Transcript document

Music: An Interdisciplinary
Combination of Physics,
Mathematics, and Biology
Steven A. Jones
This Presentation Draws From
1. General Engineering Background
2. One year graduate sequence in
acoustics (Lecture + Lab, with
thanks to Dr. Vic Anderson)
3. Thirty years of playing guitar
4. 20 Years of Research in Medical
Ultrasound
This Presentation Might Help
1. The beginning guitarist
2. The self-taught guitarist
3. The formally trained guitarist
4. Engineers who know little about
music
5. Musicians who want a new
perspective
My Musical History
1.
2.
3.
4.
5.
6.
7.
Piano Lessons (Miss Berry)
Uncles (The Banana Song)
Uncle Warren’s Guitar
Led Zeppelin
Fake Books
Classical guitar (Sor)
Lessons from
1. George Petch (and a new guitar)
2. Ron Pearl (BCGS)
3. Alan Goldspeil
Stuff I Didn’t Know When I
Started
1. Where the notes are on the guitar (vs piano).
2. Importance of rhythm
1. Triples
2. 5-tuples (hippopotamus)
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/7th (Close Encounters)
13. Major/Minor Scales
The Guitar
Thick Strings
Thin Strings
Bridge
Nut
Each string is a “vibrating string” fixed at both ends.
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
(thanks to V.C. Anderson and J.W. Miles)
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
y (0, t )  0
y ( L, t )  0
Initial Conditions
y ( x,0)  f ( x )
y ( x,0)
 v( x )
t
Plucked String
Struck String
Not for the Squeamish
• Warning: Those who tend to pass out at
the sight of math may want to leave the
room before I present the next slide. I will
let you know when it is safe to come back
into the room again.
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
1
L
  ; n  1, 2, 3, ..., 
n
Harmonics of a String
1
Node
L
  ; n  1, 2, 3, ..., 
n
2
Harmonics of a String
1
Node
3
L
  ; n  1, 2, 3, ..., 
n
2
Harmonics of a String
1
Node
3
4
L
  ; n  1, 2, 3, ..., 
n
2
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.
I.e. a string with 1st and 2nd harmonic excited
sounds different from a string with 1st and 3rd
harmonic excited.
Can excite different harmonics by plucking at
different locations (i.e. plucking at 1/3rd length
will mute the 3rd 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
F
D
B
G
E
Every Good Boy Deserves Favor (Moody Blues)
The Circle of Fifths
C
B
C#
D
A#
D#
A
E
G#
G
F#
F
Come Get Down And Eat Big Fat Cod
The Circle of Fifths
C
B
C#
D
A#
D#
A
E
G#
G
F#
F
Come Get Down And Eat Big Fat Cod
The Circle of Fifths
C
B
C#
D
A#
D#
A
E
G#
G
F#
F
Come Get Down And Eat Big Fat Cod
The Circle of Fifths
C
B
C#
D
A#
D#
A
E
G#
G
F#
F
Come Get Down And Eat Big Fat Cod
The Circle of Fifths
C
B
C#
D
A#
D#
A
E
G#
G
F#
F
Come Get Down And Eat Big Fat Cod
The Circle of Fifths
C
B
C# = Db
D
A#
D#
A
E
G#
G
F#
F
Come Get Down And Eat Big Fat Duck
The Frequencies (Piano
Keyboard)
C
E
G
C, C#, D, D#, E, F, F#, G, G#, A, A#, B, C
Do
Re Mi Fa
Sol
La
Ti
Why are there no sharps (black keys)
between E&F and B&C?
Other Questions
• Why are there 12 notes?
• Why are 7 of these “in key?”
• Why the “Circle of Fifths?” (Why not the
“Circle of Thirds?”)
• What’s all this about Major and Minor?
• What do Augmented and Diminshed Mean?
• Why are there different chords with the
same name?
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
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
1. Tones separated by nice fractional
relationships are pleasing.
2. 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, not 1.000
 2   2048 
“Error” of Frequencies
N
Ratio
Note
% Error
Fraction
0
1
1.0000
1.0595
C
C#
0
0.29
1
1 1/16
2
1.1225
D
0.23
1
3
1.1892
D#
1.90
1 1/6
4
1.2599
E
0.79
1¼
5
1.3348
F
0.12
1⅓
6
1.4142
F#
2.77
1
7
1.4983
G
0.11
8
1.5874
G#
2.37
9
1.6818
A
0.90
10
1.7818
A#
1.78
11
12
1.8877
2.0000
B
C
0.68
0
⅛
⅜
1½
1⅝
1⅔
1¾
1⅞
++/
++/++
/
++/++
++/++
/++
+++/+++
/++
+/++
/++
++/++
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. I.e., the
combination CEG is
“natural.”
Higher harmonics
decay rapidly.
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. I.e., the
combination CEG is
“natural.”
Higher harmonics
decay rapidly.
Harmonics
• Have a “pure” tone to them.
• Were not invented by Yes or Emerson,
Lake and Palmer.
• Can be combined with natural tones.
– Granados
Notes that are in key …
• Are close matches to “nice” fractional
values.
