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Modeling Tonal ModulationModeling Tonal Modulation

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Schoenberg (Harmonielehre):Schoenberg (Harmonielehre):

There is, for example, a very popular harmony treatise, in which There is, for example, a very popular harmony treatise, in which moduations are nearly exclusively made using the dominant moduations are nearly exclusively made using the dominant seventh or diminished seventh chord. And the author only seventh or diminished seventh chord. And the author only demonstrates that after each major or minor tirad any of those demonstrates that after each major or minor tirad any of those two chords can be played, and thereby go to any tonality. If I two chords can be played, and thereby go to any tonality. If I wanted that, I could have finished even earlier. In fact I am wanted that, I could have finished even earlier. In fact I am capable to show (using „gauged“ examples from liteature) that capable to show (using „gauged“ examples from liteature) that you may use any triad after any other triad. So if that reaches you may use any triad after any other triad. So if that reaches every tonality and thereby modulation has been realized, the every tonality and thereby modulation has been realized, the procedure would even be simpler. But if somebody, to tell a procedure would even be simpler. But if somebody, to tell a story, makes a journey, he would not choose the air line. The story, makes a journey, he would not choose the air line. The shortest path is the worst. The bird‘s perspective is the shortest path is the worst. The bird‘s perspective is the perspective of a bird‘s brain. If everything is blurred, everything perspective of a bird‘s brain. If everything is blurred, everything is possible. Differences disappear. And it is then irrelevant if I is possible. Differences disappear. And it is then irrelevant if I have made a moduation with a dominant or diminished seventh have made a moduation with a dominant or diminished seventh chord. The essential of a moduation is not the target, but the chord. The essential of a moduation is not the target, but the trajectory.trajectory.

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old tonality neutral/pivot

degrees(IC, VIC)

fundamentaldegrees

(IIF, IVF, VIIF)

new tonality cadencedegrees

(IIF & VF)

Arnold Schönberg: Arnold Schönberg: Harmonielehre (1911)Harmonielehre (1911)

• What is the set of What is the set of tonalitiestonalities??• What is a What is a degreedegree??• What is a What is a cadencecadence??• Which is the Which is the modulationmodulation mechanismmechanism??• How do these structures How do these structures determine the fundamental degreesdetermine the fundamental degrees??

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pitch class space pitch class space ŸŸ1212

for 12-tempered tuningfor 12-tempered tuning0

1

2

3

4

56

7

8

9

10

11

twelve diatonic scales: twelve diatonic scales: C, F, BC, F, Bbb , E, Ebb , A, Abb , D, Dbb , G, Gbb , B, E, A, D, , B, E, A, D, GG

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 band of major scale CHarmonic band of major scale C(3)(3)

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CC(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 interpretationsinterpretations

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SS(3)(3)

space of cadence parametersspace of cadence parameters

kk11((SS(3)(3)) = {II) = {IISS,, VVSS}}

kk22((SS(3)(3)) = {II) = {IISS,, IIIIIISS}}

kk33((SS(3)(3)) = {III) = {IIISS,, IVIVSS}}

kk44((SS(3)(3)) = {IV) = {IVSS,, VVSS}}

kk55((SS(3)(3)) = {VII) = {VIISS}}

kk

k(k(SS(3)(3)))

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SS(3)(3) TT(3)(3)

gluon

strong force

W+

weak force

electromagnetic force

graviton

gravitation

force = symmetry betweenforce = symmetry between SS(3)(3) and T and T(3)(3)

quantum = set of quantum = set of pitch classes = Mpitch classes = M

k k

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SS(3)(3) TT(3)(3)

k k

A Tt

Tt.A

Tt

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 a quantumquantum for the modulation (k,g) is a set for the modulation (k,g) is a set MM of pitch classes such that:of pitch classes such that:

• the symmetry g is a symmery of the symmetry g is a symmery of MM, g(, g(MM) = ) = MM• the degrees in k(the degrees in k((3)(3))) are contained in are contained in MM• MM TT is rigid, i.e., has no non-trivial symmetries is rigid, i.e., has no non-trivial symmetries• MM is minimal with the first two conditions is minimal with the first two conditions

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The

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ts Modulation theorem for 12-tempered tuningModulation theorem for 12-tempered tuning

For two different tonalities For two different tonalities SS(3)(3),, (3)(3) there exist there exist• a modulation (k,g) anda modulation (k,g) and• a quantum a quantum MM for (k,g) for (k,g) (= (= quantizedquantized modulationmodulation))

Moreover:Moreover:• M M is the union of the degrees in is the union of the degrees in SS(3)(3),, (3)(3) contained in contained in M M

which thereby define the which thereby define the triadic interpretation Mtriadic interpretation M(3)(3) of of MM• the common degrees of the common degrees of (3)(3) and and MM(3)(3) are called the are called the

modulation degrees modulation degrees of (k,g)of (k,g)• the modulation (k,g) is the modulation (k,g) is uniquely uniquely determined by the determined by the

modulation degrees.modulation degrees.

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CC(3)(3) EEbb(3)(3)

MM(3)(3)VVEEbb

VIIVIIEEbb

IIIIEEbb

IIIIIIEEbbVC

IVC

VIIC

IIC

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# fourths# fourths

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The

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tsModulation theorem (12-tempered case) for the 7-tone scales Modulation theorem (12-tempered case) for the 7-tone scales SS and triadic interpretations and triadic interpretations SS(3) (3)

(Daniel Muzzulini(Daniel Muzzulini: Musical Modulation by Symmetries. J. for Music : Musical Modulation by Symmetries. J. for Music Theory 1995Theory 1995))

q-modulation = quantized modulationq-modulation = quantized modulation

(1) (1) SS(3) (3) is rigid.is rigid.• For such a scale, there is at least one q-modulation.For such a scale, there is at least one q-modulation.• The maximum of 226 q-modulations is reached for the The maximum of 226 q-modulations is reached for the

harmonicharmonic minor scaleminor scale #54.1, the minimum of 53 q-modulations #54.1, the minimum of 53 q-modulationshappens for the scale #41.1. happens for the scale #41.1.

