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

within the 9-element monoid of dimtone perspectives and we write $* ºA 4 < ºA4$ for the 8-element group of augtone symmetries within the 16-element monoid of augtone perspectives.

We consider three actions of the group $* ºA$ on $ºA$ :

* left :º A ×* ºA --> ºA with left(s,f) := so f. right :º A*× ºA --> ºA with right(s,f) := f o s. conj :º A × ºA --> ºA with conj(s,f) := so f o s-1.

These three actions induce actions $left3,right3,conj3$ of $* ºA3$ on $ºA3$ as well as actions $left4,right4,conj4$ of $* ºA4$ on $ºA4$ (e.g. $left3(s3,f3) := (left(s,f))3)$ .

Proposition 1 Let $º º A3/left3,A3/right3$ and $º A3/conj3$ denote the sets of orbits of dimtone perspectives with respect to the actions $left3,right3$ and $conj3$ , respectively, and let $º º A4/left4,A4/right4$ and $º A4/conj4$ denote the sets of orbits of dimtone perspectives with respect to the actions $left4,right4$ and $conj4$ . In detail one has the following orbits:

$ºA3/left3$ consists of two orbits: ${(00)3,(40)3,(80)3}$ and $ºA*3$ .
$ºA3/right3$ consists of four orbits: $0 4 8 {(0)3},{(0)3},{( 0)3}$ and $* ºA 3$ .
$ºA3/conj3$ consists of four orbits: ${(00)3,(40)3,(80)3},{(04)3},{(44)3,(84)3}$
and ${(08)3,(48)3,(88)3}$ .
$ºA4/left4$ consists of three orbits: ${(00)4,(90)4,(60)4,(30)4},{(06)4,(96)4,(66)4,(36)4}$ and $* ºA4$ .
$ºA /right 4 4$ consists of seven orbits: ${(00)},{(90) },{(60) },{(30) },{(06) ,(66) } 4 4 4 4 4 4$ , ${(96),(36)} 4 4$ and $ºA* 4$ .
$ºA4/conj4$ consists of seven orbits: $0 9 6 3 0 6 {( 0)4,( 0)4,( 0)4,( 0)4},{(6)4,(6)4}$ ,
$9 3 0 6 9 3 {( 6)4,( 6)4},{(9)4},{(9)4},{3)4,(3)4}$ and $0 9 6 3 {( 3)4,( 3)4,( 3)4,( 3)4}$ .

The verification of this propositions is left to the reader.

Let $ºA* = ºA* × ºA* 3×4 3 4$ denote the group of outer tone symmetries within the monoid $º A3× 4$ of all outer tone perspectives. The actions $left3$ , $right3$ , $conj3$ and $left4$ , $right4$ , $conj4$ induce three actions

º * º º left3× 4,right3× 4,conj3×4 :A 3×4× A3×4 --> A3× 4

(e.g. $left3×4((s3,s4),(f3,f4)) := (left3(s3,f3),left4(s4,f4))$ . Obviously, the sets $ºA3× 4/left3×4,ºA3 ×4/right3×4$ and $ºA3 ×4/conj3×4$ of orbits of these three actions consist of cartesian products $O3× O4$ of orbits $O3$ and $O4$ with respect to the corresponding actions on $ºA3$ and $ºA4$ . As a consequence, the above propositions provide a complete picture of the three orbit structures on $ºA3×4$ . Furthermore, the natural isomorphy between (inner) tone perspectives and outer tone perspectives implies, that the group actions $left,right$ and $conj$ yield the same orbit structures on $ºA$ as the actions $left3×4,right3×4$ and $conj3×4$ do on on $ºA3 ×4$ . This is subsumed in the following proposition: