music: an interdisciplinary combination of physics, mathematics, and biology

Music: An Interdisciplinary Combination of Physics,

Mathematics, and Biology

Steven A. Jones

This Presentation Draws From

1.General Engineering Background2.One year graduate sequence in

acoustics (Lecture + Lab, with thanks to Dr. Vic Anderson)

3.Thirty years of playing guitar4.20 Years of Research in Medical

Ultrasound

This Presentation Might Help

1. The beginning guitarist2. The self-taught guitarist3. The formally trained guitarist4. Engineers who know little about

music5. Musicians who want a new

perspective

My Musical History

1. Piano Lessons (Miss Berry)2. Uncles (The Banana Song)3. Uncle Warren’s Guitar4. Led Zeppelin5. Fake Books6. Classical guitar (Sor)7. 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. Triples2. 5-tuples (hippopotamus)3. Other African animals

3. What is interpretation?4. Concept of Voices5. Cerebellar Function6. How keys work7. How chords work

More Stuff I Didn’t Know8. Harmonic Analysis9. What tone is10.Rubato11.Dynamics12.Major/Minor 3rd/5th/7th (Close Encounters)13.Major/Minor Scales

The Guitar

Bridge Nut

Each string is a “vibrating string” fixed at both ends.

Thin Strings

Thick Strings

Vibration of a String

The wave equation

• Behavior in time is the same as behavior in space• Wave Speed depends on tension (T) and string density

per unit length ( )

Tc

2

22

2

2

xyc

ty

y

x

t1 t2

(thanks to V.C. Anderson and J.W. Miles)

General Solution to the Equations

xctgxctftxy )(),(

Meaning: The string shape can propagate along the string in the forward and/or reverse direction.

Initially Stationary String

xgxf )(

If the string is not moving initially, must have:

0)(0),(00

xgcxfcttxyv

t

0-x +x

Forward and reverse waves are inverted copies (except for constant).

Boundary Conditions

0),(0),0(

tLytyBoundary Conditions

)()0,()()0,(

xvtxy

xfxy

Initial Conditions 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

1221 00),0( AAeAeAty titi

kxtikxti eAeAtxy 21),(

...,,3,2,1;20, nLnctLy

Assume simple harmonic motion:

tAtAkx

ekxiAeeAetxy

ir

tiikxikxti

cossinsin2Resin2,

Harmonics of a String

...,,3,2,1; nnL

1

Harmonics of a String

...,,3,2,1; nnL

1

2Node

Harmonics of a String

...,,3,2,1; nnL

1

3 2Node

Harmonics of a String

...,,3,2,1; nnL

1

43 2Node

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.

ColorFourier 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#, BDo 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#, BDo Re Mi Fa Sol La Ti

E

BG

FD

Every Good Boy Deserves Favor (Moody Blues)

The Circle of Fifths

CC#

D

D#

E

FF#GG#

A

A#

B

Come Get Down And Eat Big Fat Cod

The Circle of Fifths

CC#

D

D#

E

FF#GG#

A

A#

B

Come Get Down And Eat Big Fat Cod

The Circle of Fifths

CC#

D

D#

E

FF#GG#

A

A#

B

Come Get Down And Eat Big Fat Cod

The Circle of Fifths

CC#

D

D#

E

FF#GG#

A

A#

B

Come Get Down And Eat Big Fat Cod

The Circle of Fifths

CC#

D

D#

E

FF#GG#

A

A#

B

Come Get Down And Eat Big Fat Cod

The Circle of Fifths

CC# = Db

D

D#

E

FF#GG#

A

A#

B

Come Get Down And Eat Big Fat Duck

The Frequencies (Piano Keyboard)

C, C#, D, D#, E, F, F#, G, G#, A, A#, B, CDo Re Mi Fa Sol La Ti

Why are there no sharps (black keys) between E&F and B&C?

E GC

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

MajorC, C#, D, D#, E, F, F#, G, G#, A, A#, BMinorC, C#, D, D#, E, F, F#, G, G#, A, A#, BDiminishedC, C#, D, D#, E, F, F#, G, G#, A, A#, BAugmentedC, C#, D, D#, E, F, F#, G, G#, A, A#, B

Chords

Diminished SeventhC, 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.• Must have • Thus, r = 0.943874313• Or 1/r = 1.059463094

12112 2121 rr

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-cBut note that the second “c” doesn’t work. It’s

000.1,0679.120482187

2311

7

not

“Error” of FrequenciesN Ratio Note % Error Fraction

0 1.0000 C 0 11 1.0595 C# 0.29 1 1/16 ++/2 1.1225 D 0.23 1 ⅛ ++/++3 1.1892 D# 1.90 1 1/6 /4 1.2599 E 0.79 1 ¼ ++/++5 1.3348 F 0.12 1 ⅓ ++/++6 1.4142 F# 2.77 1 ⅜ /++7 1.4983 G 0.11 1 ½ +++/+++8 1.5874 G# 2.37 1 ⅝ /++9 1.6818 A 0.90 1 ⅔ +/++10 1.7818 A# 1.78 1 ¾ /++11 1.8877 B 0.68 1 ⅞ ++/++12 2.0000 C 0

