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

June 22, 2017 Music as a Mathematician’s 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 ♭ B ♭ D ♭ E ♭ G A B C D E F G A B C D E Pitch space (chromatic scale) F ♯ G G ♯ A A ♯ B C C ♯ D D ♯ E F F ♯ G G ♯ A A ♯ B C C ♯ D D ♯ E

( ℤ ) Pitch space F ♯ G G ♯ A A ♯ B C C C ♯ C ♯ D D D ♯ D ♯ E F E F F ♯ F ♯ G G G ♯ G ♯ A A A ♯ A ♯ B C B C ♯ D D ♯ E − 6 − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 one octave C B C ♯ ( ℤ 12 ) Pitch-class space 0 11 1 A ♯ D 10 2 A D ♯ 9 3 8 4 G ♯ E 7 5 6 G F F ♯

Pitch-class space Ti e circle of fjfu hs C C B C ♯ F G 0 0 11 1 5 7 A ♯ D A ♯ D 10 2 10 2 +1 
 +7 
 (mod 12) (mod 12) A D ♯ D ♯ A 9 3 3 9 8 4 8 4 G ♯ E G ♯ E 7 5 1 11 6 6 G F C ♯ B F ♯ F ♯

Pitch-class space Ti e circle of fjfu hs C C B C ♯ F G × 7 (mod 12) 0 0 11 1 5 7 A ♯ D A ♯ D 10 2 10 2 A D ♯ D ♯ A 9 3 3 9 8 4 8 4 G ♯ E G ♯ E 7 5 1 11 6 6 G F C ♯ B F ♯ F ♯

ℤ 12 ≅ ℤ 4 × ℤ 3 2 5 8 11 2 5 8 11 2 10 1 4 7 10 1 4 7 10 +4 
 (mod 12) 6 9 0 3 6 9 0 3 6 +3 
 (mod 12) 2 5 8 11 2 5 8 11 1 10 1 4 7 10 1 4 7 10 6 9 0 3 6 9 0 3 6 2 5 8 11 2 5 8 11 2 10 1 4 7 10 1 4 7 10

ℤ 12 ≅ ℤ 4 × ℤ 3 D F G ♯ B D F G ♯ B D 2 5 8 11 2 5 8 11 2 A ♯ C ♯ E G A ♯ C ♯ E G A ♯ 10 1 4 7 10 1 4 7 10 F ♯ A C D ♯ F ♯ A C D ♯ F ♯ 6 9 0 3 6 9 0 3 6 D F G ♯ B D F G ♯ B D 2 5 8 11 2 5 8 11 1 A ♯ C ♯ E G A ♯ C ♯ E G A ♯ 10 1 4 7 10 1 4 7 10 F ♯ A C D ♯ F ♯ A C D ♯ F ♯ 6 9 0 3 6 9 0 3 6 D F G ♯ B D F G ♯ B D 2 5 8 11 2 5 8 11 2 A ♯ C ♯ E G A ♯ C ♯ E G A ♯ 10 1 4 7 10 1 4 7 10

ℤ 12 ≅ ℤ 4 × ℤ 3 D F G ♯ B D F G ♯ B D 2 5 8 11 2 5 8 11 2 A ♯ C ♯ E G A ♯ C ♯ E G A ♯ 10 1 4 7 10 1 4 7 10 F ♯ A C D ♯ F ♯ A C D ♯ F ♯ 6 9 0 3 6 9 0 3 6 D F G ♯ B D F G ♯ B D 2 5 8 11 2 5 8 11 1 A ♯ C ♯ E G A ♯ C ♯ E G A ♯ 10 1 4 7 10 1 4 7 10 F ♯ A C D ♯ F ♯ A C D ♯ F ♯ 6 9 0 3 6 9 0 3 6 D F G ♯ B D F G ♯ B D 2 5 8 11 2 5 8 11 2 A ♯ C ♯ E G A ♯ C ♯ E G A ♯ 10 1 4 7 10 1 4 7 10

