Guerino Mazzola

U & ETH Zürich

Internet Institute for Music Science

guerino@mazzola.ch

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?

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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

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I IV VII III VI VII

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I

IV

II

VIV

III

VII

Harmonic strip of diatonic scaleHarmonic strip of diatonic scale

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))

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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

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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 “

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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.

CC(3)(3) EE bb(3)(3)

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MM(3)(3)VVEEbb

VIIVIIEEbb

IIIIEEbb

IIIIIIEEbbVC

IVC

VIIC

IIC

expe

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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

ee bb

EE bb(3)(3)

expe

rimen

ts

bb

BB(3)(3)

Inversion Inversion e e bb

expe

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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.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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ory

ff cc gg dd

ddbb aabb eebb bbbb

aa ee bb ff##

just

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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

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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

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ic m

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rhyt

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ic m

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onset

Percussion encoding

62^62^

Retro-Retro-gradegradeofof62^62^

62^

R(62^)

rhyt

hm

ic m

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3:18-5:48

rhyt

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n12/8

B.1-6m1 m1

m2m1m2m3

m1m2m3m4

m1m2m3m4m5

m1m2m3m4m5m6,m7

B.7-12m1m1

m2m1m2m3

m1m2m3m4

m1m2m3m4m5

m1m2m3m4m5m6,m7

RR

B.13-24

modulation pivotsmodulation pivots

new tonicat 9/8 of bar 21

new bar system