Guerino%20Mazzola

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

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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?

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model
Space Ÿ12 of pitch classes in 12-tempered tuning
Twelve diatonic scales C, F, Bb , Eb , Ab , Db ,
Gb , B, E, A, D, G
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model
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Harmonic strip of diatonic scale
model
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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)
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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
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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
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• 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
400
mi-b-gtsi
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experiments
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Ludwig van Beethoven op.106/Allegro/124-127 Inve
rsiondb G(3) E b(3)
experiments
450
sol-gtmi b
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Ludwig van Beethoven op.106/Allegro/188-197 Cata
strophe E b(3) D(3) b(3)
experiments
600
mi b-gtre Si min.
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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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• 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
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Modulation theorem for 7-tone scales S and
triadic coverings S(3) in just tuning (Hildegard
Radl)
just theory

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Just modulation Same formal setup as for
well-tempered tuning.
just theory
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Lemma If the seven-element scale S is
generating, a non-trivial automorphism A of S(3)
is of order 2. Proof The nerve automorphism
Nerve(A) on Nerve(S(3)) preserves the boundary
circle of the Möbius strip and hence is in the
dihedral group of the 7-angle. By Minkowskys
theorem, the composed group homomorphismltAgt
GL2(Ÿ) GL2(Ÿ3)is injective. Since GL2(Ÿ3)
48, the order is 2.
just theory
Lemma Let M et.A S(3) T(3) be a
modulator, with A ea.R. For any x Î Ÿ2, the
ltMgt-orbit isltMgt(x) e Ÿ(1R)t.x È e
Ÿ(1R)t.M(x)
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Just modulation Target tonalities for the
C-major scale.
just theory
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Just modulation Target tonalities for the
natural c-minor scale.
just theory
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Just modulation Target major tonalities from
the natural c-minor scale.
just theory
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Just modulation Target minor tonalities from
the Natural c-major scale.
just theory
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Just modulation Target tonalities for the
harmonic C-minor scale.
just theory
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Just modulation Target tonalities for the
melodic C-minor scale.
just theory
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just theory
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just theory
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rhythmic modulation
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rhythmic modulation
Classes of 3-element motives M Í Ÿ122
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rhythmic modulation
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rhythmic modulation
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12/8
rhythmic modulation
318-548