1. de l‘interprétation musicale: Champs vectoriels d‘interprétation
Les mathématiques
de l‘interprétation musicale:
Champs vectoriels
d‘interprétation
What is Performance?
Performance is the physical execution of a work of art. This work may be a written poem, a musical score or a sculpture (a sculpture is performed while you walk around it and interact with its perspectives and its presence in space). This terminology subtends that performance adds an essential part to the somewhat abstract or symbolic "text". To put it with Paul Valéry: "C'est l'execution du poème qui est le poème." The point of performance is that its added value depends on the way of executing the symbolic data, performance is a rhetoric category. In performance, we communicate the work's contents, we make clear what could, to our mind, be the signification of the symbolic score.
So why is performance essential? Couldn't we just stick to the abstract analysis of a work and try to understand it on the level of reflection? There are at least two reasons why this is not sufficient. First, the existential level of physical execution is different from the mental level of score symbols: Reading a recipe never replaces its cooking and consumption. Second, the compact sensual presentation of a fact has a degree of evidence which cannot be paralleled by intellectual meditation. After all, geometric visualization is an essential extension of the abstract mathematical concept framework and enjoys the well-known power of evidence. In a first approach, performance is to the fine arts what is geometric visualization to mathematics.
Guerino Mazzola
U & ETH Zürich
Internet Institute for Music Science

2. The Topos of Music Geometric Logic of
October 2002
1300pp, ≈ 150¤
The Topos of Music
Geometric Logic of
Concepts, Theory, and Performance
in collaboration with
Carlos Agon, Moreno Andreatta, Gérard Assayag, Jan Beran, Chantal Buteau, Roberto Ferretti, Anja Fleischer, Harald Fripertinger, Jörg Garbers,
Stefan Göller, Werner Hemmert, Michael Leyton,
Mariana Montiel, Stefan Müller, Thomas Noll, Joachim Stange-Elbe, Oliver Zahorka

3. Fields
Performance is a very complex, global phenomenon [Mazzola 2002]. It involves four globalization factors: (1) instrumental differentiation, (2) splitting the score into voices, periods, and similar groupings, (3) specifying the roles of the musical parameters (pitch, duration, onset, loudness, etc.) into basic parameters, such as onset time, vs. dependent parameters, such as duration (depending on onset data), and (4) unfolding the level of detail and sophistication of a performance as a function of its rehearsal history in the spirit of a genealogical tree.
Rhetorically speaking, each performance is a perspective view upon a complex object, and its understanding results from a sum of varied perspectives. If you perform a sculpture, you walk around it and integrate the various perspectives. This is the most common variant of the famous Yoneda lemma in mathematical category theory [Mac Lane1998]: Classification means integration of morphisms. This also suggests that the famous unicorn of hermeneutics: the ideal performance, does not exist. On the contrary, a work needs an infinity of performances to comprise its complete understanding. Summarizing, a concrete performance is one of an infinity of expressions of the hidden and ambiguous contents of a given work.

4. Fields √ √E √E(I1) √E(Ik) [q /sec] H E L h e l X x = √(X) E e pE pe
Performance theory splits into two main concerns:
1. structure theory and
2. semantic theory.
Whereas the former deals with the precise description of a given performance, the latter tries to understand the rationales of a given performances, i.e., why a performance is produced in the way it appears, for what reason an artist shapes his/her execution in one way and not in another one.
We shall first discuss structure theory. Performance is a map from symbolic to physical reality, Tempo is a classical expression for this reality switch. E e pE pe √E
T(E) = (d√E/dE)-1
[q /sec]
√E(I1)
√E(Ik) I1 Ik

5. P-Cells Product fields: Tempo-Intonation field EH E H
Z(E,H)=(T(E),S(H)) E H
P-Cells
S(H)
Local Hierarchies
Consider first the default field generated as a product of a tempo and an intonation field.
Here bothe parameters, onset and pitch play the same role, and their fields are projections of the field on the onset-pitch plane.
T(E)

6. P-Cells Parallel fields: Articulation field ED E
Z(E,D) = ¶T(E,D) = (T(E),2T(E+D)-T(E))
P-Cells E D
But not all music parameters play the same role in performance. For example, performance of durations is determined by performance of onsets. In the default setup, the performed duration is the difference between the performed onset of the event's end and the performed onset of the event's beginning. This type of dependencies entails hierarchies of parameters, and therefore, hierarchies of performance fields. Rational performance deals with the calculation of such field hierarchies by use of analytical weight functions.
(e(E),d(E,D) = e(E+D)-e(E))
T(E)

7. P-Cells Work with Basis parameters E, H, L,
and corresponding fields T(E), S(H), I(L)
Pianola parameters D, G, C
A cell hierarchy is a Diagram D in Cell such that
there is exactly one root cell
the diagram cell parameter sets are closed under union and non-empty intersection
P-Cells
¶T ¥ S
Root
¶T ¥ I ¥ S
Here is a classical example of the piano performance hierarchy ¶T
¶T ¥ I
T ¥ S
T ¥ I ¥ S
Fundament T
T ¥ I S
I ¥ S I

