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

In a slightly more advanced point of view view we now extend the repertoire of tones that may be played when a key is pressed down. To each key $k$ we associate a non-empty set $Tk$ of tones consisting of an obligatory tone $t0(k) (- Tk$ and some facultative tones. Again we assume that whenever the key $k$ is pressed down, the obligatory tone $t0(k)$ is automatically played. The facultative tones in $Tk$ , however, may be played or may not. All the sets $(Tk)k (- K$ are supposed to be pairwise disjoint. In other words, from any tone $t (- Tk$ being played we may infer the corresponding key $k$ being pressed down as well as the associated obligatory tone $t0(k)$ being played too, but not vice versa: We do not know just from $k$ being pressed down, which of the facultative tones in $Tk$ are played. We mention that the modality of choice is not part of our consideration. The reader may think of an organ player freely choosing registers in addition to a fixed (obligatory) one, or of a random algorithm or of a complicated context-dependent procedure. Formally we simply extend the direct coupling of keys $k (- K$ and obligatory tones $t (- t0(K)$ to a projection $p$ sending each tone to its associated obligatory tone:

|_| p : Tk-- > {t0(k)|k (- K} with p(t) = t0(k) for all t (- Tk. k (- K

What is the logics behind this projection? What we know about the possibly played tone sets $|_| T (_ k (- K Tk$ is that they necessarily contain their projections $p(T )$ . This knowledge offers the possibility to consider refined characteristic functions

| _| BelongsToT : Tk-- > {True,FalseBut,False}. k (- K

In addition to the two truth values $T rue$ and $F alse$ we have a third one $F alseBut$ , saying >False, but becoming True under projection<. This means in detail:

{ True for t (- T BelongsToT(t) = FalseBut for t (- /T , but p(t) (- T False for p(t) / (- T for

The meaning of $BelongsToT (t) = F alseBut$ is that the tone $t$ does not belong to the considered chord $T$ , but it is nevertheless a facultative variant of an obligatory tone $p(t)$ belonging to $T$ . In contrast, $BelongsT oT(t) = False$ means that $t$ is not in $T$ and also not a facultative variant of any tone of $T$ . The same definition works for relative set inclusions $T (_ S (_ |_| Tk k (- K$ where $p(T ) (_ T$ and $p(S) (_ S$ .

In order to relate this example to the deeper mathematical background of topos theory--as indicated in the beginning of this subsection--we translate our description into more sophisticated ones in two steps.

Step 1:: The property $p(T) (_ T$ allows us to define a monoid action $m : Z*2× T --> T$ of the multiplicative monoid $Z2 = {0,1}$ of residue classes of integers $mod 2$ (i.e. where $0 .0 = 0$ , $1.0 = 0.1 = 0$ and $1 .1 = 1.$ ) on the set $T$ . The two set maps $m0,m1 : T --> T$ are $p$ (restricted to $T$ ) and the identity map $IdT$ of $T$ . It is easy to see that this yields a monoid action. Especially we have $m0(t) = [m0 o m0](t)$ (i.e. $p(p(t)) = p(t)$ ) because each obligatory tone $t0(k)$ is always projected onto itself: $p(t0(k)) = t0(k).$