It has been argued that since the 1990s, rock music has increasingly distanced itself from the so-called “verse-chorus paradigm” (Osborn 2013) and moved towards more exploratory, contrasting sections in later parts of songs. Here, new and unusual chord progressions are often introduced (Doll 2007; Spicer 2004; Pieslak 2007). Chromatic chord sequences of this nature have already been analyzed using a neo-Riemannian methodology, with examples drawn from various pop and rock songs (Capuzzo 2004). However, the current paper will aim to combine a more conventional, functional perspective on chord relations with a transformational analysis of these, the latter serving to address points in the music, in which harmony suddenly diverges into little expected territories.

The focus of this paper is the song Mask Machine from the album Second Nature by American supergroup Flying Colors. The structure of the song conforms to the broadly defined progressive rock practices (Holm-Hudson 2002; Anderton 2010) and reflects the band’s ethos “to combine complex music with accessible songwriting” (FlyingColorsMusic, 2015). It reveals a similarity to Classical ternary forms, where a contrasting middle section provides an alternative to the overarching functional harmony by introducing less functional chromatic chord relations (Sutcliffe and Tilmouth, 2001). The Interlude and Bridge constitute such a contrasting block in the middle of the song and the sudden appearance of E@ major after E minor (the neo-Riemannian Slide operation) is key in creating harmonic contrast, while this whole block is overall more exploratory in its chordal content. A Tonnetz depiction of all the important chords in the song helps situate them within bubbles – one gravitating around E minor in the verses and another one around F major in the choruses, with movement in these restricted mostly to horizontal (hence diatonic and functional) chord relations. Thus, the sudden introduction of E@ major at the beginning of the interlude comes across as surprising, considering that coming from E minor it is a movement downwards on the tonal grid; a Slide neo-Riemannian operation, which is “profoundly uncommon in common-practice tonality but becomes surprisingly prevalent in some 20^th- and 21^st-century harmonic dialects” (Lehman, 2018).

Francisco Mignone (1897-1986) was a Brazilian composer who freely transited between classical-nationalist aesthetics and popular music, combining many influences (Martins 1990, 109). His range can be observed in his four piano Sonatinas (1949), where typical Brazilian rhythmic elements dialogue with modal and symmetric collections resulting in complex and distinctive harmonic sonorities. This paper advances an original perspective on how Mignone introduces several Brazilian rhythms, fluidly connecting them to form a colorful panel. Although not as commonly as harmony, rhythm has recently been the focus of transformational approaches (Cohn 2016; Guerra 2018; Mathias and Almada 2021) since David Lewin’s book (1987). This paper contributes to this topic by examining the two-movement fourth Sonatina, considering its peculiar rhythmic organization based on the idea of clave (Toussaint, 2013). Claves are used consistently in this Sonatina as a source of thematic rhythmic material, including references to popular genres. The claves here share two basic features: (a) they are written over ostinatos built on a secondary voice; and (b) the complex metric relations between the two layers arise from syncopation and unexpected manipulations of the clave’s building blocks.

The main clave of the first movement’s section A is based on the tresillo pattern, recurrent in many Brazilian popular genres (baião, xaxado, maxixe, etc.). The analysis demonstrates how transformations applied to non-fixed durational blocks of the clave generate variants. I isolate them and propose a possible logic chain of transformations. The resulting derived claves match some of tresillo rhythmic variants found in the music of Ernesto Nazareth, a Brazilian composer of maxixes. More complex maxixe-like claves are introduced in section B. Clave derivation also operates in the second movement addressing the samba rhythm.

Mignone’s claves constitute an original and economic compositional procedure, providing both rhythmic variety and unity through the manipulation of a few basic cycles. By analyzing Mignone’s use of claves and comparing them to those in popular Brazilian music, this study shows how these compositional procedures and elements were introduced into his concert music. This examination sheds light on the incorporation of popular Brazilian music elements into concert music during the nationalistic period.

Including jazz repertoire in transformational literature poses a particular theoretical challenge, complicated by the real-time interaction inherent to improvisation. This paper discusses the advantages of animation to solve problems that theory and still imagery alone cannot. While transformational methods have focussed little on jazz improvisation, they have contributed theoretically to post-1950s composed jazz harmony. However, predominantly static diagrams provide limited clarification of improvised events, because of their number and the challenge of interpreting changing relationships through time. Animation presents a promising solution for transformational analysis to engage interaction in improvised music.

In an MTO special issue on animation of transformational analysis, John Roeder (2009b) argues both for its technical utility and engagement with gesture and agency. (2009a) This salience increases for real-time improvisation which introduces more complex agency. Jazz’s longstanding cultural practice of playing “outside” (Givan 2007) introduces additional musical relationships between musician and composed materials (the “referent”). Despite continuing theoretical development and analytical animation, existing methods require adaptation to accommodate jazz’s interactivity.

An individual’s complex positionality is multiplied both when further group members improvising incongruously with the “referent”, and with other musicians. Real-time negotiations create interactive tensions, forming a web resistant to analysis. Despite animation’s diachronic advantage, multiple harmonic spaces pose a further challenge.

In this paper I argue for the re-development of animation for analysis of improvisation, for transformational methods and music theory more broadly. I introduce one particular solution by animating multiple musicians simultaneously on the 2D Tonnetz. Colour-coded layers enable a technical view of harmonic relationships between musicians, and between musicians and the referent. Comparing animations of the jazz quartets of John Coltrane and Ornette Coleman reveals distinct types of harmonic motion and syntax between soloists and rhythm section. Simultaneously, it enables distinguishing between kinds of interactive dynamics, through moves away from or towards both the referent and each other. The theoretical tools discussed provide clarity on persistent discourse surrounding how in the music Coltrane and Coleman manifest ideas of “transcendence” and “freedom” respectively. A better theory made visible through animation in turn allows closer engagement with the broader problem of interactivity that jazz repertoire presents.