• Match the natural harmonics of the
vibrating string (C E G D B Bb)
• Are early members of the circle of 5ths (C
G D A E B).
More Complex Relationships
• Inverse Relationships
– F is a Fourth to C
– C is a Fifth to F
• Bootstrapping
– G# is not in C, but
– E is the Major 3rd in C and
– G# is the Major 3rd of E so
– Perhaps we can get to the G# note through E
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.
Breakdown of the Fourier View
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
Stacked Chords
3 steps
(Minor 3rd)
4 steps
(Maj 3rd)
3 steps
(Minor 3rd)
4 steps
(Maj 3rd)
C, C#, D, D#, E, F, F#, G, G#, A, A#, B
F
D
B
G
E
Starting with…
the chord will be a
3
4
C
Major
4
3
D
Minor
4
3
E
Minor
3
4
3
4
4
3
F
G
Major Major
3
3
A
B
Minor Dim
Diminished Chords
3 steps
(Minor 3rd)
3 steps
(Minor 3rd)
3 steps
(Minor 3rd)
4 steps
(Maj 3rd)
In a Major chord (4-3) move
Major 3rd to Minor 3rd to get the
minor chord (3-4). Move the
major 3rd in the minor chord
down to a minor 3rd to get the
diminished chord (3-3).
C, C#, D, D#, E, F, F#, G, G#, A, A#, B
F
D
B
G
E
Starting with…
the chord will be a
3
4
C
Major
4
3♭
C
Minor
3♭
♭
3
C
Diminished
Augmented Chords
In a Major chord (4-3)
move minor 3rd to a
major 3rd to get the
augmented chord (4-4).
4 steps
(Major 3rd)
3 steps
(Minor 3rd)
4 steps
(Maj 3rd)
C, C#, D, D#, E, F, F#, G, G#, A, A#, B
F
D
B
G
E
Starting with…
the chord will be a
3
4
C
Major
4#
4
C
Augmented
Chords with Added Notes
10 steps
(Minor 7th)
11 steps
(Major 7th )
C, C#, D, D#, E, F, F#, G, G#, A, A#, B
1
2
F
D
B
G
E
3
3
4
C Major
4
Major
7th
5
6
Minor
7th
C Major 7th
7
♭
C Minor 7th
(Blues)
Ninth Chords
14 steps
(9th )
C, C#, D, D#, E, F, F#, G, G#, A, A#, B, C, C#, D
1
2
F
D
B
G
E
3
3
4
4
5
9th
C Major
A D is added in the next octave up.
C 9th
6
7
8
9
Inverted Chords
F
D
B
G
E
3
4
C Major
C Major
Inversions
Root Position
C Major
First
Inversion
C Major
Second
Inversion
The “Standard” C Chord
The C chord that is familiar to every
guitarist can be plucked as several
inversions.
The low E (6th string) is in key, but
sounds harsh on the bottom.
F
D
B
G
E
3
4
C Major
C Major
Inversion
Root Position
C Major
First
Inversion
C Major
Second
Inversion
What is Interpretation?
• If I play the notes exactly as written on the
page, am I not playing the piece correctly?
• No. Interpretation is necessary to a
successful performance and includes:
– Speeding up & slowing down (rubato, firmata)
– Increasing/decreasing volume (dynamics,
piano, forte)
– Modulating notes (vibrato)
– Ornamentation (trills, slurs)
Wrong Thinking
• I try to play the piece exactly the way
Segovia does because it would be
egotistical of me to think that I could play it
better than him.
• I (Rod Stewart) try to sing everything like
Pavarotti does because it would be
egotistical of me to think that I could sing it
better than him.
Rhythm
•
•
•
•
Important
Triples
Quintuples
Septuples
Rhythm
• We tend to think of music in terms of different
notes, but the duration of each note and the
timing of each note is just as important as its
pitch.
• Not playing notes can be just as important as
playing the right notes at the right time.
• Analogy to photography – A picture can be
ruined by additions to the image. E.g. a lunch
wrapper in front of the Venus de Milo, or a palm
tree growing out of uncle Ned’s head.
• Musical notation is explicit about when not to
play sounds.
Note Durations
1
F
D
B
G
E
Common
(4-4) time
½
+
¼
+
⅛
+
1/
16
+ 1/16 = 1
4
4
Whole
Note
One
Measure
Half
Note
Quarter
Note
Eighth
Note
Another Measure
Sixteenth
Notes
Combinatorics
Consider only quarter notes. Let the note be on or off,
and go for 2 measures in common time (8 beats).
4
4
4
4
For just these two measures, and for just quarter
notes, there are 28 = 256 combinations. A huge
amount of the variety in music stems from the
myriad of possible rhythm combinations.
Strong and Weak Beats
In common time, the first note is typically
strong. The third beat is the next strongest.
ONE two Three
four
ONE two Three
four
4
4
The measure tells us where the stress is.
To Add
•
•
•
•
•
•
•
Resolution
Rhythm
Rubato
Dynamics
Amplitude Modulation
Fingernails
Key Signatures