Music as a Mathematicians Playground Julian Hook Department of - - PowerPoint PPT Presentation

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Music as a Mathematicians Playground Julian Hook Department of - - PowerPoint PPT Presentation

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June 22, 2017 Music as a Mathematicians Playground Julian Hook Department of Music Theory Jacobs School of Music Indiana University Bloomington F G A C D F G A C D G A B D E G A

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SLIDE 1

Music as a Mathematician’s Playground

Julian Hook Department of Music Theory Jacobs School of Music Indiana University Bloomington

June 22, 2017

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SLIDE 2

G A B C D E F G A B C D E

F♯ G♭ G♯ A♭ A♯ B♭ C♯ D♭ D♯ E♭ F♯ G♭ G♯ A♭ A♯ B♭ C♯ D♭ D♯ E♭ G A B C D E F G A B C D E F♯ G♯ A♯ C♯ D♯ F♯ G♯ A♯ C♯ D♯ Pitch space (chromatic scale)

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SLIDE 3

1 2 3 4 5 6 7 8 9 10 11

G A B C D E F G A B C D E F♯ G♯ A♯ C♯ D♯ F♯ G♯ A♯ C♯ D♯ C D E F G A B C♯ D♯ F♯ G♯ A♯ C D E F G A B C♯ D♯ F♯ G♯ A♯

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 −1 −2 −3 −4 −5 −6

Pitch space Pitch-class space (ℤ) (ℤ12)

  • ne octave
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SLIDE 4

C D E F G A B C♯ D♯ F♯ G♯ A♯

1 2 3 4 5 6 7 8 9 10 11

C D E B C♯ D♯ F G A F♯ G♯ A♯

7 2 9 4 11 6 1 8 3 10 5

Pitch-class space Tie circle of fjfuhs

+1


(mod 12)

+7


(mod 12)

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SLIDE 5

C D E F G A B C♯ D♯ F♯ G♯ A♯

1 2 3 4 5 6 7 8 9 10 11

C D E B C♯ D♯ F G A F♯ G♯ A♯

7 2 9 4 11 6 1 8 3 10 5

Pitch-class space Tie circle of fjfuhs

×7 (mod 12)

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SLIDE 6

3 6 9 3 6 9 6 8 11 2 5 8 11 2 5 2 4 7 10 1 4 7 10 1 10 4 7 10 1 4 7 10 1 10 8 11 2 5 8 11 1 5 2 3 6 9 3 6 9 6 4 7 10 1 4 7 10 1 10 8 11 2 5 8 11 2 5 2

ℤ12 ≅ ℤ4 × ℤ3

+3


(mod 12)

+4


(mod 12)

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SLIDE 7

3 6 9 3 6 9 6 8 11 2 5 8 11 2 5 2 4 7 10 1 4 7 10 1 10 4 7 10 1 4 7 10 1 10 8 11 2 5 8 11 1 5 2 3 6 9 3 6 9 6 4 7 10 1 4 7 10 1 10 8 11 2 5 8 11 2 5 2

G D F G♯ B D F G♯ B D A♯ C♯ E G A♯ C♯ E G A♯ F♯ A C D♯ F♯ A C D♯ F♯ D F G♯ B D F G♯ B D A♯ C♯ E G A♯ C♯ E G A♯ F♯ A C D♯ F♯ A C D♯ F♯ D F G♯ B D F G♯ B D A♯ C♯ E G A♯ C♯ E A♯

ℤ12 ≅ ℤ4 × ℤ3

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SLIDE 8

3 6 9 3 6 9 6 8 11 2 5 8 11 2 5 2 4 7 10 1 4 7 10 1 10 4 7 10 1 4 7 10 1 10 8 11 2 5 8 11 1 5 2 3 6 9 3 6 9 6 4 7 10 1 4 7 10 1 10 8 11 2 5 8 11 2 5 2

G D F G♯ B D F G♯ B D A♯ C♯ E G A♯ C♯ E G A♯ F♯ A C D♯ F♯ A C D♯ F♯ D F G♯ B D F G♯ B D A♯ C♯ E G A♯ C♯ E G A♯ F♯ A C D♯ F♯ A C D♯ F♯ D F G♯ B D F G♯ B D A♯ C♯ E G A♯ C♯ E A♯

ℤ12 ≅ ℤ4 × ℤ3

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SLIDE 9

3 6 9 4 7 10 1 8 11 2 5

G♯ B D F E G A♯ C♯ C D♯ F♯ A Tonnetz: Leonhard Euler (1739) Hugo Riemann (1915) Richard Cohn (1997) (and many others)

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SLIDE 10

C C C C

C

Infjnite Tonnetz Finite Tonnetz

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SLIDE 11

Infjnite Tonnetz (planar graph) Finite Tonnetz (non-planar graph) complete graph K5

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SLIDE 12

3 8 1 10 5 7 2

D G C F A♯ D♯ G♯ C♯

9 8 7 11 1 2

G D C♯ C B A G♯

+7


(mod 12)

+1


(mod 12)

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SLIDE 13

6 3 6 9 6 11 2 5 11 2 5 2 4 7 10 1 4 10 10 4 7 1 4 7 10 1 10 8 2 8 11 1 5 2 3 6 9 3 6 9 6 4 7 10 1 4 10 10 8 11 2 5 8 11 5

F G♯ B D F G♯ B A♯ E G A♯ C♯ E A♯ F♯ A D♯ F♯ A D♯ F♯ D F G♯ D G♯ B D A♯ C♯ E G C♯ E G A♯ F♯ A C F♯ C D♯ F♯ D F B D F B D A♯ E G A♯ C♯ E A♯

3 9 8 8 7 1 10 11 5 7 1 2 2

G D D C♯ G C C B F A♯ D♯ A G♯ G♯ C♯

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SLIDE 14

CEG

(C major triad)

CE♭G

(C minor triad) F♯ A C E♭ F♯ D F G♯ B D B♭ C♯ E G B♭ F♯ A C E♭ F♯ D F G♯ B D

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SLIDE 15

major triads minor triads

F♯ A C E♭ F♯ D F G♯ B D B♭ C♯ E G B♭ F♯ A C E♭ F♯ D F G♯ B D

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SLIDE 16

E G C

CEG

(C major)

CE♭G

(C minor)

P

Parallel

E♭

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SLIDE 17

E G C

CEG

(C major)

ACE

(A minor)

P R

Parallel Relative

E♭ A

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SLIDE 18

E G C

CEG

(C major)

EGB

(E minor)

P R L

Parallel Relative Leituonwechsel

E♭ A B

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SLIDE 19

F♯ A C E♭ F♯ D F G♯ B D B♭ C♯ E G B♭ F♯ A C E♭ F♯ D F G♯ B D

P P P P

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SLIDE 20

F♯ A C E♭ F♯ D F G♯ B D B♭ C♯ E G B♭ F♯ A C E♭ F♯ D F G♯ B D

R R R R

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SLIDE 21

F♯ A C E♭ F♯ D F G♯ B D B♭ C♯ E G B♭ F♯ A C E♭ F♯ D F G♯ B D

L L L L

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SLIDE 22

F♯ A C E♭ F♯ D F G♯ B D B♭ C♯ E G B♭ F♯ A C E♭ F♯ D F G♯ B D

L P R R P

C♯ minor E♭ major

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SLIDE 23

F♯ A C E♭ F♯ D F G♯ B D B♭ C♯ E G B♭ F♯ A C E♭ F♯ D F G♯ B D

P R R P

C major P R C major P R A major

PR ≠ RP

E♭ major

(!)

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SLIDE 24

11 2 8 5 2 3 6 9 6 7 10 4 1 10 3 6 9 6 11 2 8 5 2

pitch-class numbers

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SLIDE 25

11 2 8 5 2 3 6 9 6 7 10 4 1 10 3 6 9 6 11 2 8 5 2

A♭ g♯ F f D d B b E e D♭ c♯ B♭ b♭ G g C c A a F♯ f♯ E♭ e♭ A♭ g♯ F f D d B b

pitch-class numbers minor triads major triads

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SLIDE 26

11 2 8 5 2 3 6 9 6 7 10 4 1 10 3 6 9 6 11 2 8 5 2

A♭ g♯ F f D d B b E e D♭ c♯ B♭ b♭ G g C c A a F♯ f♯ E♭ e♭ A♭ g♯ F f D d B b

pitch-class numbers

L P R

minor triads major triads

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SLIDE 27

A♭ g♯ F f D d B b E e D♭ c♯ B♭ b♭ G g C c A a F♯ f♯ E♭ e♭ A♭ g♯ F f D d B b

major triads minor triads

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SLIDE 28
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SLIDE 29

Brahms, Concerto
 for Violin and Cello

L P

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SLIDE 30

Brahms

Liszt,
 Consolation No. 3

L P L P

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SLIDE 31

Brahms Liszt L P L P R P

Smetana,
 Bartered Bride Overture

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SLIDE 32

Brahms Liszt Smetana L P L P R P L R

Beethoven, Symphony No. 9

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SLIDE 33

Brahms Liszt Smetana Beethoven L P L P R P L R L R

Chopin,
 Fantasy in F Minor

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SLIDE 34

Brahms Liszt Smetana Beethoven Chopin

8

L P L P R P L R L R R L P R P

Verdi,
 Il trovatore

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SLIDE 35

Brahms Liszt Smetana Beethoven Chopin Verdi L P L P R P L R L R R L P R P R P P L P L R L

Schubert,
 “Trout” Quintet

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SLIDE 36

Brahms Liszt Smetana Beethoven Chopin Verdi L P L P R P L R L R R L P R P R P P L P L R L Schubert

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SLIDE 37

Map of interkey distances: Carol Krumhansl (1983)

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SLIDE 38

Map of interkey distances: Carol Krumhansl (1983)

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SLIDE 39

Map of interkey distances: Carol Krumhansl (1983)

R P L

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SLIDE 40

Map of interkey distances: Carol Krumhansl (1983)

11 7 3 3 4 8 8 9 5 1 6 2 10

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SLIDE 41

3 6 9 8 11 2 5 4 7 10 1 8 11 2 5 9 5 1 5 4 7 10 1 1

F G♯ B D C♯ E G A♯ A C D♯ F♯ F G♯ B D F C♯ A F C♯ E G A♯ C♯

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SLIDE 42

3 6 9 8 11 2 5 4 7 10 1 8 11 2 5 9 5 1 5 4 7 10 1 1

F G♯ B D C♯ E G A♯ A C D♯ F♯ F G♯ B D F C♯ A F C♯ E G A♯ C♯ D♯EG D♯F♯G F♯GA♯ F♯AA♯ AA♯C♯ DD♯F♯ DFF♯ C♯DF

Schoenberg,
 Pierrot lunaire, “Nacht”

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SLIDE 43

dominant 7th chords half-diminished 7th chords

3-D Tonnetz: Edward Gollin (1998)

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SLIDE 44

Spiral Array: Elaine Chew (2014)

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SLIDE 45

major triads

Spiral Array: Elaine Chew (2014)

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SLIDE 46

minor triads

Spiral Array: Elaine Chew (2014)

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SLIDE 47

dominant 7th chord

Spiral Array: Elaine Chew (2014)

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SLIDE 48

G A B C D E F G A B C D E

F♯ G♭ G♯ A♭ A♯ B♭ C♯ D♭ D♯ E♭ F♯ G♭ G♯ A♭ A♯ B♭ C♯ D♭ D♯ E♭ E G B C E G A C E G A C E D F A D F B D F B D Diatonic pitch space

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SLIDE 49

C D E F G A B C♯ D♯ F♯ G♯ A♯ C D E B F G A

Chromatic pitch-class space Diatonic pitch-class space (ℤ12) (ℤ7)

C D E B C♯ D♯ F G A F♯ G♯ A♯ C G D F A E B

Chromatic circle of fjfuhs Diatonic circle of fjfuhs

1 2 3 4 5 6 7 8 9 10 11 2 4 6 1 3 5 7 2 9 4 11 6 1 8 3 10 5 1 2 3 4 5 6

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SLIDE 50

C D E F G A B C♯ D♯ F♯ G♯ A♯ C D E B F G A

Chromatic pitch-class space Diatonic pitch-class space (ℤ12) (ℤ7)

C D E B C♯ D♯ F G A F♯ G♯ A♯ C G D F A E B

Chromatic circle of fjfuhs Diatonic circle of fjfuhs

1 2 3 4 5 6 7 8 9 10 11 2 4 6 1 3 5 7 2 9 4 11 6 1 8 3 10 5 1 2 3 4 5 6

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SLIDE 51

C D E F G A B C♯ D♯ F♯ G♯ A♯ C D E B F G A

Chromatic pitch-class space Diatonic pitch-class space (ℤ12) (ℤ7)

