Many have documented the elegant way that the Euclidean algorithm models musical structures. However, the algorithm only generates structures whose IOIs (inter-onset intervals) differ by 1. But when the Euclidean algorithm is applied recursively p times, it can model many more phenomena. Musically, what is interesting to this approach is that higher-order Euclidean sets progressively generate less symmetrical structures with more IOI variation. Many musical structures (such as the harmonic minor scale, Steve Reich’s Clapping Music, and the the gahu timeline) are symmetrical to varying degrees. The results this paper generates are relevant because they provide new insight into the larger music-theoretic project that considers the geometric properties of rhythm and pitch sets.

This project employs a Max/MSP patch which intuitively illustrates the methodology through a very easy-to-use graphical user interface. Parallel to this visual approach, programs written in C++ organize data into n-ary trees, making analysis much more efficient.

This paper lays groundwork for the first generalized model of Wechsel cycles. A Wechsel cycle is a cycle of consonant triads generated by two alternating transformations, W_m and W_n.

To date, transformational theory has focused on only a small number of Wechsel cycles, the most famous being the PL (hexatonic) and the PR (octatonic), both of which involve maximal common-tone retention and parsimonious voice-leading (Cohn 2012, Hook 2022). More recent research has made rigorous the concept of a contextual inversion (of which a Wechsel is an example), especially of trichords beyond those only in T/I class (037) (Straus 2011, Visconti 2018, Yust 2019). Furthermore, theorists have begun to attend to lesser-known Wechsel cycles, such as those involving Weitzmann regions (Rings 2011) or alternating mediant (M) and SLIDE transformations (Segall 2017). Yet, there exists no concerted study of the properties of all possible Wechsel cycles, and the conditions under which those properties emerge. The purpose of this paper is make progress towards such a study.

The paper has three parts. Part I reviews the definition of a Wechsel and studies its properties. Drawing on methods of Hook 2022, I recount the Wechsel-axis theorem, which specifies the location, in pitch-class space, of the axis of inversion involved in any Wechsel of any consonant triad. I then introduce a new procedure called Wechsel inversion in which the index of a Wechsel exchanges with its mod 12 complement (W_n ↔ W_12–n). Part II constructs a taxonomy of Wechsel cycles. I propose that there are 33 Wechsel-cycle classes, each of which contains four cycles related by retrograde, cycle inversion, cycle retrograde inversion, or identity transformations. I also explain how the length of any cycle varies with the indices of its generators. Finally, Part III explores examples of Wechsel cycles in two pieces by Debussy: Reverie (1890) and Le Chevelure (1897).

This paper has two takeaways: (1) a rich theoretical context for further research on triadic transformations; and (2) an invitation to reflect further on the uniqueness and properties of the structures that those transformations engender.

The innovation in this paper introduces an extra-linear, tri-level reorganization of the fifth-related blocks within the traditional Tonnetz model to illuminate the complementary and inversional tonal structures found in the song "Pure Imagination" by Leslie Bricusse and Anthony Newley in the film "Willy Wonka and the Chocolate Factory" (1971); the paper develops the mathematical concepts of tiles and tiling in the Tonnetz to expand on the music-theoretic techniques proposed by Candace Brower (2008). Brower’s Escherian-style paradoxes of pitch space are a nod to Wonka himself.

The musical "Tile of Imagination," an I-shaped Tonnetz tile proposed in this paper, constructs an enharmonically conformed Tonnetz space that preserves more music-theoretic information related to tonality, arguably the original purpose of a tonnetz, than the standard parallelogram tiles currently used in the literature. Hook (2022) discusses the standard tile, and his discussion provides the tools for challenging its current shape. The I-shaped tile is constructed so that the tile boundaries acquire music-theoretic meaning: in particular, tonal function. The proposed Tonnetz tile is monohedral, monomorphic and enantiomorphic while still preserving the well-known toroidal topology of the Tonnetz. The geometric model, for example, grounds the diatonic part with five flats (for Wonka, the world of pure imagination) in the collection with no accidentals (the real world). This paper explains through the geometry of convex and concave shapes how the I-shaped tile systematically changes tonal function of pitches and chords in the text-setting of "Pure Imagination." The inversional symmetry and axis of symmetry appearing within the tile itself provides some new insights into another analytic technique, long associated with studies of Bartók, with the song "Pure Imagination" illustrating how pitch-class inversion can be used tonally rather than atonally as a compositional technique to layer Wonka’s true intentions in the music. This paper continues the work of Lehman (2018) in showing how harmony in film, and especially chromaticism, evokes the qualities of cinematic wonder. The I-shaped tile associated with the opening of Mozart's overture to The Marriage of Figaro (K. 492) serves as your musical combination to unlock this world of pure imagination.

While the unique gamut of the Musica and Scolica enchiriadis is highly systematic and self-consistent, it proves an imperfect fit for the polyphony notated in the treatises themselves. The authors variously ignore, amend, conceal, explain by way of miracle, or simply omit passages of noncongruence between theory and practice. This article seeks to address two well-known deficiencies with the enchiriadis gamut: its infamous lack of octave equivalency and the absence of common pitches in practice. Drawing upon musical examples from the treatises, I will show how contemporaneous polyphonic practice involved palpable tension between theory, practice, and musical notation. I will then expand the enchiriadis gamut into a Tonnetz that solves its practical problems while maintaining its theoretical strengths. Yet for its size and scope, this pitch space is firmly rooted in medieval pitch modelling, merely extending 9^th-c. methods to their logical conclusion.