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

and listings of isomorphism classes: In 1973 Allen Forte (Allen, 1973) established the list of 352 orbits of chords of pitch classes under the translation group $T = eZ12 12$ and the 224 orbits of chords under the group $TI = eZ12.± 1 12$ of translations and inversions. In 1978, George Halsey and Edwin Hewitt (Halsey and Hewitt, 1978) succeeded to give a recursive formula for enumeration of translation orbits of chords in finite abelian groups, and enumerating the translation orbit number for chords in cyclic groups of cardinality $n < 24$ . In 1980, the author (Mazzola, 1981) calculated the list of 158 affine orbits of chords in $Z12$ , the list of 26 affine orbits of three-element motives in $2 Z12$ , and the list of 45 three-element motives in $Z5× Z12$ . In 1989, Hans Straub and Egmont Köhler (Straub, 1989; Köhler, 1988) gave the list of all affine 216 four-element motive orbits in $2 Z12$ . In his works starting from 1991 to the present, Harald Fripertinger (loc.cit.) has given enumeration formulas for chord orbit numbers in $Zn$ under $Tn,TIn$ , and the full affine group, also for $n$ -phonic $k$ -series, all-interval series, motives in $Zm × Zn$ , and Vuza canons in $Zn$ . He has calculated lists of affine motive orbits in $2 Z12$ for motives up to cardinality 6.

The usage of classification has been annotated above, but there is one particular result which we cannot withhold from the reader: Fripertinger’s formulas yield the impressive number of

2230741522540743033415296821609381912

affine orbits of 72-element motives in $Z212$ . This is of order $1036$ against the estimated order of $1011$ stars in a galaxy! So the musical universe is a serious competitor against the physical universe, in its quantity as well as in its quality of a spiritual antagonist.

Let us briefly review the Pólya-de-Bruijn enumeration methods applied by Fripertinger. We typically work in the space $F = Zn$ . A subset $C < F$ is identified with its characteristic function $xC : F --> 2 = {0,1}$ . For a permutation $g$ in a subgroup $G$ of the full group $-G-->L(F )$ of affine automorphisms of $F$ , we have the cycle index $cyc(g) = (c ,...c),f = card(F ) 1 f$ , with $c = i$ number of cycles of cardinality $i$ . Take the indeterminates $X ,...X 1 f$ and set $Xg = Xc1...Xcf 1 f$ . Then the cycle index polynomial is defined by

Z(G) = card(G)- 1 sum Xg. G

Consider now »Pólya weights« $w(0),w(1) (- Q[x]$ and for a characteristic function $x : F --> 2$ , the product $prod pw(x) = t (- F w(x(t))$ which is invariant under the canonical action of $G$ on $2F$ . Then, the configuarion counting series is defined by

sum C(G, F,w) = pw(x). 2F/G

With these definitions, we have the following results:

For weights $w(0) = 1,w(1) = x$ , the $xk$ -coefficient of $C(G,F, w)$ is the number of $G$ -orbits of $k$ -element sets in $F$ .
For the constant weights $w(0) = w(1) = 1$ , we have $C(G, F,w) = card(2F/G)$ .