The Z-Relation in Theory and Practice

Jeremiah Goyette

View slideshow at:
http://jeremiahgoyette.com/defense-presentation/

A pair of pc sets...which have identical interval vectors but which are not transpositionally or inversionally equivalent will be called a Z-related pair.

Allen Forte, The Structure of Atonal Music (1973), p. 21

23 pairs of z-related sets in mod12

Z-relation first discovered in music theory by:

David Lewin	1960. The Intervallic Content of a Collection of Notes, Intervallic Relations between a Collection of Notes and its Complement: an Application to Schoenberg’s Hexachordal Pieces. Journal of Music Theory 4/1: 98-101.
Howard Hanson	1960. The Harmonic Materials of Twentieth-Century Music. New York: Appleton-Century-Crofts.

Some prominent critics/commentators:

John Clough	1965. Pitch-Set Equivalence and Inclusion (A comment on Forte’s Theory of Set-Complexes). Journal of Music Theory 9/1: 163-180.
George Perle	1982. Letter from George Perle. Journal of the American Musicological Society 35/2: 373-377.
	1990. Pitch-Class Set Analysis: An Evaluation. The Journal of Musicology 8/2: 151-172.

Sets in the Z-relation will sound similar because they have the same interval-class content, but they won't be as closely related to each other as sets that are members of the same set class.

Joseph Straus, Introduction to Post-Tonal Theory
(2nd ed., 2000), p. 80

Alban Berg, op. 2 no. 4, mm. 18-22.

All-interval tetrachords sharing trichords

[016] shared

[026] shared

A few questions with which to grapple

What criteria must be met in order so that two sets of different set classes share the same interval-class content?
Do all groups of z-related sets satisfy the same criteria, or are there different types of z-relation?
Is it possible to enumerate the z-related sets; in other words, is it possible to calculate the number of z-related sets without having to count them either by hand or with a computer?
Given a z-related pitch-class set, how can one derive the z-related partner?
How can one tell if a random pitch-class set is potentially a z-related set?

Robert Morris

Several works describing aspects of z-relation, including Composition with Pitch Classes (1987) and Class Notes for Advanced Atonal Music Theory (2001).
By way of his ZC-relation, observed certain key properties of z-related hexachords, such as that they derive from particular subset unions.
Observed some characteristics of the all-interval tetrachords, such as that they both arise from a non-overlapping union of [06] and [03] dyads, and that together they form an octatonic collection.

Stephen Soderberg

"Z-Related Sets as Dual Inversion" JMT (1995).
First substantial theory of z-relation in music theory.
Aim is to generate z-related sets in various moduli, not just mod12.
Methodology involves extracting z-related pairs from the 'Q-grid', a pitch-class matrix comprised of overlapping cyclic collections.
Dual inversion – depending on which pitch classes have been chosen in the Q-grid, the z-related partner is found by inverting each of the pitch classes in one of two ways.

New information

1. Fourier transform:

Introduced by David Lewin (1959 and 2001), and expanded by Ian Quinn (2007).
Sets that are z-related have Fourier coefficients with the same magnitudes; that is, they are equally imbalanced on all of the Fourier balances.
Consequently, sets that are z-related contain identical cyclic-collection subsets (e.g., [06] or [048] in mod12).

Fourier balances (Quinn 2007)

balance 1	balance 2	balance 3
balance 4	balance 5	balance 6

New information

2. Parallels to crystallography:

Z-related sets are analogous to what crystallographers call homometric sets.
Homometricity was first recognized in 1930 by Linus Pauling and M. D. Shappell as an obstacle in the analysis of atomic structures.
Crystallographers have posited several mathematical explanations for homometricity, which range in scope and complexity.

Homometric pair

Patterson, Arthur Lindo. Ambiguities in the X-ray Analysis of Crystal Structure. Phys. Rev. 65 (1944): 195-201.

Some insights to be observed

Homometric sets exist in a variety of periodic and linear spaces.
There are indeed different types of homometric sets, such as Bagchi and Hosemann's (1953) distinction between homomorphs and pseudohomometric sets.
Certain homometric pairs can be expressed as algebraic expressions. All homometric groups can be expressed as polynomial equations (Rosenblatt and Seymour 1982).
Work on homometric sets shows that the z-related sets arise from concrete subset relationships, and thus offers a transformational perspective for the z-relation.

My contributions

Survey of Fourier transform, and its implications for the z-relation
Generation of algebraic z-relation formulas

General formula (and criteria) for z-related pairs with sets that have one cyclic-collection subset.
Pumping (from O'Rourke, et al. 2008), which adds pitch classes to the general formula to generate larger cardinality z-related pairs with more than one cyclic-collection subset.
The 'reciprocal set union', which describes certain aggregate-forming z-related pairs (sets with cardinality of half the modulus).