# Représentations et formalisation des structures musicales

** Representation and formalization of musical structures **

### Samedi 13 décembre 2003

Ircam, Salle I. Stravinsky

Entrée libre dans la mesure des places disponibles

## Programme de la journée

- 10h - 10h15 -
**Moreno Andreatta**: Présentation de la journée - 10h15 - 11h00 -
**John Rahn**(Escola Superior de Musica de Catalunya): Chloe’s Friends: A Symposium on Good Application of Mathematics to Music - 11h15 - 12h00 -
**Oren Kolman**(Kings College, London) : Reverse music theory. How much abstract mathematics does tranformational theory require? - 12h15 - 13h00 -
**Guerino Mazzola**(University of Zürich) et**Moreno Andreatta**(Ircam) : Applying Denotator Theory to the Case of Klumpenhouwer and Related Transformational Networks

- 14h30 - 15h15 -
**Jon Wild**(Harvard University): Tiling Gapped and Warped Harmonic Space - 15h30 - 16h15 -
**Dave Meredith**(City University, London) : Pitch Spelling Algorithms - 16h15 - 17h00 -
**Olivier Lartillot**(Ircam) : Computational musical pattern discovery based on a modelling of listening strategies

## Abstracts

#### John Rahn

**Chloe’s Friends: A Symposium on Good Application of Mathematics to Music**

Chloe's Friends is a discussion among a group of intellectuals at a drinking party after a lecture in ancient Athens. The discussion rambles from subject to subject, touching shallowly and briefly on number mysticism and numerology, spaces, Pythagoreanism, and neo-Epicurianism (ill-informed proto-quantum theory and string theory), before the arrival of the lecturer, an older but eminent theorist named Megakephalos. He clears up some of the lines of questioning derived loosely from his lecture that night and but fails to go into it in any depth as it was derived from the work of a mathematician from Zurich (aka Guerino Mazzola) and he (Meg) is feeling tired and a bit drunk. But Meg does set the discussion going on the path of examining the nature of a good application of math to music.

The discussion settles down more seriously with a particular example of this, in some depth, as a young composer tries to figure out the good and bad aspects of some mathematical work involving sequences of wreath products as models for generative growth and perception (from Michael Leyton's recent book). He takes them all through a number of issues and they examine some of the consequences. They cover most of the issues before they all get too tired and drunk and the party breaks up.

#### Oren Kolman

**Reverse music theory. How much abstract mathematics does tranformational theory require?**

Some uses of abstract mathematics in transformational music theory are examined. We discuss the role of the Axiom of Choice (AC) in the study of the structural properties of classes of generalized interval systems (GISs) and transformational networks (TNs). Appealing to examples in David Lewin's monograph *Generalized Musical Intervals and Transformations*, we illustrate how abstract concepts can explicate common principles of GIS and TN construction. The incursion of non-finitistic methods in contemporary transformational theory prompts a brief introduction to the project of reverse music theory. We describe how the choice of logic (classical, intuitionistic, modal, temporal, etc.) and the computational complexity of first-order theories of classes of GIS are related. Some technical conjectures and philosophical questions relating to finitism and moderate realism in transformational theory are formulated in the final section.]

Could you tell me what level of technicality is appropriate? Should I pitch the talk at a broad but sophisticated audience interested in mathematical music theory, but perhaps without specific expertise in logic, group theory, GIS and transformational theory (so I shall give some definitions and examples even)? I expect everyone knows more than I do, which is very worrying indeed.

#### Guerino Mazzola and Moreno Andreatta

**Applying Denotator Theory to the Case of Klumpenhouwer and Related Transformational Networks**

We present some basic aspects of Klumpenhouwer transformation theory by describing an abstraction process leading to the formalization of this analytical approach in the functorial conceptual model of denotators. After summarizing the main idea about recursive construction in music theory, we show that a Klumpenhouwer network is formally equivalent to a very special limit of a diagram and give natural generalizations thereof. We end with the discussion of some problematic aspects of a group-theoretical model of music theoretical constructions.