G ♯ B D F Tonnetz: 8 11 2 5 Leonhard Euler (1739) E G A ♯ C ♯ Hugo Riemann (1915) Richard Cohn (1997) 4 7 10 1 (and many others) C D ♯ F ♯ A 0 3 6 9

In fj nite Tonnetz Finite Tonnetz C C 0 0 C 0 C C 0 0

In fj nite Tonnetz Finite Tonnetz (planar graph) (non-planar graph) complete graph K 5

D 2 D +1 
 +7 
 2 (mod 12) (mod 12) G 1 C ♯ 7 C 0 C 0 F 11 B 5 10 A ♯ D ♯ 9 A 3 G ♯ 8 G ♯ 8 C ♯ 7 G 1

D F G ♯ B D F G ♯ B D 2 5 8 11 2 5 8 11 2 A ♯ C ♯ E G A ♯ C ♯ E G A ♯ 10 1 4 7 10 1 4 7 10 F ♯ A C D ♯ F ♯ A C D ♯ F ♯ 6 9 0 3 6 9 0 3 6 D F G ♯ B D F G ♯ B D 2 5 8 11 2 5 8 11 1 A ♯ C ♯ E G A ♯ C ♯ E G A ♯ 10 1 4 7 10 1 4 7 10 F ♯ A C D ♯ F ♯ A C D ♯ F ♯ 6 9 0 3 6 9 0 3 6 D F G ♯ B D F G ♯ B D 2 5 8 11 2 5 8 11 2 A ♯ C ♯ E G A ♯ C ♯ E G A ♯ 10 1 4 7 10 1 4 7 10

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

A C F ♯ E ♭ F ♯ major triads minor triads D F B D G ♯ E G B ♭ C ♯ B ♭ A C F ♯ E ♭ F ♯ D F B D G ♯

CEG E (C major) G CE ♭ G (C minor) P P arallel E ♭ C

CEG E (C major) G R P A P arallel E ♭ C ACE R elative (A minor)

B EGB (E minor) CEG L E (C major) G R P A P arallel E ♭ C R elative L ei tu onwechsel

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

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

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

A C F ♯ E ♭ F ♯ C ♯ minor D F B D G ♯ R P E G B ♭ C ♯ B ♭ L E ♭ major R A C P F ♯ E ♭ F ♯ D F B D G ♯

A C F ♯ E ♭ F ♯ D F B D G ♯ C major P R E G B ♭ C ♯ B ♭ E ♭ major P R C major R A C F ♯ E ♭ F ♯ P R P A major D F B D G ♯ ( ! ) PR ≠ RP

6 9 0 3 6 pitch-class numbers 2 5 8 11 2 10 1 4 7 10 6 9 0 3 6 2 5 8 11 2

6 9 0 3 6 D F A ♭ B pitch-class d g ♯ b f numbers 2 5 8 11 2 B ♭ D ♭ E G b ♭ c ♯ e g major 10 1 4 7 10 triads F ♯ A C E ♭ f ♯ a c e ♭ 6 9 0 3 6 minor D F A ♭ B triads d f g ♯ b 2 5 8 11 2

6 9 0 3 6 D F A ♭ B pitch-class d g ♯ b f numbers 2 5 8 11 2 B ♭ D ♭ E G b ♭ c ♯ e g major 10 1 4 7 10 triads L R F ♯ A C E ♭ f ♯ a c e ♭ P 6 9 0 3 6 minor D F A ♭ B triads d f g ♯ b 2 5 8 11 2

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

Brahms, Concerto 
 P for Violin and Cello L

Brahms P L P L Liszt, 
 Consolation No. 3

Brahms P L P L Liszt R P Smetana, 
 Bartered Bride Overture

Brahms P L Beethoven, Symphony No. 9 R L P L Liszt R P Smetana

Brahms P L Beethoven R L L P R L Chopin, 
 Fantasy in F Minor Liszt R P Smetana

Brahms P L Beethoven R L L P R L Chopin Liszt R P Smetana R Verdi, 
 P Il trovatore 8 L R P

