1. IV EXPERIMENTS
IV.4 (Thu Apr 05) The modulation landscape in Beethoven’s op. 106

2. ! Sonata scheme for the allegro movement in Beethoven‘s op.106
according to Erwin Ratz: Einführung in die musikalische Formenlehre. Universal Edition, Wien 1973 !

3. Fast normal modulation: B♭ ® G♭
7:06

4. Normal modulation: G ® E ♭
4:50

5. Modulation of „catastrophe“ type: E ♭(3) ® D(3)~ b(3)
6:00

6. Erwin Ratz‘ (1973) and Jürgen Uhde‘s (1974) theses
Ratz: The sphere of tonalities of op. 106 is polarized in a“world“ centered around B-flat major, the main tonality of the sonata, and an „antiworld“ centered around b-minor.
Uhde*: When one changes between the worlds of Ratz— an event that happens twice in the allegro movement—then the modulation processes become dramatic. They are completely different from other modulations, and Uhde calls them „catastrophes“.
b minor
B-flat major

7. Thesis:. The modulation structure of op. 106 is governed
Thesis: The modulation structure of op. 106 is governed by the symmetries of the diminished seventh chord C# -7 = {c#, e, g, b♭} carrying the admitted modulation forces.
C(3)
B♭ (3)
E♭(3)
D♭(3)
G♭ (3)
E(3)
A(3)
G(3)
F(3)
A♭(3)
B(3)
D(3) ~ b(3)
Exposition
Developement
Recapitulation
Coda

8. = maximal symmetry group separating D from B♭
C(3)
F(3)
B♭ (3)
E♭(3)
A♭(3)
D♭(3)
G♭ (3)
B(3)
E(3)
A(3)
D(3)
G(3)
Sym(C# -7)
= maximal symmetry group separating D from B♭

9. inversion of tonalities?
C(3)
B♭ (3)
E♭(3)
D♭ (3)
G♭ (3)
E(3)
A(3)
G(3)
F(3)
A♭(3)
B(3)
D(3) ~ b(3)
How to calculate
inversion of tonalities?
Invert tonic of old tonality, you get third of new tonality.

10. Modulators in op. 106/allegro
Exposition
Recapitulation
Development
Coda
B♭ ® G G ® E♭ ® D/b ® B ® B♭ B♭ ® G♭ ® G ® B♭ B♭
symmetries of transposition!
e Ig * Id/d# * Ib♭ Ia/a♭ e3

11. Fast normal modulation: B♭ ® G♭
Inversionb♭
VIG♭
IB♭
Fast normal modulation: B♭ ® G♭
IIIG♭  VG♭
7:06 I IV II VI V
III
VII
a-flat in II  V  VII
b-flat in III  VI  I
f in III  V  VII

12. Ludwig van Beethoven: op.130/Cavatina/# 41
Inversione♭ : E♭(3) ® B(3)
4:00
oppressive

13. e♭
E♭(3) b
B(3)
Inversion e♭

14. Inversiond♭ : G(3) ® E♭(3)# #
4:50 d♭ g

15. C-minor melodic Gruppen und Kategorien in der Musik, p.107 a g f e♭ db♭
C-minor melodic
fundamental
instead of tone of double duration
replaces tone of a higher octave
Gruppen und Kategorien
in der Musik, p.107

16. Transposition-3: B♭(3) ® G(3)
VIIG Ia
VIIG

17. Limited transpositions by a minor third
motivic zigzag

18. motivic zigzag in op.106
1 0 ( ) b

19. b

20. L 3