C D E B C♯ D♯ F G A F♯ G♯ A♯ C G D F A E B

Chromatic circle of fjfuhs Diatonic circle of fjfuhs

1 2 3 4 5 6 7 8 9 10 11 2 4 6 1 3 5 7 2 9 4 11 6 1 8 3 10 5 1 2 3 4 5 6

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SLIDE 52

1 3 5 6 4 4 6 8 2 5 2 3 1 2 4 6 5 6 1 3 4 2

E G B D C A C E G F D F A C B G B D F E C E G B F

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SLIDE 53

1 3 5 6 4 4 6 8 2 5 2 3 1 2 4 6 5 6 1 3 4 2

E G B D C A C E G F D F A C B G B D F E C E G B F

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SLIDE 54

1 3 5 6 4 4 6 8 2 5 2 3 1 2 4 6 5 6 1 3 4 2

E G B D C A C E G F D F A C B G B D F E C E G B F

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SLIDE 55

1 3 5 6 4 4 6 8 2 5 2 3 1 2 4 6 5 6 1 3 4 2

E G B D C A C E G F D F A C B G B D F E C E G B F

complete graph K7

C E D F A G B

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SLIDE 56

G A B C D E F G A B C D E F♯ G♯ A♯ C♯ D♯ F♯ G♯ A♯ C♯ D♯ C E A B C♯ D♯ F♯ G♯ A♯

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 −1 −2 −3 −4 −5 −6

D F G Pitch space (ℝ) x y z 3-voice pitch space (ℝ3)

x = 2 y = 5 z = 7 (2, 5, 7)

(D, F, G)

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SLIDE 57

G A B C D E F G A B C D E F♯ G♯ A♯ C♯ D♯ F♯ G♯ A♯ C♯ D♯ C D E F G A B C♯ D♯ F♯ G♯ A♯

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 −1 −2 −3 −4 −5 −6

Pitch space (ℝ)

(2, 5, 7)

(D, F, G) D Octave equivalence:

(14, 5, 7)

(D, F, G)

~

(2, 5, 7)

(D, F, G) Permutational equivalence:

(7, 2, 5)

(G, D, F)

~

(2, 5, 7)

(D, F, G) Cardinality equivalence:

(2, 5, 7, 7)

(D, F, G, G)

~

Equivalence relations on pitch space: Callender, Quinn, & Tymoczko (2008)

slide-58
SLIDE 58

2-voice pitch space (ℝ2) x y

slide-59
SLIDE 59

12 12

(0, 0)

(C, C)

(0, 12)

(C, C)

(12, 0)

(C, C)

(12, 12)

(C, C)

2-voice pitch space (ℝ2) x y with octave equivalence

(2, 5)

(D, F)

(5, 2)

(F, D)

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SLIDE 60

DF

with permutational
 equivalence with octave equivalence

CC CC CC

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SLIDE 61

DF

CC CC CC

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SLIDE 62

DF

F♯F♯ CC CC

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SLIDE 63

DF

F♯F♯ F♯F♯ CC CC

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SLIDE 64

DF F♯F♯ F♯F♯ CC CC

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SLIDE 65

CC BC♯ A♯D D♯A EG♯ FG F♯F♯ F♯F♯ FG EG♯ D♯A BC♯ CC A♯D FF EF♯ D♯G DG♯ A♯C BB AC♯ EE D♯F DF♯ C♯G AB A♯A♯ G♯C D♯D♯ DE C♯F CF♯ G♯A♯ AA GB DD C♯D♯ CE FB GA G♯G♯ F♯A♯ C♯C♯ CD BD♯ EA♯ F♯G♯ GG FA FF♯ EG D♯G♯ AD BC A♯C♯ EF D♯F♯ DG G♯C♯ A♯B AC D♯E DF C♯F♯ GC AA♯ G♯B DD♯ C♯E CF F♯B G♯A GA♯ C♯D CD♯ BE FA♯ GG♯ F♯A CC♯ BD A♯D♯ EA F♯G FG♯

