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

perspectives. Each one is given by a multiplication factor $a (- Z12$ , together with a translation summand $b (- T$ and we write $ba$ in order to denote the corresponding tone perspective $b a : T-- > T$ with $b a(t) = at+ b$ . The 144 tone perspectives form a monoid $ºA$ with respect to the operation $o$ of concatenation. One has $d b cb+d co a = (ca)$ .

There are 6 submodules $K (_ T$ , namely the two trivial ones: $T$ and $0T = {0}$ and 4 proper submodules: $2T$ , $3T$ , $4T$ and $6T$ . For each submodule $K (_ T$ consider the submonoid $ºA(K) = {f (- ºA |f(K) (_ K}$ of selfperspectives of $K$ , consiting of those tone perspectives mapping $K$ into itself. Furthermore, we will be concerned with the factor module $T/K$ , whoose elements are affine subspaces $t+ K$ associated with $K$ .

The following definition and lemma focus on the proper submodules $3T$ and $4T$ and the corresponding factor modules $T/3T$ and $T/4T$ .

Definition 1 (Outer decomposition of the 12-tone module)

The factor module $T := T/3T 3$ is called the outer 3-cycle of the 12-tone module $T$ and its elements are called inner 4-cycles or dimtones.
The factor module $T4 := T/4T$ is called the outer 4-cycle of the 12-tone module $T$ and its elements are called inner 3-cycles or augtones.
The direct sum $T3×4 := T3 o+ T4$ is called the outer decomposition of the 12-tone-module.

Remark 1 The technical terms ”dimtone” and ”augtone” shall, on the one hand, refer to the traditional terms ”diminished seventh chord” and ”augmented triad”. On the other hand we do not intend to refer to the operations of diminution and augmentation in this definition.

Lemma 1 Let $?3$ and $?4$ denote the projection maps from $T$ onto $T3$ and $T4$ respectively and consider their product map $?3×4$ onto $T3×4$ , i.e.

?3 : T --> T3 with t3 := t+ 3T, ?4 : T --> T4 with t4 := t+ 4T, ?3×4 : T --> T3× 4 with t3×4 := (t3,t4).

Then the following holds:

The tone perspectives $04$ and $09$ induce module injections $?*4 : T3-- > T$ and $?*9 : T4 --> T$ such that $(?*4)3 = idT3$ and $(?*9)4 = idT4$ .
$?3× 4$ is an isomorphism of $Z12$ -modules. Its inverse map is given through $(?,?)*4×*9 : T3×4 --> T$ with $(t,s)*4×*9 := 4t+ 9s$ .

Proof:

The value $0 4(t+ 3k) = 4t$ does not depend on $k$ , and,