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

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7.1 Modules in Musicology

This algebraic structure was chosen for theoretical and practical reasons, from a mathematical standpoint as well as from a musicological one. We will present some examples that show certain modules that play an outstanding role in TMM.

Example: A very important module in MMT is the monoid ring $Z <ASCII >$ . This construction lets us use the universal ASCII code, that is, the set of ASCII characters that we need in order to represent any situation in which words appear (but which must be adjusted to our theory, based on the presheaf $Mod@$ ). We construct $/\<ASCII >$ with an alphabet $A$ and a commutative coefficient ring $/\$ . Then $<A>$ is the free monoid of all words $(a1,a1a2,...,a1...an,...)$ generated by $A$ . The monoid ring, $/\ <A>$ are all the sums $sum p (- <A>cpp$ with only a finite number of coefficients $cp /= 0$ in $/\$ .

The ring $Z <ASCII >$ is constructed over the integer ring $Z$ . For the names of denotators and forms of Simple type, we will usually only use $cp = 1$ for one determined word $p$ , which gives us $<A> < Z <ASCII >$ ; however there are situations in which $Z<ASCII >$ can be used as a $Z$ - module. This could be, for example, when there are different multiplicities of »sub-forms« used in a given form (for example, Pianoscore form, Figure 1).

Example: The form $EulerM odule$ has its coordinator in the module $3 Q$ over $Q$ :

$EulerM odule-- >$ Simple $3 (Q )$

$Identity$

Let $o = (1,0,0),q = (0,1,0),t = (0,0,1)$ be the canonical base of $Q3$ , where $o$ is the axis of octaves (as a musical interval), $q$ of the fifths and $t$ of thirds. Each point in the space $EulerM odule$ , $(a,b,c)$ , can be written as $p = a .o+ b.q +c .t$ . In the same way, the major third and fifth can be associated with the points $q -o$ and $t- 2o$ (see Mazzola, 2002, Appendix A.2), that represent the minor fourth and sixth respectively. Also

p = (a + b+ 2c).o+ b.(q- o)+ c.(t- 2o) = (a,b,c).

This tells us that any note (which, in turn, can be conceived as an interval, or difference of two notes, because the note and the interval are associated with a vector) can be generated by the octave, the fifth and the major third by means of interchange, juxtaposition and division.

Example. Another example of important modules for MMT are direct sums, that is, forms which are themselves constructed by products of forms of Simple type (diagrams without arrows in category theory language).

Let $F -->$ Simple $(M ),$ $G -->$ Simple $(N),$ $M, N (- Mod$ .

Then, on the one hand, we have

$F × G --> Limit(M, N )$

$@(M o+ N)~=@M × @N$