La séance a été dédiée à certains problèmes concernant les hauteurs dans un espace musical généralisé. En particulier, on s'est concentré sur le réseau d'Euler dans lequel il est possible d'introduire plusieurs notions de distance entre hauteurs et, plus en général, entre accords. On a vu aussi comment généraliser le phénomène de la modulation à des systèmes de micro-intervalles à l'aide de structures topologico-combinatoires issues de la théorie mathématique des complexes simpliciaux.
While listening to music we do not just notice the occurance of sound, but we actively experience the music. In my talk I will argue in favour of a tone system that models this active involvement of the mind on the basis of a dialogic principle. The dialogic principle naturally leads to a relativistic model of tone apperception. It includes the interplay of virtual and actual apperception. An economy principle says that the amount of decision in the dialog is less or equal to the amount of hight (to be explained through experience).
We present the tonal modulation theorem for 12-tempered and just scales, including the extensions calculated by Daniel Muzzulini and Hildegard Radl, concluding with an example of a rhythmical modulation in jazz, i.e., the application of the theory when rotated from pitch to onset. This theory is contained in the book "The Topos of Music", chapter 27, to be released on October 2002 by Birkhäuser, Boston-Basel.
Two dimensions of this active tone system are related to a chromatic plane and two are related to a quintic plane. These are interrelated in a non-trivial way. One possibility to interpret the quintic plane is to identify it with the Eulernet. But in this case, the Eulernet is not meant as a model for just intonation, but rather as a model for virtual apperception. Its dynamic counterpart, i.e. the space of actual (quintic) apperceptive acts turns out to be a curved lattice. I recall some of my calculations presented in the last year in order to provide arguments in favour the presented approach.