(2) (2) SS(3) (3) isn‘t rigid.isn‘t rigid.• For the scales #52 and #55, there are q-modulations except forFor the scales #52 and #55, there are q-modulations except for

transposition t = 1, 11;transposition t = 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 the 6 other types have at least one q-modulation.All the 6 other types have at least one q-modulation.

• The maximum of 114 q-modulations happens for the The maximum of 114 q-modulations happens for the melodicmelodic minor minor scalescale #47.1. Among the scales with q-modulations for all t, #47.1. Among the scales with q-modulations for all t,

the the major scalemajor scale #38.1 has a minimum of 26. #38.1 has a minimum of 26.

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The

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tsSonata schemeSonata scheme

for the allegro movementfor the allegro movementin Beethoven‘sin Beethoven‘s

op.106op.106according to Erwin Ratz:according to Erwin Ratz:

Einführung in dieEinführung in diemusikalische Formenlehre.musikalische Formenlehre.

Universal Edition, Wien 1973Universal Edition, Wien 1973

!!

!!

!!

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7:06

Fast normal modulation:Fast normal modulation: B Bbb GG bb

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4:50

Normal modulation:Normal modulation: G G EE bb

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Modulation of „catastrophe“ typeModulation of „catastrophe“ type: EE bb(3) (3) DD(3)(3)~~ bb(3) (3)

6:00

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Erwin Ratz‘ (1973) and Jürgen Uhde‘s (1974) thesesErwin Ratz‘ (1973) and Jürgen Uhde‘s (1974) theses

Ratz: Ratz: The sphere of tonalities of op. 106 is polarized in a“The sphere of tonalities of op. 106 is polarized in a“worldworld“ centered “ centered around B-flat major, the main tonality of the sonata, around B-flat major, the main tonality of the sonata, and an „and an „antiworldantiworld“ centered around b-minor.“ centered around b-minor.

Uhde*: When one changes between the worlds of Ratz— an event that Uhde*: When one changes between the worlds of Ratz— an event that happens twice in the allegro movement—then the modulation happens twice in the allegro movement—then the modulation processes become dramatic. They are completely different from processes become dramatic. They are completely different from other modulations, and Uhde calls them „other modulations, and Uhde calls them „catastrophescatastrophes“. “.

b minorb minorb minorb minorB-flat majorB-flat majorB-flat majorB-flat major

* Uhde J: Beethovens Klaviermusik III. Reclam, Stuttgart 1974* Uhde J: Beethovens Klaviermusik III. Reclam, Stuttgart 1974

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CC(3)(3)

BBbb (3)(3)

EEbb(3)(3)

DDbb(3)(3)

GGbb (3)(3)

EE(3)(3)

AA(3)(3)

GG(3)(3)

Thesis:Thesis: The modulation structure of op. 106 is governedThe modulation structure of op. 106 is governedby the symmetries of the diminished seventh chord by the symmetries of the diminished seventh chord CC## -7-7 = {c = {c##, e, g, b, e, g, bbb} } carrying the admitted modulation forces. carrying the admitted modulation forces.

FF(3)(3)

AAbb(3)(3)

BB(3)(3)

DD(3) ~ (3) ~ bb(3) (3)

ExpositionExposition

RecapitulationRecapitulation

DevelopmentDevelopment

CodaCoda

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CC(3)(3)

FF(3)(3)

BBbb (3)(3)

EEbb(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)

Sym(Sym(CC## -7-7))= max. symmetry = max. symmetry group separating D group separating D from Bfrom B

Sym(Sym(CC## -7-7))= max. symmetry = max. symmetry group separating D group separating D from Bfrom B

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CC(3)(3)

BBbb (3)(3)

EEbb(3)(3)

DDbb(3)(3)

GGbb (3)(3)

EE(3)(3)

AA(3)(3)

GG(3)(3) FF(3)(3)

AAbb(3)(3)

BB(3)(3)

DD(3) ~ (3) ~ bb(3) (3)

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ee-3 -3 IIg g * I * Id/dd/d## * * IIbbbb IIa/aa/abb ee3 3

Modulators in op. 106/allegroModulators in op. 106/allegro

ExpositionExposition RecapitulationRecapitulation DevelopmentDevelopment CodaCoda BBbb G G G G EEbbD/b D/b BBbbBBbbGGbb G G BBbb BBbb

symmetries of transposition!symmetries of transposition!

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7:06

Fast normal modulation:Fast normal modulation: B Bbb GGbbIIBBbb

VIVIGGbb

IIIIIIGGbb V VGGbb

InversionInversionbbbb

I

IV

II

VIV

IIIVII

a-flat in II a-flat in II V V VII VIIb-flat in III b-flat in III VI VI I If in III f in III V V VII VII

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The

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tsLudwig van Beethoven: op.130/Cavatina/Ludwig van Beethoven: op.130/Cavatina/# 41 # 41

InversionInversioneebb : EEbb(3) (3) BB(3)(3)

4:00

oppressive

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ee bb

EEbb(3)(3)

bb

BB(3)(3)

Inversion Inversion e e bb

Transposition TTransposition T44

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The

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tsInversionInversionddbb : GG(3) (3) EEbb(3)(3)

ddbb

gg

gg

#124 - 125 #126 - 1274:50

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The

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tsTranspositionTransposition-3-3: BBbb(3) (3) GG(3)(3)

VIIG

VIIG