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Guerino Mazzola
U & ETH Zürich
Internet Institute for Music Science
www.encyclospace.org
Just and Well-tempered Modulation TheoryJust and Well-tempered Modulation Theory
mod
el
Old Tonality Neutral
Degrees(IC, VIC)
Modulation Degrees
(IIF, IVF, VIIF)
New Tonality CadenceDegrees
(IIF & VF)
Arnold Schönberg: Arnold Schönberg: Harmonielehre (1911)Harmonielehre (1911)
• 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?
mod
el
Space Space ŸŸ1212 of pitch classes in of pitch classes in
12-tempered tuning12-tempered tuning0
1
2
3
4
56
7
8
9
10
11
Twelve diatonic scales: Twelve diatonic scales: C, F, BC, F, Bb b , E, Eb b , A, Ab b , D, Db b , G, Gb b , B, E, , B, E, A, D, GA, D, G
Scale Scale = part of = part of ŸŸ1212 C
mod
elCC(3)(3)
FF(3)(3)
BBbb (3)(3)
EE bb(3)(3)
AAbb(3)(3)
DDbb(3)(3)
GGbb (3)(3)
BB(3)(3)
EE(3)(3)
AA(3)(3)
DD(3)(3)
GG(3)(3)
DiaDia(3)(3)
triadic triadic
coveringscoverings
mod
el SS(3)(3)
Space of cadence parameters
k1(SS(3)(3)) = {IIS, VS}k2(SS(3)(3)) = {IIS, IIIS}k3(SS(3)(3)) = {IIIS, IVS}k4(SS(3)(3)) = {IVS, VS}k5(SS(3)(3)) = {VIIS}
k
k(SS(3)(3))
mod
el
SS(3)(3) TT(3)(3)
gluon
strong force
W+
weak force
electromagneticforce
graviton
gravitation
force = symmetry between S(3) and T(3)
quantum = set of pitch classes = M
k k
mod
el
SS(3)(3) TT(3)(3)
k k
A et
et.A
et
modulation modulation SS(3) (3) TT(3) (3) = „cadence + symmetry “= „cadence + symmetry “ modulation modulation SS(3) (3) TT(3) (3) = „cadence + symmetry “= „cadence + symmetry “
mod
el
SS(3)(3) TT(3)(3)
k k
Given a modulation k, g:Given a modulation k, g:SS(3) (3) (3)(3)Given a modulation k, g:Given a modulation k, g:SS(3) (3) (3)(3)
g
MM
A quantum for (k,g) is a set MM of pitch classes such that:
• the symmetry g is a symmetry of MM, g(MM) = MM• the degrees in k((3)(3)) are contained in MM• MM TT is rigid, i.e., has no proper inner symmetries• MM is minimal with the first two conditions
mod
elModulation Theorem for 12-tempered CaseModulation Theorem for 12-tempered Case
For any two (different) tonalities SS(3)(3),, (3)(3) there is• a modulation (k,g) and • a quantum MM for (k,g)
Further:
• M M is the union of the degrees in SS(3)(3),, (3)(3) contained in M, M, and thereby defines the triadic covering MM(3)(3) of
MM• the common degrees of (3)(3) and MM(3)(3) are called the
modulation degrees of (k,g)• the modulation (k,g) is uniquely determined by the
modulation degrees.
expe
rimen
tsLudwig van Beethoven: op.130/Cavatina/Ludwig van Beethoven: op.130/Cavatina/# 41 # 41
Inversion Inversion e e bb : EE bb(3) (3) BB(3)(3)
4:00
mi-b->si
expe
rimen
tsLudwig van Beethoven: op.106/Allegro/Ludwig van Beethoven: op.106/Allegro/#124-127#124-127
InversionInversionddbb : GG(3) (3) EE bb(3)(3)
ddbb
gg
gg
#124 - 125 #126 - 1274:50
sol->mi b
Ludwig van Beethoven: op.106/Allegro/Ludwig van Beethoven: op.106/Allegro/#188-197#188-197CatastropheCatastrophe : EE bb(3) (3) DD(3)(3)~~ bb(3) (3)
expe
rimen
ts
6:00
mi b->re=Si min.
expe
rimen
tsTheses of Erwin Ratz (1973) and Jürgen Uhde (1974)Theses of Erwin Ratz (1973) and Jürgen Uhde (1974)
Ratz: Ratz: The „sphere“ of tonalities of op. 106 is polarized into a The „sphere“ of tonalities of op. 106 is polarized into a „world“ centered around B-flat major, the principal tonality„world“ centered around B-flat major, the principal tonalityof this sonata, and a „antiworld“ around B minor. of this sonata, and a „antiworld“ around B minor.
Uhde: When we change Ratz‘ „worlds“, an event happening twiceUhde: When we change Ratz‘ „worlds“, an event happening twicein the Allegro movement, the modulation processes becomein the Allegro movement, the modulation processes becomedramatic. They are completely different from the other dramatic. They are completely different from the other modulations, Uhde calls them „catastrophes“. modulations, Uhde calls them „catastrophes“.
B minorB minorB minorB minorB-flat majorB-flat majorB-flat majorB-flat major
CC(3)(3)
BBbb (3)(3)
EE bb(3)(3)
DDbb(3)(3)
GGbb (3)(3)
EE(3)(3)
AA(3)(3)
GG(3)(3)
expe
rimen
tsThesis:Thesis: The modulation structure of op. 106 is governed byThe modulation structure of op. 106 is governed by
the inner symmetries of the diminished seventh the inner symmetries of the diminished seventh chordchord
CC## -7-7 = {c = {c##, e, g, b, e, g, bbb} } in the role of the admitted modulation forces. in the role of the admitted modulation forces.