#### Jon Wild

**Tiling Gapped and Warped Harmonic Space**

We recap some results of tiling problems in a non-cyclic homogeneous pitch-space. We then attempt to generalise to subsets of the usual chromatic, considering the subsets variously as _warped_ or _gapped_. Tiling sets may be obtained from different kinds of equivalence classes; transformations are illustrated that map one space to itself or to another space, and that map one equivalence class of tiles to another, creating isomorphism classes of tilings. Special attention is given to tilings of the diatonic implied in various music-theoretical historical sources.

#### Dave Meredith

**Pitch Spelling Algorithms**

In this talk I focus on the problem of constructing a reliable pitch spelling algorithm - that is, an algorithm that computes the correct pitch names (e.g., C#4, Bb5 etc.) of the notes in a passage of tonal music, when given only the onset-time, MIDI note number and possibly the duration of each note. Such an algorithm is an essential component of any system for computing a notated score from a MIDI or audio file. The performance of music information retrieval systems can also be improved by encoding the pitch names of the notes in the databases that they search.

In this talk I compare my own ps13 algorithm with the pitch spelling algorithms of Cambouropoulos, Temperley and Longuet-Higgins. I show that, in terms of the number of notes spelt correctly, ps13 performs significantly better than the other algorithms (99.33% of 1.73 million notes spelt correctly). However, when the algorithms are evaluated in terms of the number of intervals spelt correctly, Temperley's algorithm performs best (99.45% of 1.73 million intervals). If intervals and notes are considered together, ps13 again comes out on top (99.25% of 3.46 million notes and intervals). The performance of each algorithm also seems to depend in an interesting way on the style of the music being processed.

#### Olivier Lartillot

**Computational musical pattern discovery based on a modelling of listening strategies**

We propose a system for automatic motivic analysis based on an exhaustive search for repeated patterns in both melodic and rhythmical dimensions. The analysis is undertaken directly in a multidimensional representation of the score. This score is considered as a two-side memory : a syntagmatic memory links successive notes through syntagmatic relationships, whereas an associative memory relates notes and syntagmatic relationships that share identities along each different parameter and their intersections. The syntagmatic concatenations of all kinds of associative relationships forms pattern classes in a tree-like structure. The combinatorial efficiency of the process is founded on a principle of unique description of pattern classes by maximisation of specificity. Other issues such as the discovery and representation of suffixes, general patterns, periodic patterns, and also patterns of patterns, are introduced.

## A short bibliography

- Gretchen C. Foley: "Arrays and K-nets. Transformational relationships within Perle’s Twelve-Tone Tonality", Music Theory Midwest, Mai 2003
- Henry Klumpenhouwer: "The Inner and Outer Automorphisms of Pitch-Class Inversion and Transposition: Some Implication for Analysis with Klumpenhouwer Network", Integral, Vol.12, 1998, pp. 81-93.
- Oren Kolman: "Transfert Principles for Generalized Interval Systems" (to appear in the next issue of Perspectives of New Music).
- Olivier Lartillot: Olivier Lartillot, Fondements d'un système d'analyse musicale computationnelle suivant une modélisation cognitiviste de l'écoute (temporary title), doctoral thesis, Paris-6 University, in preparation. http://www.ircam.fr/equipes/repmus/lartillot
- David Lewin: Generalized Musical Intervals and Transformations, Yale University Press, 1987
- David Lewin: "Klumpenhouwer Networks and Some Isographies that involve them", Music Theory Spectrum, 1990, 12/1, pp.83-120.
- Michael Leyton: A Generative Theory of Shape, Springer, 2001 (http://www.rci.rutgers.edu/~mleyton/Generative.htm)
- Guerino Mazzola: Topos of Music. Geometric concepts of Logic, Theory, and Performance, Birkhäuser, 2002 (http://www.ifi.unizh.ch/groups/mml/publications/special4.php4)
- Dave Meredith: "Pitch spelling algorithms. Proceedings of the Fifth Triennial ESCOM Conference, Hanover University of Music and Drama, Germany, pp.204-207 (http://www.titanmusic.com/index.html)
- John Rahn: "Chloe’s Friends: A Symposium on Good Application of Mathematics to music" (to appear in the next issue of Perspectives of New Music).
- John Wild: "Tessellating the chromatic" (to appear in the next issue of Perspectives of New Music).