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SLIDE 66

CC BC♯ A♯D D♯A EG♯ FG F♯F♯ F♯F♯ FG EG♯ D♯A BC♯ CC A♯D FF EF♯ D♯G DG♯ A♯C BB AC♯ EE D♯F DF♯ C♯G AB A♯A♯ G♯C D♯D♯ DE C♯F CF♯ G♯A♯ AA GB DD C♯D♯ CE FB GA G♯G♯ F♯A♯ C♯C♯ CD BD♯ EA♯ F♯G♯ GG FA FF♯ EG D♯G♯ AD BC A♯C♯ EF D♯F♯ DG G♯C♯ A♯B AC D♯E DF C♯F♯ GC AA♯ G♯B DD♯ C♯E CF F♯B G♯A GA♯ C♯D CD♯ BE FA♯ GG♯ F♯A CC♯ BD A♯D♯ EA F♯G FG♯

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SLIDE 67

CC BC♯ A♯D D♯A EG♯ FG F♯F♯ F♯F♯ FG EG♯ D♯A BC♯ CC A♯D FF EF♯ D♯G DG♯ A♯C BB AC♯ EE D♯F DF♯ C♯G AB A♯A♯ G♯C D♯D♯ DE C♯F CF♯ G♯A♯ AA GB DD C♯D♯ CE FB GA G♯G♯ F♯A♯ C♯C♯ CD BD♯ EA♯ F♯G♯ GG FA FF♯ EG D♯G♯ AD BC A♯C♯ EF D♯F♯ DG G♯C♯ A♯B AC D♯E DF C♯F♯ GC AA♯ G♯B DD♯ C♯E CF F♯B G♯A GA♯ C♯D CD♯ BE FA♯ GG♯ F♯A CC♯ BD A♯D♯ EA F♯G FG♯

slide-68
SLIDE 68

CC BC♯ A♯D D♯A EG♯ FG F♯F♯ F♯F♯ FG EG♯ D♯A BC♯ CC A♯D FF EF♯ D♯G DG♯ A♯C BB AC♯ EE D♯F DF♯ C♯G AB A♯A♯ G♯C D♯D♯ DE C♯F CF♯ G♯A♯ AA GB DD C♯D♯ CE FB GA G♯G♯ F♯A♯ C♯C♯ CD BD♯ EA♯ F♯G♯ GG FA FF♯ EG D♯G♯ AD BC A♯C♯ EF D♯F♯ DG G♯C♯ A♯B AC D♯E DF C♯F♯ GC AA♯ G♯B DD♯ C♯E CF F♯B G♯A GA♯ C♯D CD♯ BE FA♯ GG♯ F♯A CC♯ BD A♯D♯ EA F♯G FG♯

slide-69
SLIDE 69

CC BC♯ A♯D D♯A EG♯ FG F♯F♯ F♯F♯ FG EG♯ D♯A BC♯ CC A♯D FF EF♯ D♯G DG♯ A♯C BB AC♯ EE D♯F DF♯ C♯G AB A♯A♯ G♯C D♯D♯ DE C♯F CF♯ G♯A♯ AA GB DD C♯D♯ CE FB GA G♯G♯ F♯A♯ C♯C♯ CD BD♯ EA♯ F♯G♯ GG FA FF♯ EG D♯G♯ AD BC A♯C♯ EF D♯F♯ DG G♯C♯ A♯B AC D♯E DF C♯F♯ GC AA♯ G♯B DD♯ C♯E CF F♯B G♯A GA♯ C♯D CD♯ BE FA♯ GG♯ F♯A CC♯ BD A♯D♯ EA F♯G FG♯

Möbius strip of 2-voice chords: Dmitri Tymoczko (2006)

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SLIDE 70

CC BC♯ A♯D D♯A EG♯ FG F♯F♯ F♯F♯ FG EG♯ D♯A BC♯ CC A♯D FF EF♯ D♯G DG♯ A♯C BB AC♯ EE D♯F DF♯ C♯G AB A♯A♯ G♯C D♯D♯ DE C♯F CF♯ G♯A♯ AA GB DD C♯D♯ CE FB GA G♯G♯ F♯A♯ C♯C♯ CD BD♯ EA♯ F♯G♯ GG FA FF♯ EG D♯G♯ AD BC A♯C♯ EF D♯F♯ DG G♯C♯ A♯B AC D♯E DF C♯F♯ GC AA♯ G♯B DD♯ C♯E CF F♯B G♯A GA♯ C♯D CD♯ BE FA♯ GG♯ F♯A CC♯ BD A♯D♯ EA F♯G FG♯

