Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

Theorem 1 The solution space of (3) is a linear fibration over some appropriate affine space.

Corollary 1 The dimension of the non-empty fibers of this fibration is generically minimal, and it increases along some finite union of proper algebraic subvarieties (the loci where the minors of the coefficients’ matrices vanish). This means that the non-empty fibers $Fib(e.)$ , i.e., the solution spaces defined by (2) over a given output set $e.= (ez)z$ are all generically isomorphic, i.e., isomorphic when restricted to appropriate open subschemes. However, the configuration of the specialization subschemes is not evident and depends on the particular vector summands. Also, the condition for non-empty fibers is not evident in general.

How can we interpret this result in musicological terms? A quantitative measurement of a performance can be done by recording the values of the parameters that characterize a note, i.e. loudness, duration, and so on. In the last paragraph we have denoted this data by the vectors $e z$ . Equation $(3)$ explains that in order to produce a given values of the parameters from the weight functions, one has to find suitable values for all $cmzk y,x$ . It would be nice if the choice of this coefficients could be small, because this would mean that in order to produce a given performance one does not have a lot of freedom in the choice of system parameters. However from this model this is not the case. In fact, corollary 1 tells us that for a fixed performance one has either no one or else infinitely many possibilities to choose system parameters which produce the given performance. Moreover, we learn that all performances with non-empty fibers $F ib(e) .$ have open dense subsets $U (e ) < F ib(e) . .$ which are all isomorphic with each other, so these generic subsets $U (e) .$ are qualitatively equivalent. In other words, the non-empty fibers only differ on their special loci apart from generic open subschemes $U (e ) .$ .

In musical terms, a fiber $F ib(e) .$ could be called a critical fiber because its points are the possible background parameters--in the present stemmatic model--which lead to the given performance output data $e .$ , such as local tempi, articulations, dynamics, detunings, etc. So the fiber really includes the possible ways of understanding why a performer is playing his actual performance. In fact, finding out which parameters the interpreter could have used is (or should be) the core activity of a music critic. Having generically isomorphic fibers means that in any two critical fibers of two given performances, there are “dominant” open sets of “criticisms” (i.e. points in the fiber!) which are isomorphic with each other. This does not mean that the criticisms are the same for the two given performances, but that their structural contexts my be identified. Again, this does not mean that the relevant criticisms may be identified--on the contrary: maybe, this isomorphic contexts just describe the criticisms which are what everybody could say if no supplementary information about the specific performance culture of the interpreter is known.

So we should not discard this model as insignificant, instead we need to look at it in the right way. In fact, it is like pretending to explain the geometry of smooth plane curves, just using lines. The theory of lines can be useful for local questions, but not for global ones. The same thing happens here. The great flexibility of this model enables us to adapt it to the questions that one tries to answer, but they