FF(3)(3)
AAbb(3)(3)
BB(3)(3)
DD(3) ~ (3) ~ bb(3) (3)
gene
raliz
atio
nModulation Theorem for 12-tempered 7-tone Modulation Theorem for 12-tempered 7-tone Scales Scales SS and triadic coverings and triadic coverings SS(3) (3) (Muzzulini)(Muzzulini)
q-modulation = quantized modulationq-modulation = quantized modulation
(1) (1) SS(3) (3) is rigid.is rigid.• For every such scale, there is at least one q-modulation.For every such scale, there is at least one q-modulation.• The maximum of 226 q-modulations is achieved by theThe maximum of 226 q-modulations is achieved by the
harmonicharmonic scale #54.1, the minimum of 53 q-modulations scale #54.1, the minimum of 53 q-modulationsoccurs for scale #41.1. occurs for scale #41.1.
(2) (2) SS(3) (3) is not rigid.is not rigid.• For scale #52 and #55, there are q-modulations except for t = For scale #52 and #55, there are q-modulations except for t = 1, 11;1, 11;
for #38 and #62, there are q-modulations except for t = 5,7. for #38 and #62, there are q-modulations except for t = 5,7. All 6 other types have at least one quantized modulation.All 6 other types have at least one quantized modulation.
• The maximum of 114 q-modulations occurs for the The maximum of 114 q-modulations occurs for the melodicmelodicminorminor scale #47.1. Among the scales with q-modulations for scale #47.1. Among the scales with q-modulations for
all t, the diatonic all t, the diatonic majormajor scale #38.1 has a minimum of 26. scale #38.1 has a minimum of 26.
just
the
ory
Modulation theorem for 7-tone scales Modulation theorem for 7-tone scales SS and triadic and triadic coverings coverings SS(3) (3) in just tuningin just tuning (Hildegard Radl)(Hildegard Radl)
ff cc gg dd
aa ee bb
log(5)log(5)
log(3)log(3)
eebb
aabb
bbbb
ff##
ddbb
SS(3)(3)
TT(3)(3)
Just modulation: Just modulation: Same formal setup as for Same formal setup as for well-temperedwell-temperedtuning.tuning.
just
the
ory
A
eett
et.A
Lemma: Lemma: If the seven-element scale SIf the seven-element scale S is generating, a non-trivial is generating, a non-trivial automorphism Aautomorphism Aof Sof S(3) (3) is of order 2.is of order 2.
Proof: Proof: The nerve automorphism Nerve(A) on Nerve(SThe nerve automorphism Nerve(A) on Nerve(S(3)(3)) ) preserves the boundary circle of the Möbius strip and preserves the boundary circle of the Möbius strip and hence is in the dihedral group of the 7-angle.hence is in the dihedral group of the 7-angle.By Minkowsky‘s theorem, the composed group By Minkowsky‘s theorem, the composed group homomorphismhomomorphism
A> A> GL GL22((ŸŸ) ) GLGL22((ŸŸ33))
is injective. Since #GLis injective. Since #GL22((ŸŸ33) = 48, the order is 2.) = 48, the order is 2.
Lemma: Lemma: Let M = Let M = et.A: S: S(3) (3) T T(3) (3) be a modulator, with A = be a modulator, with A = ea.R. For any x ŸŸ22, the <M>-orbit is, the <M>-orbit is
<M>(x) = e <M>(x) = e ŸŸ(1+R)t(1+R)t.x .x e e ŸŸ(1+R)t(1+R)t.M(x).M(x)
just
the
ory
Just modulation: Just modulation: Target tonalities for the C-major scale.Target tonalities for the C-major scale.
bbbb
aabb eebb bbbb**ddbb
ff gg dd
aa ee bbddbb**
just
the
ory
Just modulation: Just modulation: Target tonalities for the natural c-minor scale.Target tonalities for the natural c-minor scale.
bbbb
aabb eebb bbbb**ddbb
ff gg dd
aa ee bbddbb**
just
the
ory
Just modulation: Just modulation: Target Target majormajor tonalities from the natural c- tonalities from the natural c-minorminor scale.scale.
bbbb
aabb eebb bbbb**ddbb
ff gg dd
just
the
ory
Just modulation: Just modulation: Target Target minorminor tonalities from the Natural c- tonalities from the Natural c-majormajor scale.scale.
bbbb ff gg dd
aa ee bbddbb**
just
the
ory
Just modulation: Just modulation: Target tonalities for the harmonic C-minor scale.Target tonalities for the harmonic C-minor scale.
bbbb
aabb eebb bbbb**ddbb
ff gg dd
aa ee bbddbb**
gg## dd##
ff##
a*a*
ffbbbbbbbb
ggbb
eebb**
just
the
ory
Just modulation: Just modulation: Target tonalities for the melodic C-minor scale.Target tonalities for the melodic C-minor scale.
bbbb
aabb eebb
ff gg dd
aa ee
just
the
ory
9 10 1211 13 14 15 16
1 2 43 5 6 7 8
25 26 2827 29 30 31 32
17 18 2019 21 22 23 24
no modulationsno modulations infinite modulationsinfinite modulations limited modulationslimited modulations
four modulationsfour modulations
major, natural, harmonic, melodic minormajor, natural, harmonic, melodic minor
just
the
ory
Classes of 3-element motives M Classes of 3-element motives M ŸŸ1212
22
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
genericgeneric
rhyt
hm
ic m
od
ula
tio
n
onset
Percussion encoding
62^62^
Retro-Retro-gradegradeofof62^62^
62^
R(62^)
rhyt
hm
ic m
od
ula
tio
n