- Social culture is constituted by a community in which individuals and
institutions occupy different positions in a structure of interdependences, that
regulate their actual behavior through manifold kinds of stimulation and
restriction. For our considerations of a scientific community we are especially
interested in the regulation of research and communication on the background
of specialization and division of labour.
- Material culture is constituted by all kinds of artifacts produced and
consumed by its members, like musical instruments, computers, etc. –
including all kinds of sign vehicles, like scores and datafiles. We are especially
interested in the conditions under which computer programs and electronic
musical corpora may contribute to a new experimental paradigm.
- Mental culture is constituted by knowledge domains, natural languages,
musical and other sign systems, theories, etc. We are especially interested in
the way theories and other knowledge domains may coexists and/or influence
each other.
Music Theory has to be characterized as an open substructure of a larger
surrounding culture having many operlaps with and ramifications into musical,
scientific and technological domains. We may presuppose the penetration
of “alien” disciplines into music theory as something natural with regard
to the cultural mechanism. But we focuss our attention to phenomena of
globality inside the music-theoretical subculture though its unquestionable
openness.2
- With respect to ongoing discussions about globalization of knowledge through internet
technology one should not appraise globality naively. Global accessibility to information will
perhaps support a more general process of knowledge globalization towards a new type of
encyclopedism. But such a process will heavily depend on further fundamental research.
Mazzola (cf. [7], [8]) argues in favour of a programmatic role of music within such a movement.
We nevertheless prefer to continue our attention to the needs of Music Theory.
|
It is very popular to illustrate globality with a network-metaphor: Everything can be
linked to everything else. But there is a difference between a mere reference from one
object to another and two objects being “glued” along a shared substructure. The latter
happens when geographers reconstruct the globe from an atlas of overlapping maps. The
idea of gluing local maps is behind the mathematical concept of global structures. The
overlap of the maps allows a controlled transition from one coordinate system to the
other.
Mathematical models of global structure can be applied to the music-theoretical
domain in two ways:
- in order to conceive musical structures as global ones,
- in order to understand knowledge about musical structures as global.
Of course, without being forced by the music-theoretical content, one would not leave a
suitable coordinate system. Therefore we start with a simple musical example of a chord
sequence resisting against interpretation within a specific local coordinate system: the Euler
Tone-Net.3
- Positions in the Euler Tone-Net correspond to octave classes in just tuning, cf. [6], [11].
|
This coordinate system is explicitly or