1. IV EXPERIMENTS
IV.3 (Thu March 29) Modeling tonal modulation

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

3. Arnold Schönberg: Harmonielehre (1911)
old tonality
neutral
degrees
(IC, VIC)
fundamental
(pivot)
degrees
(IIF, IVF, VIIF)
new tonality
cadence
degrees
(IIF & VF)
What is the set of tonalities?
What is a degree?
What is a cadence?
Which is the modulation mechanism?
How do these structures determine the fundamental degrees?

4. C scale = part of Ÿ12 pitch class space Ÿ12 for 12-tempered tuning1 2 3 4 5 6 7 8 9 10 11
scale = part of Ÿ12 C
twelve diatonic scales: C, F, Bb , Eb , Ab , Db , Gb , B, E, A, D, G

5. I II
III IV V VI
VII

6. Harmonic band of major scale C(3)IV II VI V
III
VII

7. Dia(3) C(3) G(3) F(3) Bb (3) D(3) triadic interpretations E b(3) A(3)
Ab(3)
Db(3)
Gb (3)
B(3)
E(3)
A(3)
D(3)
G(3)
Dia(3)
triadic interpretations

8. 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

9. force = symmetry between S(3) and T(3)
gluon
strong force W+
weak force
g photon
electromagnetic force
graviton
gravitation
quantum = set of
pitch classes = M
S(3)
T(3)
force = symmetry between
S(3) and T(3) k

10. modulation S(3) ® T(3) = „cadence + symmetry “
et.A
S(3)
T(3) k et
modulation S(3) ® T(3) = „cadence + symmetry “

11. Given a modulation k, g:S(3) ® T(3)
a quantum for the modulation (k,g) is a set M of pitch classes such that:
the symmetry g is a symmery of M, g(M) = M
the degrees in k(T(3)) are contained in M
M Ç T is rigid, i.e., has no non-trivial symmetries
M is minimal with the first two conditions M
S(3)
T(3) k g

12. Modulation theorem for 12-tempered tuning
For two different tonalities S(3), T(3) there exist
a modulation (k,g) and
a quantum M for (k,g)
(= quantized modulation)
Moreover:
M is the union of the degrees in S(3), T(3) contained in M which thereby define the triadic interpretation 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.

13. VC
IVC
VIIC
IIC
VEb
VIIEb
IIEb
IIIEb
M(3)
C(3)
E b(3)

14. # fourths

15. These fundamental degrees
coincide with Schoenberg's
everywhere, where he
discusses modulation.

17. q-modulation = quantized modulation
Modulation theorem (12-tempered case) for the 7-tone scales S and triadic interpretations S(3) (Daniel Muzzulini: Musical Modulation by Symmetries. J. for Music Theory 1995)
q-modulation = quantized modulation
(1) S(3) is rigid.
For such a scale, there is at least one q-modulation.
The maximum of 226 q-modulations is reached for the harmonic minor scale #54.1, the minimum of 53 q-modulations happens for the scale #41.1.
(2) S(3) isn‘t rigid.
For the scales #52 and #55, there are q-modulations except for transposition t = 1, 11; for #38 and #62, there are q-modulations except for t = 5,7. All the 6 other types have at least one q-modulation.
The maximum of 114 q-modulations happens for the melodic minor scale #47.1. Among the scales with q-modulations for all t, the major scale #38.1 has a minimum of 26.

18. Here, we illustrate the theory on the jazz CD „Synthesis“ which I recorded in Its entire structure, in harmony, in rhythmics, and in melodics, was deduced and constructed by use of the composition software presto® (written for Atari computers, but now also working on Atari emulations on Mac OS X), and starting from the 26 classes of three-element motives.
This composition (a grant from the cultural department of the city of Zurich) was not recognized as a computer-generated music by the jazz critics. Only the piano part was played by myself, the entire bass and percussion part was played by synthesizers, driven by the presto® application via MIDI messages.
During the production of this composition (Synthesis is a four-part, 45-minute piece), I never felt inhibited in my piano playing, in the contrary, it was a great pleasure to collaborate with complex structures of rhythm or melody, objects of a complexity that human percussionists would never be able to play from a score.
presto®

20. 1 2 3 4 5 6 7 8 9
generic 10 11 12 13 14 15 16 17 18
The classification of all possible concept types and the instances within one fiexed type is a ‚wild‘ problem. Let us just look at all isomorphism types of three-element motives with pitch modulo 12 and onset modulo 12 in the integer valued coordinate spaces. Here, isomorphism means that one is allowed to reflect, shift or rotate a given motive. The ‚orbits‘ of these isomorphism actions are the classes, there are 26 of them. I shall come back to this example below when discussing a jazz CD. 19 20 21 22 23 24 25 26
motif classes with 3-elements M Í Ÿ122

22. percussion parameters
time
percussion parameters
62^
retro-
grade of
62^
R(62^)

23. 12/8
M.1-6 m1 m2 m3 m4 m5
m6, m7
M.7-12 m1 m2 m3 m4 m5
m6, m7 R
M.13-24
3:18-5:48
modulation pivots
2nd tonic
at 9/8 of b. 21
2nd system of bars

24. 3:18-5:48