Conference Agenda

American composer and educator Howard Hanson’s 1960 treatise Harmonic Materials of Modern Music: Resources of the Tempered Scale receives scant mention in comprehensive literature reviews on pitch-class set theory. Contemporary writings by theorists George Perle (1955, 1962), David Lewin (1959, 1960), and Allen Forte (1964, 1973) historically enjoy greater circulation, reducing Hanson's treatise to a stepping-stone in set theory development (Bernard 1997). Hanson was the first to enumerate all sets under transpositional and inversional equivalence, reducing them to 224 at the 1951 general meeting of the American Philosophical Society [APS] (Cohen 2004; Verdi 2007; Berry and Van Solkema 2014). An in-progress accession order of the Howard Hanson Collection—compiled by archivist David Peter Coppen at the Sibley Music Library—presents new archival evidence that extends traditional history of theory narratives. Our findings place Hanson’s pioneering proto-set theory work in the Eastman School of Music as early as 1940, suggesting his theories were taught to a generation of composers and theorists through 1964.

Historical testimonials by composer-theorists William Bergsma and Robert V. Sutton (Russell-Williams 1988) confirm the widespread use of Hanson’s theories in Eastman classrooms. The primary focus of this paper is the undergraduate application of Hanson’s theory, with a previously unknown and unpublished projected theory textbook manuscript intended as an addition to the tenth volume of the New Scribner Music Library. Our methodology follows Charles Atkinson and Edward Nowacki’s untangling of the multiple layers of the Alia musica (Atkinson 2008; Nowacki 2020). Hanson’s 1960 text acts as the core treatise of his fully formed theory, allowing comparison between the 1960 proof copy and the 1951 APS presentation. Moreover, the core treatise provides the backdrop for the undergraduate projected theory textbook assembled between 1960 and 1966.

The diminished-seventh chord, 4-28(0369), generates the octatonic scale, 8-28(0134679t), in several ways: via complementation (Morris 1982), via transpositional combination (Cohn 1988), or via a newly recognized transformation related to Lewin’s use of binary numbers to express “changes of state” in music by Babbitt (Lewin 1995). I define the circular binary state of a set class as a necklace of binary values representing notes as 1’s and missing notes as 0’s, the rotational or inversional presentation of which is arbitrary. Traversing the circular binary state of any set-class and computing the intervals between adjacent binary values (i.e. 0 for no change or 1 if there is a change) generates the circular binary state of a new “descendant” set class. This transformation implements the cellular automaton known as Rule 90 (Wolfram 1983); taking a single pitch class as a starting point would thus replicate a mod-2 version of the Sierpiński triangle until its expanding edges encounter each other a tritone away from the starting point.

Transforming all 224 set classes in this way yields seven separate families of interconnected set classes that have not been discussed in the literature. Four of the seven families spiral inward toward a central foursome of set classes, endlessly generating each other in a little loop; one family reaches a central pair of set classes; and two other families reach a single endpoint, set-class 8-28 in one case and the null set in the other case. The evolving verticalities at the opening of Ligeti’s Lux aeterna provide an interesting illustration of possible analytical payoffs of this transformation and its associated set-class families. The occurrence of sets from different families at the outset of Lux aeterna is shown to coincide with the relative durations of distinct verticalities in certain salient ways. Further, the implementation of this transformation in modular universes other than mod-12 reveals familial structures that are totally different from the mod-12 case and could be applied to the study of melodies or harmonies in various scalar environments, rhythms involving a consistent beat or subdivision in various meters, or changes in instrumentation (as in Lewin 1995).

Based on the mathematical theories of Hook 2008 and Lam 2020, this paper offers novel equations that describe, from the ground up, how solfège methods interacts with key. The expanded coordinates of a double diatonic pitch class (sig, tonic, mode, spc, pos, deg encodes, in order, the key signature, the key note, the mode, and a note's fixed-do, la-minor, and do-minor solfege. This paper proposes a set of unified solfège equations (representable on a solfege matrix),

spc – tonic = deg,
spc – sig = pos,
pos – mode = deg,
such that mode is an element of DIA(0).
There are three degrees of freedom in such a space, suggesting a rich set of independent coordinates that can serve as interval spaces (GISs). The paper suggests critical revisions to tonal theory and scale theory, suggesting that all constructs defined on scale degrees have counterparts defined on scale positions (la-minor solfege). The following key points follow from the unified solfège equations: (1) the three solfège systems together imply the key, (2) the three solfège systems represent ndependent dimensions of a tone, (3) tonality is pluralistic and perspectival, and (4) different solmizations of a passage relate by intervals that determine the key. The paper concludes by linking subspaces of double diatonic pitch class space with Julian Hook's signature transformations and Stephen Rings's heard scale degrees.