The rise of pentatonicism in the 19th century (Day-O’Connell (2007)) can be understood not only as a breakdown of classical tonality but as the exposure of recursively-related interval structures. This paper demonstrates through linear algebra applied to basic scale patterns that the characteristic polynomials of a left-shifted circulant matrix constructed from the scale-step interval vectors of the pentatonic and diatonic collections are divisible by the 5th and 7th Fibonacci polynomials respectively when these polynomials are considered under a Wick rotation. This work extends techniques developed by Amiot and Sethares (2011), and the new result implies a nested structure within the scale-step interval patterns based on the famous recursion of the Fibonacci sequence. This recursion can then be used as a tool of music-theoretic analysis that contrasts with the Discrete Fourier Transform (DFT).

The approach of this paper is purely heuristic, and modern scale theory explains how the pentatonic and diatonic collections are related as complete copies inside of the other like Russian nesting dolls. The paper considers the simple nested structure used by Dvořák in the second movement of Symphony No. 9 (1893) “From the New World” to tell the story of Hiawatha’s wooing of Minnehaha through music, a transfer of the main theme’s pentatonicism via a recursively-framed diatonicism from Hiawatha to Minnehaha’s father, the Arrowmaker. Dvořák scholar Beckerman (2003) eschewed a purely programmatic analysis of the second movement but a close reading conforms to Dvořák’s compositional choices. The diatonic frame (the outer structure) is “opened up” at critical moments by the ternary form to reveal its pentatonic interval space (the inner structure) for several distinct musical effects within the program.

Hemitonicism is a system of pitch-class set analysis developed by Russian theorists Valentina Kholopova and her brother Yuri Kholopov. The system (described in English in Ewell 2013) focuses on sets that contain at least one semitone, and was developed independently of – but contemporaneously with – American pc-set theory. One aspect of this system is the identification of hemitonic fields, which Ewell defines as “the continuous filling-in…of some portion of the chromatic scale.”

While the Kholopovs primarily use hemitonic fields as mini-aggregates, focusing on how each hemitonic field is created and completed in a piece, my paper serves to look at small hemitonic fields within larger pc sets. This approach, which also intersects with Quinn’s fuzzy transformations and O’Donnell’s dual transformations, enables us to see relationships between sets that are not set class-equivalent, but share the same constituent hemitonic fields – in essence, large sets that have the same number of subsets of the same size. Using a clockface representation, familiar from post-tonal theory pedagogy, of these pc sets affords us an attractive visualization of the way in which these small hemitonic fields can be said to shift positions to create new pc sets.

In this paper, I will apply this shifting hemitonic field analysis to passages from two concertos by John Williams. Though Williams is known primarily as a composer of film music, he has composed a significant amount of concert music throughout his career, including at least one concerto for nearly every orchestral instrument. This substantial body of repertoire has been relatively untouched in the scholarly literature (notable exceptions include Schneller 2018 and Grieving 2025). My focus will be on his first and most recent concertos: the Flute Concerto (1969) and the Piano Concerto (2025).

Johanna Beyer’s early compositions demonstrate a unique compositional approach: in each piece, Beyer composes a single melodic line, and then varies that line for the duration of the piece. From one statement of the melody to the next, a transformational process can be observed—pitches are transposed, and added or deleted pitches cause the lines to grow or shrink. I propose that Levenshtein distance, or edit distance, can be used to codify the specific transformations used from one variation to another, and to measure the similarity between two variations, creating formal structure.

Levenshtein distance is a string metric that measures similarity by counting the number of single-character edits necessary to transform one string of characters into another. Possible operations include insertions, deletions, or substitutions. Edit distance has been used in music information retrieval and perceptual studies measuring melodic and rhythmic similarity; however, edit distance has not yet been applied to music theory and analysis, despite its resemblance to transformational theory. In Beyer’s music, each melodic line is a “string,” and similar single-character edits can be made: pitches can be inserted, deleted, transposed, or reordered. I will demonstrate how a similarity metric such as Levenshtein distance aids with interpretive choices and with issues of form by determining which phrases are most closely related to one another. Using Beyer’s music as a case study, I propose that Levenshtein distance is one solution to a limitation of transformational theory, allowing for transformations between two melodies of differing lengths through insertion and deletion functions.