VEXPERIMENTS

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V EXPERIMENTS V.10 (W May 06) Summary, planning
the second paper.
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I Introduction About the space, where all that
happens, in particular gesture -gt process -gt
facts. It gives a good orientation in musical
ontology WHERE AM I WITH MY MUSIC?
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tonal modulation G maj E-flat maj
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to make
to refer to
to be the case
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Hjelmslevs Babuschka-Principle
Expression
Signification
Content
Connotation
Motivation
Meta System
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The Oniontology of Music
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• Definition
• The art of collaboration is characterized by
• the flow of a graphically distributed identity
• by means gestural communication
• within a collaborative space (collaboratory)

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The Rubato Composer Space
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II Concept Spaces The universal language of
denotators and its implementation. It offers to
those, who work with music objects, conceptual
tools HOW TO CONCEIVE MUSICAL IDEAS?
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w(x) W(Sp(x)) S M Sp(x), m l(M) l(M)p
m p 2
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Retis program of analytical motif theory Rudolf
Reti The Thematic Process in Music (McMillan,
New York 1951, repr. Greenwood, Westport 1978)
Program is the analytical construction of a
germinal motif of a composition. Michael
Kopfermann Comments on Retis approach. in
Robert Schumann (I,) II (H.-K. Metzger R. Riehn
eds.) edition textkritik, Munich 1982. p. 297
construction of germinal motif from immanent
analysis. Example Kinderszenen, p. 276.
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• ... is everywhere in the composition...
• presence
• content
• weight presence x content

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... is everywhere in the composition...
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... is everywhere in the composition...
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Sylvain Auroux La sémiotique des encyclopédistes
(1979)

• Three encyclopedic caracteristics of general
  validity
• unité (unity) grammar of synthetic discourse
  philosophy
• intégralité (completeness) all facts
  dictionary
• discours (discourse) encyclopedic ordering
  representation

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ramification type completeness
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• FM-Synthesis

Support
Modulator
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D
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Dodecaphonic Series
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III Models Models are preliminary forms of
musical laws in the Planck time. They offer a
roadmap to musical revolutions WHERE SHOULD I GO?
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eternal laws in music Pythagoras, Euler,
Helmholtz, Schenker, Lewin...
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Scherer, Jacques Le Livre de Mallarmé.
Gallimard, Paris 1957

• The necessity of a proof of existence
• The mathematical attitude
• The problem of self-referentiality in the arts
• I write, therefore I am.
• The multiple perspective proof
  (symmetry-correspondence, commutative
  diagram!)

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• A model is a principle for building musical
  formulas (yielding schemes for compositions)
• it must be such that it has been understood as a
  double negation, i.e.,
• 1. it is embedded in a space of variants
  (negation) and
• 2. can be distinguished from them by its
  characteristic position (negation of negation,
  anthropic principle)

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(K/D) is a strong dichotomy, i.e., there is a
unique symmetry (invertible) ya.xb of torus
exchanging K and D, i.e.,y5.x2 This is the
autocomplementarity fonction AC AC(0) 2AC(3)
5AC(4) 10AC(7) 1AC(8) 6AC(9)
11AC2 Id
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• ? x ?.k
• AC?(X) 5.X (8x ?.2)unique symmetry fixing
  the tangent space in x
• Take g such that
• ? ? g(Ke)
• AC? exchanges g(Ke), g(De)
• g(De) ? Ke maximal with 1., 2.

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All ? ? g(De) ? Ke are permitted
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Allowed Transitions for Major Scale
The Topos of Music Table O.2 pp.1217/18
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Parallel fifths are strictly forbidden
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Jonathan Winson hippocampal gate hypothesis
Elton John and Diana
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Proposition Among the 34 dichotomy
classes, there are 6 strong classes. The
distances between the members of one half (or the
other) of such a dichotomy are class invariants
and characterize these classes
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K 0, 3, 4, 7, 8, 9, 11 (add leading 11 to
consonances)
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K 0, 3, 4, 7, 8, 9, 11 mela Nr. 13
gayakapriyaclass 60 ragas -gt melakarta 72
scales scale mela Nr. 15 0, 3, 4, 7, 8 , 9,
1 mayamalavagaulaclasse 61
Conterpoint with major dichotomy on exotic
scales! Write progam for contrapuntal deformation
(K/D)-gt(I/J)!
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Piano 1 uses these address changes
one single address change U?U
Piano 2 uses these address changes
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IV Operationalization Performance theory deals
with the transformation of symbolic reality into
physical reality. It opens the discourse about
the fine-tuned shaping of symbols HOW SHOULD WE
PERFORM?
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instrumentalize
gesture of sound
unfold
formula
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x v(X)
X
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• In order to model piano performance, we work
  with
• the base parameters E, H, L,
• corresponding fields T(E), S(H), I(L),
• and the duration parameter D (pianola)
• A cellular hierarchy is a diagram D in Cell such
  that
• there exists exactly one cell source
• the sets of parameters of the cell diagram
  are closed under union and non-empty
  intersection

