Transcript Mathematical Models of Tonal Modulation and Application to Beethoven‘s op. 106 Guerino Mazzola U & ETH Zürich Internet Institute for Music Science [email protected] www.encyclospace.org.

Mathematical Models of
Tonal Modulation and
Application to
Beethoven‘s op. 106
Guerino Mazzola
U & ETH Zürich
Internet Institute for Music Science
[email protected]
www.encyclospace.org
Contents
• A Modulation Model
• Experiments with Beethoven
• Generalizations
• Open Questions
Model
Arnold Schönberg: Harmonielehre (1911)
Old Tonality
Neutral
Degrees
(IC, VIC)
Modulation
Degrees
(IIF, IVF, VIIF)
New Tonality
Cadence
Degrees
(IIF & VF)
• What is the considered set of tonalities?
• What is a degree?
• What is a cadence?
• What is the modulation mechanism?
• How do these structures determine the
modulation degrees?
Model
11
0
Space Z12 of pitch classes in
12-tempered tuning
1
2
10
C
9
3
8
Scale = part of Z12
4
7
6
5
Twelve diatonic scales: C, F, Bb , Eb , Ab , Db , Gb , B, E, A,
D, G
Model
I
II
III
IV
V
VI
VII
Model
Harmonic strip of diatonic scale
II
VI
V
IV
I
VII
III
C(3)
Model
G(3)
F(3)
Bb (3)
D(3)
Dia(3)
A(3)
E b(3)
triadic
coverings
Ab(3)
E(3)
B(3)
Db(3)
Gb (3)
Model
S(3)
k1(S(3)) = {IIS, VS}
k2(S(3)) = {IIS, IIIS}
k3(S(3)) = {IIIS, IVS}
k4(S(3)) = {IVS, VS}
k5(S(3)) = {VIIS}
k
k(S(3))
Space of cadence parameters
Model
gluon
W+
g
strong force
weak force
electromagnetic
force
graviton
gravitation
quantum = set of
pitch classes = M
S(3)
T(3)
force = symmetry between
S(3) and T(3)
k
k
et
Model
A
et.A
S(3)
T(3)
k
et
k
modulation S(3)  T(3) = „cadence + symmetry “
Model
Given a modulation k, g:S(3) 
T(3)
A quantum for (k,g) is a set M of pitch classes such that:
• the symmetry g is a symmetry of M, g(M) = M
• the degrees in k(T(3)) are contained in M
• M T is rigid, i.e., has no proper inner symmetries
• M is minimal with the first two conditions
M
S(3)
T(3)
g
k
k
Model
Modulation Theorem for 12-tempered Case
For any two (different) tonalities S(3), T(3) there is
• a modulation (k,g) and
• a quantum M for (k,g)
Further:
• M is the union of the degrees in S(3), T(3) contained in
M, and thereby defines the triadic covering M(3) of M
• the common degrees of T(3) and M(3) are called the
modulation degrees of (k,g)
• the modulation (k,g) is uniquely determined by the
modulation degrees.
IVC
IIEb
Model
VIIEb
IIC
M(3)
VC
C(3)
VIIC
VEb
IIIEb
E b(3)
Ludwig van Beethoven: op.130/Cavatina/# 41
Experiments
Inversion e b : E b(3)  B(3)
Ludwig van Beethoven: op.130/Cavatina/# 41
Experiments
Inversion e b : E b(3)  B(3)
b
bb
eb
Inversion e b
ab
E b(3)
g
f
B(3)
Ludwig van Beethoven: op.106/Allegro/#124-127
Experiments
Inversiondb : G(3) E b(3)
#124 - 125
#126 - 127
g
db
g
Ludwig van Beethoven: op.106/Allegro/#188-197
Generalization
Catastrophe : E b(3) D(3)~ b(3)
Experiments
Theses of Erwin Ratz (1973) and Jürgen Uhde (1974)
Ratz: The „sphere“ of tonalities of op. 106 is polarized into a
„world“ centered around B-flat major, the principal tonality
of this sonata, and a „antiworld“ around B minor.
Uhde: When we change Ratz‘ „worlds“, an event happening twice
in the Allegro movement, the modulation processes become
dramatic. They are completely different from the other
modulations, Uhde calls them „catastrophes“.
B-flat major
B minor
Experiments
Thesis: The modulation structure of op. 106 is governed by
the inner symmetries of the diminished seventh
chord
C# -7 = {c#, e, g, bb}
in the role of the admitted modulation forces.
C(3)
G(3)
F(3)
Bb (3)
D(3) ~ b(3)
E b(3)
A(3)
Ab(3)
E(3)
B(3)
Gb (3)
Db(3)
Generalization
• All 7-scales in well-tempered pitch classes
-> Daniel Muzzulini/Hans Straub
• Diatonic, melodic, and harmonic scales in just tuning
-> Hildegard Radl
• Applications to rhythmic modulation
-> Guerino Mazzola
• Ongoing research: Modulation for generalized tones
-> Thomas Noll
Generalization
Modulation Theorem for 12-tempered 7-tone
Scales S and triadic coverings S(3) (Muzzulini)
q-modulation = quantized modulation
(1) S(3) is rigid.
• For every such scale, there is at least one q-modulation.
• The maximum of 226 q-modulations is achieved by the
harmonic scale #54.1, the minimum of 53 q-modulations
occurs for scale #41.1.
(2) S(3) is not rigid.
• For scale #52 and #55, there are q-modulations except for t = 1, 11;
for #38 and #62, there are q-modulations except for t = 5,7.
All 6 other types have at least one quantized modulation.
• The maximum of 114 q-modulations occurs for the melodic
minor scale #47.1. Among the scales with q-modulations for
all t, the diatonic major scale #38.1 has a minimum of 26.
Questions
• Develop methodology and software for
systematic experiments on given corpora of
musical compositions (recognize tonalities and
modulations thereof!)
• Modulation in other musical dimensions, such
as motive, rhythm, and global object spaces.
• Modulation in other concept spaces for tonality
and harmony, e.g., self-addressed pitch, or
harmonic topologies following Riemann.
• Generalizing the quantum/force analogy in the
modulation model, develop a general theory of
musical dynamics, i.e., the theory of musical
interaction forces between general structures.