8. l Typology mother ¶T daughter granddaughter Stemma T Z(¶T,l) Tl
Genealogical Trees of Rehearsal
The fourth global component of performance stems from the fact that performance is never a one-step process. Artists have to practice, to rehearse, to develop their valid performance through a cascade of intermediate performances, starting with the mechanical prima vista performance on the given score. This is formalized in the so-called stemma. This is a genealogical tree, starting from the mechanical prima vista performance, and successively splitting into performances of the score's parts (voices, periods, etc.) and generated from the previous performances by specific performance operators which are loaded with selected weight functions. For example, a child of a mother performance may be defined by the application of a metrical weight to the mother's tempo curve.
granddaughter Tl
Z(¶T,l)
Stemma

9. Typology ???? ??!! Describe typology of shaping operators!
Big Problem:
Describe typology of shaping operators!
Emotions, Gestures, Analyses
????
??!!
Typology
w(E,H,…)
There are three main rationales for performance: emotion, gesture, and ratio.
Emotion has strongly been preconized by Alf Gabrielsson. He maintains that "we may consider emotion, motion and music as being isomorphic" [Gabrielsson1995]. While this conjecture may please psychologists, it is completely useless to scientific investigation. In fact, such an isomorphism is a piece of poetic literature as long as the components: emotions, gestures, and music, are not described in a way to make this claim verifiable. Presently, there is no hope for a realistic and exhaustive description of emotions. Same for gestures, and as to music, the mathematical categories of local and global musical objects are so incredibly complicated that the mere claim sounds like a cynical joke. For example, the number of isomorphism classes of 72-element motives in pitch and onset (modulo octave and onset period) is ^36 [Fripertinger1993]. How could the claimed isomorphism fit in this virtually infinite arsenal?
Gestural categories as a rationale for performance have been advanced in approaches [Kronmann1987] which maintain that musical retards, for example, share a structure of Newtonian mechanics. Such approaches cannot, however, explain the agogic phenomena within a motivic movement, or the dynamical differentiation within a chord, for example. Moreover, the gestural motivation for a determined instance of performance is extremely complex: How could one deduce Glenn Gould's performane when knowing his beautiful dance of fingers, arms, and body?
This is why we shall stick to rational semantics in performance, it is the easiest and most explicit rationale. This means that we have to investigate the score text by means of metrical, rhythmical, motivic, harmonic, contrapuntal etc. analyses and to correlate these findings to the expressive shaping of performance. This is also a traditional and important requirement of rhetorics: to convey the text's meaning, and not personal emotions or gestures. Theodor W. Adorno has strongly recommended such an analytical performance approach [Adorno1963]. It is an interesting question, whether traditional performances have much to do with analytical performance, and if not, how such a performance would sound like! We shall give an example of such a performance in this talk.
w(E,H,…) H E

10. Calculations RUBATO® software:
Calculations via Runge-Kutta-Fehlberg methods for numerical ODE solutions
Calculations
Field integration on RUBATO

11. Typology ? Zw = Qw(E,D).Z Tempo Operators T(E) w(E) Tw(E) = w(E).T(E)
Deformation of the articulation field hierarchy ¶T
¶Tw T Tw Z Zw T Tw ?
Sundberg's Performance Grammars
The way how a given bunch of analytical weights acts on the successive refinement of performance on the stemma has been termed "performance grammar" by Johan Sundberg, one of the fathers of modern performance theory [Friberg1991]. The yoga of performance grammars is this: It could appear as if performance theory would just be interested in the production and simulation of "good" performances. But this is not very interesting. If we listen to a performance, we are not seeking to understand this particular result, but we want to recognize a possible system which leads to the present output. If a specific performance can be produces without any systematic insight, it is useless. Performance research is the search for systems which generate interesting performances. This is what a grammar should look like. It is a scientific requirement: The "experiment" should be reproducible, not just a matter of chance.
Qw(E,D) =
w(E)
w(E+D)—w(E) w(E+D)
Zw = Qw(E,D).Z
Qw = J(√w)-1 „w-tempo“

12. Typology RUBATO®: Scalar operator Linear action Qw on
ED-tangent bundle
Typology
Direction of field changes
Numerical integration control

13. Inverse Theory Restriction Lie type Affine transport
Inverse Performance
Control of performance need not restrict to its productive perspective, it is also important to control given performances, i.e., to understand the mechanisms which could lead to such a performance. This is the subject of inverse performance theory.
To begin with, such a theory has to offer means for calculating the final performance fields of a supposed stemma (the fields at the stemma's leaves).

14. Inverse Theory Stefan Müller: EspressoRubette
In his ongoing PhD thesis, Stefan Müller has implemented the EspressoRubette which calculates all the performance fields of a MIDI-formatted recording with respect to the given score file (also in MIDI format) [Müller2002].

15. Inverse Theory Restriction Restriction Lie type Lie type Sum
Affine transport

16. Inverse Theory Roberto Ferretti Lie operator parameters: weights,
directions
fiber(Z.)
Inverse Theory
Given such an output of performance fields, we may consider all possible parameters which yield this output, i.e., all possible weights, stemma parameters and performance operator parameters.
In a special case of such a background parameter system, the locally linear grammars, Roberto Ferretti has calculated the algebraic varieties of parameters which yield a fixed given output. It turns out [Ferretti2002] that generically, such varieties are isomorphic, but under reasonable restrictions, the parameter varieties help distinguishing different performances on the level of algebro-geometric structures.
Output
fields Z.
Affine
transport
parameters
Roberto Ferretti

17. Inverse Theory
For instance, this approach reveals more global coherence in Martha Argerich's performance of Schumann's famous "Träumerei" than in Vladimir Horowitz's performance [Mazzola1993]. A statistical approach to inverse performance theory is exposed in [Beran2000].