Enumeration in Music Theory

Tone rows and tropes

“…Section 2, or Pólya [1937], Pólya and Read [1987], de Bruijn [1964, 1971,1972, Harary and Palmer [1966], Kerber [1999]), it is possible to determine the number of orbits of tone rows in a purely algebraic way. (For similar and other applications of these methods to the enumeration of tone rows, see Reiner [1985], Fripertinger [1991Fripertinger [ , 1992aFripertinger [ , 1992b, Read [1994Read [ , 1997.) The most important methods for the enumeration of orbits of tone rows are collected in Sections 2.3 and 2.4.…”

“…There exist 12 6 = 924 different hexachords in the set H = {A ⊂ Z 12 | |A| = 6}. In Reiner (1985), Fripertinger (1992aFripertinger ( , 1992b, and Read (1997), the numbers of orbits of k-subsets of Z 12 were computed for different group actions. (If a group G acts on a set X, then the natural action of g ∈ G on a k-subset A of X, for 1 ≤ k ≤ |X |, is given by g * A := {ga | a ∈ A}.)…”

How Rare Is Symmetry in Musical 12-Tone Rows?

“…For an alternate derivation, see [11]. Proposition 1 appears in [12]. Whereas these sources give formulas for all n, we restrict our attention to the case n = 12.…”

“…The proof of Proposition 1 in [12] is a nice application of Burnside's lemma, but our model provides a simpler derivation, using only standard facts from undergraduate algebra.…”

“…To calculate the double cosets DaD for a in S12, execute the following: G: =SymmetricGroup (12); D:=Group( (1,2,3,4,5,6,7,8,9,10,11,12),(1,12) (2,11) (3,10) (4,9) Even when cyclic shift is allowed, just 1.16% of all row classes are symmetric. Yet 9% of the rows used by Berg display this property (2 of 23).…”