Harmonics of a StringN Octave N/O Note

1 1 1 C2 2 1 C3 2 1 ½ G *4 4 1 C5 4 1 ¼ E *6 4 1.5 G7 4 1 ¾ Bb *8 8 1 C9 8 1 1/8 D10 8 1 ¼ E11 8 1 3/8 F# (ish)12 8 1 ½ B13 8 1 5/8 G# (ish)

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 StringN Octave N/O Note

1 1 1 C2 2 1 C3 2 1 ½ G *4 4 1 C5 4 1 ¼ E *6 4 1.5 G7 4 1 ¾ Bb *8 8 1 C9 8 1 1/8 D10 8 1 ¼ E11 8 1 3/8 F# (ish)12 8 1 ½ B13 8 1 5/8 G# (ish)

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

17 Tone Scale

0 1.0000 C  1 1.0416    2 1.0850    3 1.1301 D 1.12254 1.1771 D# 1.18755 1.2261    6 1.2772 E 1.257 1.3303 F 1.3338 1.3857 F# 1.3759 1.4433    

10 1.5034 G 1.511 1.5660    12 1.6311 G# 1.62513 1.6990 A 1.66714 1.7697 A# 1.7515 1.8434 B 1.87516 1.9201    17 2.0000    

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

4

42

2

2

xyE

ty

gyrationofradius• Boundary Conditions

– Clamped• Displacement = 0 (y)• Slope = 0 (1st derivative wrt x)

– Free• Bending Moment = 0 (2nd derivative wrt x)• Shear Force = 0 (3rd derivative wrt x)

Vibrating Bar• Allowed Frequencies Are Roots of:

12cos2cosh ll

These are not integer multiples of one another.Sound is different from Bach’s “Metallic”

etc.;25.2;2494.1;2597.0

3

2

1

lll

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 InstrumentsPiano 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

C, C#, D, D#, E, F, F#, G, G#, A, A#, B

E

BG

FD

C Major

D Minor

E Minor

G Major

F Major

A Minor

B Dim

4 steps (Maj 3rd)

3 steps (Minor 3rd)

4 steps (Maj 3rd)

3 steps (Minor 3rd)

4 3

3 4

3 4

4 3

4 3

3 4

3 3

Starting with… the chord will be a

Diminished Chords

C, C#, D, D#, E, F, F#, G, G#, A, A#, B

E

BG

FD

C Major

C Minor

C Diminished

4 steps (Maj 3rd)

3 steps (Minor 3rd)

3 steps (Minor 3rd)

3 steps (Minor 3rd)

4 3

3 4

Starting with… the chord will be a

3 3♭♭♭

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).

Augmented Chords

C, C#, D, D#, E, F, F#, G, G#, A, A#, B

E

BG

FD

C Major

C Augmented

4 steps (Maj 3rd)

3 steps (Minor 3rd)

4 steps (Major 3rd)

3 4

Starting with… the chord will be a

4 4

#

In a Major chord (4-3) move minor 3rd to a major 3rd to get the augmented chord (4-4).

Chords with Added Notes

C, C#, D, D#, E, F, F#, G, G#, A, A#, B

E

BG

FD

C Major C Major 7th

10 steps (Minor 7th)

11 steps (Major 7th )

3 4

C Minor 7th (Blues)

Major 7th

Minor 7th

♭

1 2 3 4 5 6 7

Ninth Chords

C, C#, D, D#, E, F, F#, G, G#, A, A#, B, C, C#, D

E

BG

FD

C Major C 9th

14 steps (9th )

3 4

9th

1 2 3 4 5 6 7 8 9

A D is added in the next octave up.

Inverted Chords

E

BG

FD

C Major C Major Inversions Root Position

3 4

C Major First Inversion

C Major Second Inversion

The “Standard” C Chord

E

BG

FD

C Major C Major Inversion Root Position

3 4

C Major First Inversion

C Major Second Inversion

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.

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

E

BG

FD

Whole Note

Half Note

Quarter Note

Eighth Note

44

Sixteenth Notes

½ + ¼ + ⅛ + 1/16 + 1/16 = 11

One Measure

Another Measure

Common (4-4) time

Combinatorics

44

Consider only quarter notes. Let the note be on or off, and go for 2 measures in common time (8 beats).

44

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.

44

ONE two Three four ONE two Three four

The measure tells us where the stress is.

To Add

• Resolution• Rhythm• Rubato• Dynamics• Amplitude Modulation• Fingernails• Key Signatures