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violin default hierarchy
Attention different notation I instead of S
here (from book The Topos of Music)
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presto
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stemma
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Roberto FerrettiJan Beran
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V Experiments We give two extended examples of
experiments by musical composition. They are made
to test theoretical models on the level of sound
objects CAN CREATION BE RATIONALIZED ?
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Physicists and mathematicians are far in advance
of musicians in realizing that their respective
sciences do not serve to establish a concept of
the universe conforming to an objectively
existent nature. They are fully cognizant of the
fact that, conversely, their task is to make an
existing conception of the universe conform to
the largest number of observations demonstrable
by scientific experiments. Musical systems
(...) have not been created by nature or by some
mystical Supreme Being, but have been produced by
man to render music possible within a certain
sphere. (...) As the study of axioms eliminates
the idea that axioms are something absolute,
conceiving them instead as free propositions of
the human mind, just so would this musical theory
free us from the concept of major/minor tonality
(...) as an irrevocable law of nature. Whereas
geometric axioms are sufficiently justified if
their combinations prove them to be both
independent of and compatible with each other,
the accuracy of musical axioms can be proved
exclusively by their fitness for musical
practical use. (...) A system of musical axioms
can never been established in theory until it has
been demonstrated in practice.
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This yields a generic concept of an
experiment The generic concept of an experiment
is that of a symmetric dialog of question and
answer between two partner instances, which are
seeking the answers from the traces of a
navigation through topoi of specifically known
worlds and thereby exerting mutual
influence. Not a one-sided inquiry or
interrogation gt uncertainty principle in quantum
physics The question influences and changes
reality... So the question What is music? is
an experiment of the mind. This all the more that
we still live in Plancks early time.
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Robert Gjerdingen An Experimental Music
Theory? in N. Cook and M. Everest, eds.
Rethinking Music , vol. 2, pp. 161170, (Oxford
Oxford Univ. Press, 1999). Today, whenever I
attend a meeting of music theorists. I am struck
by the conviction with which old beliefs are
invoked as eternal verities. Tonality, like
Aristotles principle of the soul, is asserted
as an agent in the world. Voice-leading, like
Aristotles heat, is given causal force as an
explanation of why musical things are the way
they are. The waxing and waning of these
principles seemingly create music history, govern
style, and, proceeding a little further in this
way, determine hearing itself. Recent attempts
to problematize these verities have resulted in
complexified dogma. The basic terms of the
debate, however, remain unchanged. As a result,
music-theoretical discourse has become largely
music-exegetical in content. The
self-stabilizing, corroborating effect of
interdependent premisses precludes fundamental
revisions, major discoveries, or even accidental
breakthroughs. (American philosopher Hilary
Putnam Mathematics as an experimental science)
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• Composition principles
• Syntagmatic- global-local generalizes
  segmentationmusical manifolds the science of
  ambiguity, not ambiguous science- (general)
  modulation landscapes and journeys- (general)
  contrapuntal syntagms- Boolean operations
• Paradigmatic (para nearby, digma reference)-
  Digmatic transformations - correspondenceslocal
  principle! How to glue together local charts of
  musical manifolds, Grids global organization
  of transformations- Para alterations,
  quantizing (general) morphing - topological
  deformations

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Roman Jakobsons poetical function projecting
the pardigmatic axis onto the syntagmatic axis.
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Sonate für Klavier AutG(Messiaen III)\DIA(3)
(1981) Gruppen und Kategorien in der
Musik Heldermann, Berlin 1985 Construction on 58
pages 99 bars, 12/8 meter, C-major
Lessence du bleu Acanthus, Bern 2002 CD
Patrizio Mazzola
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Erwin Ratz (1973) and Jürgen Uhdes (1974) theses
Ratz The sphere of tonalities of op. 106 is
polarized in aworld centered around B-flat
major, the main tonality of the sonata, and an
antiworld centered around b-minor.
Uhde When one changes between the worlds of
Ratz an event that happens twice in the
allegro movementthen the modulation processes
become dramatic. They are completely different
from other modulations, and Uhde calls them
catastrophes.
Uhde J Beethovens Klaviermusik III. Reclam,
Stuttgart 1974
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Arnold Schönberg Harmonielehre (1911)
old tonality neutral degrees (IC, VIC)
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 pivot
  degrees?

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Given a modulation k, gS(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

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Thesis The modulation structure of op. 106 is
governed by the symmetries of the diminished
seventh chord C -7 c, e, g, bb
carrying the admitted modulation forces.
Exposition
Developement
Recapitulation
Coda
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Inversiondb G(3) E b(3)
450
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motivic zigzag in op.106
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op. 3
op. 106
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Thesis The modulation structure of op. 3 is
governed by the symmetries of the augmented
triad C c, f, a in the role of
admitted modulation forces.
Exposition
Developement
Recapitulation
Coda
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Motivic zigzag scheme
minor third Messiaen 2 scale limited
transposition
major third Messiaen 3 scale limited
transposition
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Dodecaphonism strategy with local symmetries
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grid of translations ? grid diagram of symmetries
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Generalization to several dimensions?
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Intercultural Morphing
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708
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Charles Baudelaire Les fleurs du mal
(1861) LA?MORT?DES?ARTISTESCombien?faut-il?de?f
ois?secouer?mes?grelotsEt?baiser?ton?front?bas??m
orne?caricature?Pour?piquer?dans?le?but??de?mysti
que?nature?Combien??ô?mon?carquois??perdre?de?jav
elots?Nous?userons?notre?âme?en?de?subtils?compl
ots?Et?nous?démolirons?mainte?lourde?armature?Av
ant?de?contempler?la?grande?CréatureDont?linfern
al?désir?nous?remplit?de?sanglots?
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Roman Jakobsons poetical function revisited!
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Prosody musical performance of written
language!
0 -gt
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The second paper should present a critical
discussion of the topic What is the difference
between the experiment realized by a musical
creation/composition and an experiment in the
sciences? This discussion must be based upon all
parts I - V of the course. Make appropriate
references for citations (the PPT files of this
course, the references given in the syllabus,
etc.). Paper is due Wednesday, 9 May (bring to
my office, please)
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• What is the topic about?
• Analyze the problem!
• Propose answers to the questions!
• Describe consequences for your approach to
  music!
• Formal hints
• Make subtitles such as The Problem, My
  Analysis, Consequences, etc., ending with
  References or Literature
• Make appropriate citations, such as BOOK98
  referring to the book in the References BOOK98
  Aristoteles Topic. NY, 555 b.c.