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

For $f = ba$ and $(f3,g4) = ((ba)3,(dc)4)$ we compute,

(f ) = ((ba) ,(ba) ) = 4b+9b4a+ 9a = ba = f, ((f3×,4g*4)×*9 ) =3(4b+49d4*4a×*+99c) = ((ba),(dc)) = (f ,g ). 3 4 *4×*9 3× 4 3×4 3 4 3 4

Finally, the product map $? = ? ×? :º A --> ºA o+ ºA 3× 4 3 4 3 4$ is a monoid morphism, because its factors $? :º A --> ºA 3 3$ and $? :º A --> ºA 4 4$ are monoid morphisms. $[]$

Remark 2 Besides the monoid structure of $ºA,ºA ,ºA 3 4$ and $ºA 3×4$ these sets carry the structure of a $Z 12$ -module (e.g., addition in $ºA$ is given by $ba+ dc = b+d(a+ c)$ and scalar multiplication by $k(ba) = kb(ka)$ . The above calculations make clear that the monoid morphisms $? ,? 3 4$ and $? 3×4$ are, at the same time, linear module morphisms.
However, the additive module structure and the multiplivative monoid structure do not combine into a ring structure, because the distributivity law is not fulfilled: Left distributivity is fulfilled:

b1 b2 d (a1+a2)d+b1+b2 ( aa11d++b1 a2) o ca2=d+b2 b1 (a1d+a2)bc2 d = (a1c)+ (a2c) = a1 o c+ a2 o c.

But right distributivity is not fulfilled:

d b b c(b +b)+d dc o (b1a1 +d2a2)b = 1cb+2d c(a1 +cab2+)d c(b+b )+2d c o 1a1 + co 2a2 = 1 (ca1)+ 2 (ca2) = 1 2 c(a1 + a2).

Some contructions for rings like ideals are also meaningful in the present situation. A useful generalization of right and left ideals in rings to a monoid like $º A$ are sieves and cosieves in the category $º A$ (a monoid is a category consisting of one object, while its elements are interpreted as arrows). It appears in our situation of the ”pseudo-ring” $º A$ , that the cosieves share the properties of ”left-ideals”, while only a few out of many sieves show the additive properties of ”right-ideals”.

Definition 3 A set $R (_ ºA$ of tone perspectives is said to be an $ºA$ -sieve, if $R o ºA = R$ , i.e. $r (- R$ implies $ro f (- R$ for all $f (- º A$ . A set $L (_ ºA$ of tone perspectives is said to be an $ºA$ -cosieve, if $ºA o L = L$ , i.e. $l (- L$ implies $f o l (- L$ for all $f (- ºA$ .

While there are thousands of $ºA$ -sieves, there are only 6 $ºA$ -cosieves, namely the submodules $T(kZ12) (_ ºA$ (for $k (- {0,1,2,3,4,6})$ (see Remark 2). All $ºA$ -sieves are semigroups, but not vice versa (e.g., the only sieve $R$ among the submonoids $M (_ ºA$ is $ºA$ itself, because $01 (- R$ always implies $01 o f = f (- R$ for all $f (- ºA$ ).

2.2 Tone Symmetries

Among the 144 tone perspectives there are 48 invertible ones, namely those having multiplicative units in $Z*12 = {1,5,7,11}$ as multiplication factors. They form a group $ºA* < ºA$ and are called tone symmetries.
Similarly, we write $* ºA 3 < ºA3$ to denote the 6-element group of dimtone symmetries