Transcript No Slide Title

Just and Well-tempered
Modulation Theory
Guerino Mazzola
U & ETH Zürich
Internet Institute for Music Science
[email protected]
www.encyclospace.org
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 Ÿ12 of pitch classes in
12-tempered tuning
1
2
10
C
9
3
8
Scale = part of Ÿ12
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)
4:00
mi-b->si
experiments
b
eb
E b(3)
Inversion e b
B(3)
Ludwig van Beethoven: op.106/Allegro/#124-127
experiments
Inversiondb : G(3) E b(3)
4:50
sol->mi b
#124 - 125
#126 - 127
g
db
g
Ludwig van Beethoven: op.106/Allegro/#188-197
experiments
Catastrophe : E b(3) D(3)~ b(3)
6:00
mi b->re=
Si min.
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
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.
just theory
Modulation theorem for 7-tone scales S and triadic
coverings S(3) in just tuning (Hildegard Radl)
log(5)
b
b
a
e
b
f
c
g
d
b
a
b
f#
d
e
b
log(3)
just theory
Just modulation:
Same formal setup as for
well-tempered
tuning.
A
S(3)
et
T(3)
et.A
just theory
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 Minkowsky‘s theorem, the composed group
homomorphism
<A>  GL2(Ÿ) GL2(Ÿ3)
is injective. Since #GL2(Ÿ3) = 48, the order is 2.
Lemma: Let M = et.A: S(3)  T(3) be a modulator, with A =
ea.R. For any x Ÿ2, the <M>-orbit is
<M>(x) = e Ÿ(1+R)t.x e Ÿ(1+R)t.M(x)
just theory
Just modulation:
Target tonalities for the C-major scale.
db*
a
bb
f
db
e
b
g
ab
eb
d
bb*
just theory
Just modulation:
Target tonalities for the natural c-minor scale.
db*
a
bb
f
db
e
b
g
ab
eb
d
bb*
just theory
Just modulation:
Target major tonalities from the natural c-minor
scale.
bb
f
db
g
ab
eb
d
bb*
just theory
Just modulation:
Target minor tonalities from the Natural c-major
scale.
db*
a
bb
f
e
b
g
d
just theory
Just modulation:
Target tonalities for the harmonic C-minor scale.
eb*
g#
d#
e
b
f#
g
d
db*
a
bb
f
gb
db
ab
bbb
fb
eb
a*
bb*
just theory
Just modulation:
Target tonalities for the melodic C-minor scale.
a
bb
e
f
g
ab
eb
d
just theory
a
e
f#
b
f
c
g
d
db
ab
eb
bb
major, natural, harmonic, melodic minor
just theory
no modulations
infinite modulations
limited modulations
four modulations
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
rhythmic modulation
rhythmic modulation
1
2
3
4
5
6
7
8
9
10
11
12
generic
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Classes of 3-element motives M  Ÿ122
rhythmic modulation
rhythmic modulation
Percussion encoding
62^
62^
Retrograde
of
62^
R(62^)
onset
rhythmic modulation
12/8
3:18-5:48
B.1-6
m1
m1
m2
m1
m2
m3
m1
m2
m3
m4
m1
m2
m3
m4
m5
m1
m2
m3
m4
m5
m6,m7
B.7-12
m1
m2
m3
m4
m5
m6,m7
m1
m2
m3
m4
m5
m1
m2
m3
m4
m1
m2
m3
m1
m2
m1
R
B.13-24
modulation pivots
new bar system
new tonic
at 9/8 of bar 21