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

furthermore we have $(4t)3 = t3$ . Similarily, the value $0 9(t+ 4k) = 9t$ does not depend on $k$ , and $(9t)4 = t4$ .

$((t,s)*4×*9)3×4 = (4t+ 9s)3×4 = (t+ 3(t+ 3s),s+ 4(t+ 2s)$ . The last pair represents the same element of $T3×4$ as $(t,s)$ does. Conversely, we have $(t3×4)*4×*9 = (t3,t4)*4×*9 = (4t+ 9t) = t$ . $[]$

According to this lemma, one has a natural identification of the augtone-module $T4$ with the dimtone $3T = {0,3,6,9}$ as well as of the dimtone-module $T3$ with the augtone $4T = {0,4,8}$ , such that they can be viewed as retracts of $T$ . The situation is different in the case of the two-element module $T2 = T/2T$ when compared with the two-element tritone-chord $6T = {0,6}$ . They are isomorphic as $Z12$ -modules, but $6T$ is not a retract of $T$ . Analogously, $2T$ is not a retract of $T$ .

Definition 2 Let $ºA 3$ and $ºA 4$ denote the monoids of affine endomorphisms of the $Z 12$ -modules $T 3$ and $T 4$ respectively. The elements of $ºA 3$ are called dimtone perspectives and the elements of $ºA 4$ are called augtone perspectives.
Let $ºA := ºA o+ ºA 3×4 3 4$ denote the direct product of the monoids $ºA 3$ and $ºA 4$ . Its elements are called outer tone perspectives.

Lemma 2 Each tone perspective $f (- º A$ induces a dimtone perspective $f3 (- ºA3$ as well as an augtone perspective $f4 (- ºA4$ by virtue of $f3(t3) := (f(t))3$ and $f4(t4) := (f(t))4$ . The mappings $?3 :º A --> ºA3$ and $?4 : ºA-- > ºA4$ are surjective monoid morphisms. Especially, the restriction of $?3$ to $ºA(4T)$ (and of $?4$ to $ºA(3T)$ ) yields a monoid isomorphism $~ ºA(4T) = ºA3$ (and $~ ºA(3T)= ºA4$ ).

Proof: Take $f = ba$ . Then

f3((t+ 3k)3) = (f(t +3k))3 = (a(t+ 3k)+ b)3 = (at+ b)3 = (f (t))3 = f3(t3).

Hence the definition of $f3$ does not depend on the representatives of an argument. Further, $(f o g)3(t3) = (f(g(t))3 = f3((g(t))3) = (f3 o g3)(t3)$ and obviously $(01)3$ is the identity in $ºA3$ . Finally, two tone perspectives $ba$ and $dc$ induce the same dimtone perspective, if and only if both differences $b- d$ and $a- c$ are multiples of $3$ . An inspection of $ºA(4T) = {ba|a,b (- 0,4,8}$ shows that these nine tone perspectives represent nine different dimtone perspectives, just because $0,4,8$ represent different residue classes modulo $3$ . Hence we are done with $f3$ . The same line of arguments works for $f4$ . $[]$

The following lemma shows that tone perspectives are in a natural 1-1-correspondence with the outer tone perspectives.

Lemma 3 The mapping $?3×4 :º A --> ºA3× 4$ with $f3×4 := (f3,f4)$ is an isomorphism of monoids.

Proof: We construct the inverse mapping $? :º A -- > ºA *4×*9 3× 4$ as follows: Let $f = b+3k(a+ 3l)$ and $g = d+4m(c + 4n)$ be arbitrary representatives of an argument $(f ,g) 3 4$ . Its image $(f,g ) (- º A 3 4 *4×*9$ has to be independent of the variables $k,l,m, n$ . Indeed, we set

(f3,g4)*4×*9 := 4(b+3k)+9(d+4m)(4(a+ 3l) +9(c+ 4n)) = 4b+9d(4a+ 9c).

Hence the mapping is well-defined. Now we check that it is inverse to $?3× 4$ .