Guerino Mazzola

1

• Guerino Mazzola
• U ETH Zürich
• Internet Institute for Music Science
• guerino_at_mazzola.ch
• www.encyclospace.org

2

• A Modulation Model
• Experiments with Beethoven
• Generalizations
• Open Questions

Contents
3
Arnold Schönberg Harmonielehre (1911)
Old Tonality Neutral Degrees (IC, VIC)
Modulation Degrees (IIF, IVF, VIIF)
New Tonality Cadence Degrees (IIF VF)
Model

• 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?

4
Model
Space Z12 of pitch classes in 12-tempered tuning
Twelve diatonic scales C, F, Bb , Eb , Ab , Db ,
Gb , B, E, A, D, G
5
Model
6
Harmonic strip of diatonic scale
Model
7
C(3)
G(3)
F(3)
Model
Bb (3)
D(3)
Dia(3) triadic coverings
E b(3)
A(3)
Ab(3)
E(3)
Db(3)
B(3)
Gb (3)
8
Model
k1(S(3)) IIS, VS k2(S(3)) IIS,
IIIS k3(S(3)) IIIS, IVS k4(S(3)) IVS,
VS k5(S(3)) VIIS
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Model
10
Model
modulation S(3) T(3) cadence symmetry
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Given a modulation k, gS(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 Ç T is rigid, i.e., has no proper inner
  symmetries
• M is minimal with the first two conditions

Model
12

• 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.

Model
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Model
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Ludwig van Beethoven op.130/Cavatina/ 41
Inversion e b E b(3) B(3)
Experiments
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Ludwig van Beethoven op.130/Cavatina/ 41
Inversion e b E b(3) B(3)
Experiments
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Ludwig van Beethoven op.106/Allegro/124-127 Inve
rsiondb G(3) E b(3)
Experiments
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Ludwig van Beethoven op.106/Allegro/188-197 Cata
strophe E b(3) D(3) b(3)
Generalization
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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.
Experiments
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.
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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.
Experiments
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• All 7-scales in well-tempered pitch classes -gt
  Daniel Muzzulini/Hans Straub
• Diatonic, melodic, and harmonic scales in just
  tuning -gt Hildegard Radl
• Applications to rhythmic modulation -gt Guerino
  Mazzola
• Ongoing research Modulation for generalized
  tones -gt Thomas Noll

Generalization
21

• 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.

Generalization
22

• 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.

Questions