Voice 1: Voice 2: G

A B C D E F G G♯ A♯ C♯ D♯ F♯ D D D D D D D D D D D D D

slide-71
SLIDE 71

CC BC♯ A♯D D♯A EG♯ FG F♯F♯ F♯F♯ FG EG♯ D♯A BC♯ CC A♯D FF EF♯ D♯G DG♯ A♯C BB AC♯ EE D♯F DF♯ C♯G AB A♯A♯ G♯C D♯D♯ DE C♯F CF♯ G♯A♯ AA GB DD C♯D♯ CE FB GA G♯G♯ F♯A♯ C♯C♯ CD BD♯ EA♯ F♯G♯ GG FA FF♯ EG D♯G♯ AD BC A♯C♯ EF D♯F♯ DG G♯C♯ A♯B AC D♯E DF C♯F♯ GC AA♯ G♯B DD♯ C♯E CF F♯B G♯A GA♯ C♯D CD♯ BE FA♯ GG♯ F♯A CC♯ BD A♯D♯ EA F♯G FG♯

Voice 1: Voice 2: G

A B C D E F G G♯ A♯ C♯ D♯ F♯ D D D D D D D D D D D D D

DG

slide-72
SLIDE 72

CC BC♯ A♯D D♯A EG♯ FG F♯F♯ F♯F♯ FG EG♯ D♯A BC♯ CC A♯D FF EF♯ D♯G DG♯ A♯C BB AC♯ EE D♯F DF♯ C♯G AB A♯A♯ G♯C D♯D♯ DE C♯F CF♯ G♯A♯ AA GB DD C♯D♯ CE FB GA G♯G♯ F♯A♯ C♯C♯ CD BD♯ EA♯ F♯G♯ GG FA FF♯ EG D♯G♯ AD BC A♯C♯ EF D♯F♯ DG G♯C♯ A♯B AC D♯E DF C♯F♯ GC AA♯ G♯B DD♯ C♯E CF F♯B G♯A GA♯ C♯D CD♯ BE FA♯ GG♯ F♯A CC♯ BD A♯D♯ EA F♯G FG♯

Voice 1: Voice 2: G

A B C D E F G G♯ A♯ C♯ D♯ F♯ D D D D D D D D D D D D D

DG♯ DG

slide-73
SLIDE 73

CC BC♯ A♯D D♯A EG♯ FG F♯F♯ F♯F♯ FG EG♯ D♯A BC♯ CC A♯D FF EF♯ D♯G DG♯ A♯C BB AC♯ EE D♯F DF♯ C♯G AB A♯A♯ G♯C D♯D♯ DE C♯F CF♯ G♯A♯ AA GB DD C♯D♯ CE FB GA G♯G♯ F♯A♯ C♯C♯ CD BD♯ EA♯ F♯G♯ GG FA FF♯ EG D♯G♯ AD BC A♯C♯ EF D♯F♯ DG G♯C♯ A♯B AC D♯E DF C♯F♯ GC AA♯ G♯B DD♯ C♯E CF F♯B G♯A GA♯ C♯D CD♯ BE FA♯ GG♯ F♯A CC♯ BD A♯D♯ EA F♯G FG♯

Voice 1: Voice 2: G

A B C D E F G G♯ A♯ C♯ D♯ F♯ D D D D D D D D D D D D D

DG♯ AD DG

slide-74
SLIDE 74

CC BC♯ A♯D D♯A EG♯ FG F♯F♯ F♯F♯ FG EG♯ D♯A BC♯ CC A♯D FF EF♯ D♯G DG♯ A♯C BB AC♯ EE D♯F DF♯ C♯G AB A♯A♯ G♯C D♯D♯ DE C♯F CF♯ G♯A♯ AA GB DD C♯D♯ CE FB GA G♯G♯ F♯A♯ C♯C♯ CD BD♯ EA♯ F♯G♯ GG FA FF♯ EG D♯G♯ AD BC A♯C♯ EF D♯F♯ DG G♯C♯ A♯B AC D♯E DF C♯F♯ GC AA♯ G♯B DD♯ C♯E CF F♯B G♯A GA♯ C♯D CD♯ BE FA♯ GG♯ F♯A CC♯ BD A♯D♯ EA F♯G FG♯

