there no sharps (black keys)
between E&F and B&C?
Chords
Major
C, C#, D, D#, E, F, F#, G, G#, A, A#, B
Minor
C, C#, D, D#, E, F, F#, G, G#, A, A#, B
Diminished
C, C#, D, D#, E, F, F#, G, G#, A, A#, B
Augmented
C, C#, D, D#, E, F, F#, G, G#, A, A#, B
Chords
Diminished Seventh
C, C#, D, D#, E, F, F#, G, G#, A, A#, B
Note that C7dim is the same as D#7dim, F#7dim and
A7dim.
This is an ambiguous chord and can resolve into
many possible chords.
There are only 4 diminished 7th chords.
Frequencies Used in Music
Frets on a Guitar
• Each fret shortens the string by the same
percentage (r) of it’s current length.
• Frets must get closer together.
• Takes 12 frets to get to ½ the length.
1 12
12


• Must have r  1 2  r  1 2
• Thus, r = 0.943874313
• Or 1/r = 1.059463094
Postulates
Tones separated by nice fractional
relationships are pleasing.
Tones separated by complicated fractional
relationships are less pleasing.
These postulates are the basis of “Just
Intonation”
(Slogan: “It’s not just intonation, its Just
intonation!”)
Pythagorean Scale
• Pythagorus proposed the scale cdefgab,
based on a series of “perfect 5ths”
• c=1
• g = cx3/2 (i.e. 1 ½)
• d = (gx3/2)/2 = c x (9/4)/2 (i.e. 1 1/8)
• a = (dx3/2) = c x (27/16) (i.e. 1 11/16)
Circulate through c-g-d-a-e-b-f-c
But note that the second “c” doesn’t work. It’s
 37   2187 
 11   
  1.0679
 2   2048 
Values of Frequencies
N
Ratio
Note
0
1
2
3
4
5
6
7
8
9
10
11
12
1.0000
1.0595
1.1225
1.1892
1.2599
1.3348
1.4142
1.4983
1.5874
1.6818
1.7818
1.8877
2.0000
C
C#
D
D#
E
F
F#
G
G#
A
A#
B
C
Approximation
1
1 1/16
1 1 /8
1¼
1 1/3
1 3/8? (=1.375)
1½
1 5/8? (=1.625)
1 2 /3
1¾
1 7/8 (=1.875)
Pythagorean
1 1 /8
1.266
1 1/3
1½
1.687
1.898
Harmonics of a String
N
Octave N/O
Note
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
2
4
4
4
4
8
8
8
8
8
8
C
C
G
*
C
E
*
G
Bb
*
C
D
E
F# (ish)
B
G# (ish)
1
1
1½
1
1¼
1.5
1¾
1
1 1 /8
1¼
1 3 /8
1½
1 5 /8
When you play a
C, you are also
playing G, E, Bb,
etc. in different
amounts and in
Just Intonation.
Harmonics
• Were not invented by Yes or Emerson,
Lake and Palmer.
• Can be combined with natural tones.
– Granados
Historical Notes
• 12 tone system (Even Tempered Scale)
relatively recent invention (ca. 1700s).
• Bach used “Well Tempered Scale”
• We don’t hear Toccata & Fugue the way it
was written.
Why 12 Notes?
• Even temperament would not work as well
with other spacings (besides 1/12)
• Works pretty well with a spacing of 17.
– Other countries use a 17 tone system.
– Google: “17 tone” music
0
1.0000
C
1
1.0416
2
1.0850
3
1.1301
D
1.1225
4
1.1771
D#
1.1875
5
1.2261
6
1.2772
E
1.25
7
1.3303
F
1.333
8
1.3857
F#
1.375
9
1.4433
10
1.5034
G
1.5
11
1.5660
12
1.6311
G#
1.625
13
1.6990
A
1.667
14
1.7697
A#
1.75
15
1.8434
B
1.875
16
1.9201
17
2.0000
17 Tone
Scale
Color
• Determined by the weights of the higher
harmonics.
• Musette ….
– Bach denotes the “crisper” sound as
“metallic.”
– What is “metallic?”
Vibrating Bar
• Equation
• Boundary Conditions
• Harmonics
– Not integer multiples
– Sound speed depends on frequency
– A cacaphony of sounds
– Nodes
– Damping
Vibrating Bar
• Equation
2 y
 2E 4 y

2
t
 x 4
• Boundary Conditions
  radius of gyration
– Clamped
• Displacement = 0 (y)
• Slope = 0
(1st derivative wrt x)
– Free
• Bending Moment = 0
• Shear Force = 0
(2nd derivative wrt x)
(3rd derivative wrt x)
Vibrating Bar
• Allowed Frequencies Are Roots of:
cosh2l  cos2l   1
1  0.597 2l ;
 2  1.494 2l ;
 3  2.5 2l ; etc.
These are not integer multiples of one another.
Sound is different from Bach’s “Metallic”
Vibrating Bar
A tuning fork is a bent vibrating free-free bar
held at the center node.
• Higher Modes Damp Out Quickly
• 1st mode provides a pure tone.
Tuning Fork
• When you strike a tuning fork, at first the
tone sounds harsh, but then it’s very very
pure.1
1 With apologies to James Joyce.
Fourier View Breakdown
Notes are finite in time.
Stopping Strings
Damping
Notes are not discrete frequencies – they are
broadened.
Breakdown of the Fourier View
(Goodbye Fourier Series, Hello
Fourier Transform)
A#
A# (smeared)
Neurophysiology
• It is hard to imagine being able to make
the complicated movements of playing a
musical instrument.
• Much of music is performed by the
cerebellum
• Purkinje cells adapt and change as we
learn.
• Practice from different starting points.
Capabilities of Instruments
Piano
Guitar
Violin
Flute
Harmonica
Note Range
+++
++
+
+
-
Dynamic Range
+++
+
++
++
-
Note Duration
++
+
+++
+++
++
Vibrato (FM)
-
++
++
+
+
Tone
+
++
++
+
+
Tremolo (AM)
-
++
Harmonics
-
++
+
Multiple Notes
+++
++
+
-
+
Keys
+++
++
++
+
-
Hammer-on/off
-
+++
++
-
-
Slide
-
++
+++
+
-
Note Bending
-
++
++
++
+++
-
The guitar is a remarkably mediocre instrument.
Dissonance
Major Chord (e.g. C E G) is Pleasant.
It is the unpleasant sounds that give the
most pleasure.
• It feels good when it stops hurting.
• In the context of the familiar, the unfamiliar
holds the most interest.
– Example: Bach’s Prelude in Dm
To Add
•
•
•
•
•
•
•
Resolution
Rhythm
Rubato
Dynamics
Amplitude Modulation
Fingernails
Chords (Major, Minor, Inversion, 3rd, 5th,
Diminished, Augmented).