Geometries & Gestures

WEDNESDAY 24 June 2015, 6pm

Arts 2 Lecture Theatre



Seven Selections from Rational Melodies (1982)
Twenty-one pieces for any instrument
Tom Johnson
Carlos Vaquero, flute
Chord Geometries in Italian Pop
Moreno Andreatta, piano
Giants’ Steps
Structures to gestures
Guerino Mazzola, piano


Program Notes


Rational Melodies (1982) may be played by any instrument, in any octave or transposition. It is not necessary to play the whole set, and performers are welcome to group selected melodies into suites however they like. The pieces are intended primarily for soloists, though they may also be performed by groups of instruments, playing in unison or alternating in antiphonal patterns.

Rationality, or more precisely, deductive logic, has seldom been the controlling factor in musical composition. Composers are usually more interested in inspiration, intuition, feelings, self-expression. Lately, however, there has been a tendency for composers to give up individual control over every note, and rely on factors outside themselves. Pieces have been controlled by the wind, by chance, by the idiosyncrasies of tape recorders, or by unpredictable variations in electronic circuitry, for example, and it seems to me that composing by rigorous adherence to logical premises involves a similar way of thinking. ~ T.J.


Chord Geometries in Italian Pop. By taking some typical Paolo Conte’s harmonic progressions as a starting point, I will show the relevance of a geometric approach in the organization of the formal structure of some recent songs. A special emphasis will be given to Hamiltonian Cycles within the Tonnetz, as they have been used in a series of poetry-based “Hamiltonian Songs” (from Aprile, with text by Gabriele D’Annunzio to La sera non è più la tua canzone, with text by Mario Luzi). Videos and musical examples are available at the author’s music web page: ~ M.A.


Giants’ Steps is a composition that has been created following a mathematical analysis of Coltrane’s famous composition Giant Steps. The analysis shows that Coltrane essentially arranged his harmonies (all seventh chords) around the inversion symmetry of C-major, i.e. inversion at D. Two large groups of nine and eleven chords, respectively, build his big architecture. The second group even realizes a totally symmetric trajectory of chords following the third Messiaen scale, which is the complement of the augmented C triad, {C, E, G-sharp}. From our analysis it follows that Coltrane’s construction can be given a polarity of a fast part, being opposed to a blues-y part that makes evident the Love Supreme motif {F, A-flat, F, B-flat}. Our composition uses all of these analytical facts and morphs them, following Boulez’s creative analysis, to an improvisation that flies around these structures and eventually dissolves them into gestural dynamics of artistic presence. ~ G.M.




Tom Johnson, born in Colorado, received BA and MMus degrees from Yale University, and studied composition privately with Morton Feldman. After 15 years in New York, he moved to Paris, where he has lived since 1983. His compositions are considered to fall into the minimalist genre, and they often use formulas, permutations, predictable sequences, and various mathematical models. Johnson is well known for his operas: The Four Note Opera (1972) continues to be presented in many countries, and Riemannoper has been staged more than 30 times in German-speaking countries since its premier in Bremen in 1988. Often played non-operatic works include Bedtime Stories, Rational Melodies, Music and Questions, Counting Duets, Tango, Narayana’s Cows, and Failing: a very difficult piece for solo string bass. As performer he frequently plays his Galileo, a 40-minute piece written for a self-invented percussion instrument. Johnson received the French national prize in the victoires de la musique in 2001 for Kientzy Loops. The latest orchestra score is 360 Chords, premiered in July 2008 by Musica Viva in Munich. ~ More at


Carlos Vaquero is a flautist and a doctoral candidate in the Institute for Logic, Language and Computation at the University of Amsterdam. His research concerns the analysis and modelling of music performance. After receiving the diploma “Título Superior de Música” in Flute (Salamanca, 2002) he earned degrees in Music Technology (BA, Koninklijk Conservatorium, 2006), Music Performance (BM, Conservatorium van Amsterdam, 2006) and Sound and Music Computing (MSc, Universitat Pompeu Fabra, 2012). He has worked as a software test engineer, as a producer at the Dutch Public Broadcasting Corporation and as a freelance flautist, audio engineer and designer. His artistic interests include the performance of baroque and contemporary music. He has performed or presented work in different venues and festivals throughout Australia, Italy, The Netherlands, U.K. and Spain.


Moreno Andreatta is a pianist and a Centre national de la recherche scientifique (CNRS) researcher in the Music Representation Team at the Institut de Recherche et Coordination Acoustique/Musique (IRCAM), where he coordinates the Acoustique, traitement du signal, informatique, appliqués à la musique Masters Program. Andreatta received his PhD in computational musicology in the Musique, Histoire, Société Doctoral Program organized by the Ecole des Hautes Etudes en Sciences Sociales (EHESS), IRCAM, the Ecole Normale Supérieure (ENS) and the Conservatoire National Supérieur de Musique et de Danse of Paris (CNSMDP). He is currently Vice President of the Society of Mathematics and Computation in Music. His research activities focus on the relationships between mathematics and music. The algebraic, categorical and topological methods of his research have recently found applications in the study of popular music, in particular around song writing and improvisation. ~ More at


Guerino Mazzola earned his Ph.D. in Mathematics from Zurich University, where he also qualified as a professor in algebraic geometry with Peter Gabriel and in computational science with Peter Stucki. Mazzola has profiled the European school of mathematical music theory since 1980 and has written six books on the subject, among them The Topos of Music, published by Birkhäuser, and proposed by the American Mathematical Society as the mathematics book of the year 2005. Mazzola’s approach to music includes sophisticated mathematics of topos theory, but also classical tools from group theory to homotopy theory. His other books include: La vérité du beau dans la musique (2007) on the philosophy of music, Flow, Gesture, and Spaces in Free Jazz – Towards a Theory of Collaboration (2009) applies mathematical gesture theory to free jazz, Musical Performance (2010) is the first comprehensive treatment of performance theory, Musical Creativity (2011) co-authored with Joomi Park and Florian Thalmann describes creativity in a tutorial for students, Computational Counterpoint Worlds (2014), co-authored with Octavio Agustin and Julien Junod, and Computational Musicology in Hindustani Music (2014), co-authored with Soubhik Chakraborty et al. ~ More at


Mathematics and Computation in Music