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

In Riemann’s spirit (Riemann, 1873, 1893, 1912), the harmonic »consonance perspective« between the constant dominant triad $Dominant : 0~>F iP iM od12(1,5,2)$ and the constant tonic triad $T onic : 0~>F iPiM od12(0,4,1)$ is defined by the monoid

Trans(Dominant, Tonic) < Z12@Z12,

a self-addressed chord generated by the transporter set of all morphisms
$u : Dominant --> T onic$ .

This self-addressed chord is related to the above stabilizer as follows: Consider the tensor multiplication embedding

ox e : Z12@Z12 >-> Z12[e]@Z12[e] : eu.v '--> e(u+e.0).(v ox Z12[e]).

Then we have a »Grand Unification« theorem (Noll (1995), see also Noll (1999) or Mazzola (2001b) for more details):

Theorem 8 With the above notations, we have

T rans(Dominant, T onic) = ox e -1Trans(Ke,Ke).

This means that the Fux and Riemann theories are intimately related by this denotator-theoretic connections. At present, it is not known to what extent this structural relation has been involved in the historical development from contrapuntal polyphony to harmonic homophony.

5.2 Conclusion and Preview

If we review the overall power of mathematics in the description, analysis and performance of music, it turns out that it has a unique unifying character: Seemingly disparate subjects become related and comparable only through the universal language and methods of modern mathematics. Moreover, the operationalization of the abstract theories on the technical level of computers and software is an immediate and very important empirical and theoretical consequence of mathematization. For the first time, models and experimental setups can be applied in a scientific, i.e., precise and objective framework. Finally, the embedding of the historically grown existing theories in the mathematical concept framework preconizes a natural extension of facticity to fictitious variants, thereby opening the way to the comprehension of the crucial question of musicology: Why do we have this music and no other?

Of course, there will be other musics. But mathematical methods and associated technologial tools will undoubtedly play a dominant role it their discovery and exploration, be it on the level of instrumental realization, be it on the very concept space which transcends pure intuition and catalyzes fantasy to an unprecedented degree.