Voice 1: Voice 2: G

A B C D E F G G♯ A♯ C♯ D♯ F♯ D D D D D D D D D D D D D

A♯D A♯D DG♯ AD DG

slide-75
SLIDE 75

CC BC♯ A♯D D♯A EG♯ FG F♯F♯ F♯F♯ FG EG♯ D♯A BC♯ CC A♯D FF EF♯ D♯G DG♯ A♯C BB AC♯ EE D♯F DF♯ C♯G AB A♯A♯ G♯C D♯D♯ DE C♯F CF♯ G♯A♯ AA GB DD C♯D♯ CE FB GA G♯G♯ F♯A♯ C♯C♯ CD BD♯ EA♯ F♯G♯ GG FA FF♯ EG D♯G♯ AD BC A♯C♯ EF D♯F♯ DG G♯C♯ A♯B AC D♯E DF C♯F♯ GC AA♯ G♯B DD♯ C♯E CF F♯B G♯A GA♯ C♯D CD♯ BE FA♯ GG♯ F♯A CC♯ BD A♯D♯ EA F♯G FG♯

Voice 1: Voice 2: G

A B C D E F G G♯ A♯ C♯ D♯ F♯ D D D D D D D D D D D D D

A♯D A♯D DG♯ DD CD AD DG C♯D BD

slide-76
SLIDE 76

CC BC♯ A♯D D♯A EG♯ FG F♯F♯ F♯F♯ FG EG♯ D♯A BC♯ CC A♯D FF EF♯ D♯G DG♯ A♯C BB AC♯ EE D♯F DF♯ C♯G AB A♯A♯ G♯C D♯D♯ DE C♯F CF♯ G♯A♯ AA GB DD C♯D♯ CE FB GA G♯G♯ F♯A♯ C♯C♯ CD BD♯ EA♯ F♯G♯ GG FA FF♯ EG D♯G♯ AD BC A♯C♯ EF D♯F♯ DG G♯C♯ A♯B AC D♯E DF C♯F♯ GC AA♯ G♯B DD♯ C♯E CF F♯B G♯A GA♯ C♯D CD♯ BE FA♯ GG♯ F♯A CC♯ BD A♯D♯ EA F♯G FG♯

Voice 1: Voice 2: G

A B C D E F G G♯ A♯ C♯ D♯ F♯ D D D D D D D D D D D D D

A♯D A♯D DG♯ DF♯ DE DD CD AD DG DF DD♯ C♯D BD

Topology of orbifolds: William Tiurston (1980)

slide-77
SLIDE 77

EEE FFF F♯F♯F♯ GGG G♯G♯G♯

Twisted prism of 3-voice chords: Dmitri Tymoczko (2006)

CCC C♯C♯C♯ DDD E♭E♭E♭ EEE G♯G♯G♯ AAA B♭B♭B♭ BBB CCC

slide-78
SLIDE 78

G♯G♯G♯ AAA B♭B♭B♭ BBB CCC CCC C♯C♯C♯ DDD E♭E♭E♭ EEE G♯G♯G♯

slide-79
SLIDE 79

A♭ g♯

E

c C e G g

E♭

b B d♯ F♯ f♯

D

b♭ B♭ d F f

D♭

a A c♯

slide-80
SLIDE 80

A♭ g♯

E

c C e G g

E♭

b B d♯ F♯ f♯

D

b♭ B♭ d F f

D♭

a A c♯

L P R

slide-81
SLIDE 81

11 2 8 5 2 3 6 9 6 7 10 4 1 10 3 6 9 6 11 2 8 5 2

A♭ g♯ F f D d B b E e D♭ c♯ B♭ b♭ G g C c A a F♯ f♯ E♭ e♭ A♭ g♯ F f D d B b

L P R

A♭ g♯

E

c C e G g

E♭

b B d♯ F♯ f♯

D

b♭ B♭ d F f

D♭

a A c♯

L P R

b e G g

slide-82
SLIDE 82

Questions welcome: juhook@indiana.edu

Tiank you!