Brahms P L Beethoven Schubert, 
 “Trout” Quintet R L L P P R R P L L Chopin P Liszt L R P R Smetana L R P Verdi L R P

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

Map of interkey distances: Carol Krumhansl (1983)

Map of interkey distances: Carol Krumhansl (1983)

R P L Map of interkey distances: Carol Krumhansl (1983)

1 9 5 8 4 0 8 3 11 7 3 6 2 10 Map of interkey distances: Carol Krumhansl (1983)

F G ♯ B D F 5 8 11 2 5 C ♯ E G A ♯ C ♯ 1 4 7 10 1 A C A D ♯ F ♯ 9 0 3 6 9 F G ♯ B D F 5 8 11 2 5 C ♯ E G A ♯ C ♯ 1 4 7 10 1

F G ♯ B D F 5 8 11 2 5 C ♯ E G A ♯ C ♯ Schoenberg, 
 1 4 7 10 1 F ♯ GA ♯ AA ♯ C ♯ D ♯ EG Pierrot lunaire, “Nacht” D ♯ F ♯ G F ♯ AA ♯ A C A D ♯ F ♯ 9 0 3 6 9 DD ♯ F ♯ DFF ♯ F G ♯ B D F 5 8 11 2 5 C ♯ DF C ♯ E G A ♯ C ♯ 1 4 7 10 1

dominant 7th chords half-diminished 7th chords 3-D Tonnetz: Edward Gollin (1998)

Spiral Array: Elaine Chew (2014)

major triads Spiral Array: Elaine Chew (2014)

minor triads Spiral Array: Elaine Chew (2014)

dominant 7th chord Spiral Array: Elaine Chew (2014)

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

( ℤ 12 ) Chromatic pitch-class space Chromatic circle of fjfu hs C C B C ♯ F G 0 0 11 1 5 7 A ♯ D A ♯ D 10 2 10 2 A D ♯ D ♯ A 9 3 3 9 8 4 8 4 G ♯ E G ♯ E 7 5 1 11 6 6 G F C ♯ B F ♯ F ♯ ( ℤ 7 ) Diatonic pitch-class space Diatonic circle of fjfu hs C C 0 0 B D F G 6 3 1 4 2 1 5 6 A E B D 3 5 4 2 G F E A

( ℤ 12 ) Chromatic pitch-class space Chromatic circle of fjfu hs C C B C ♯ F G 0 0 11 1 5 7 A ♯ D A ♯ D 10 2 10 2 A D ♯ D ♯ A 9 3 3 9 8 4 8 4 G ♯ E G ♯ E 7 5 1 11 6 6 G F C ♯ B F ♯ F ♯ ( ℤ 7 ) Diatonic pitch-class space Diatonic circle of fjfu hs C C 0 0 B D F G 6 3 1 4 2 1 5 6 A E B D 3 5 4 2 G F E A

( ℤ 12 ) Chromatic pitch-class space Chromatic circle of fjfu hs C C B C ♯ F G 0 0 11 1 5 7 A ♯ D A ♯ D 10 2 10 2 A D ♯ D ♯ A 9 3 3 9 8 4 8 4 G ♯ E G ♯ E 7 5 1 11 6 6 G F C ♯ B F ♯ F ♯ ( ℤ 7 ) Diatonic pitch-class space Diatonic circle of fjfu hs C C 0 0 B D F G 6 3 1 4 2 1 5 6 A E B D 3 5 4 2 G F E A

C E G B D 2 4 6 1 3 F A C E G 5 0 2 4 6 B D F A C 1 3 5 0 2 E G B D F 4 6 1 3 5 F C E B G 0 2 4 6 8