This articleoutlines a newform oftransformational theorythat universalizesthe basic logicof voice-leading geometry. It defines musical spaces asgroupoids, categories in which all arrows are invertible. This allows theorists to use spaces like the Tonnetz without presupposing dualism, revealing that thereareatleastfourdistinctobjectscalled "the"Tonnetz. Italso provides principled reasons forapplying topological descriptions to musical spaces: Familiar spaces such as the pitch-class circle and the Tonnetz are shown not to have the topology typically ascribed to them. Transformations are defined as equiva-lence classes ofarrows; these are unconstrained and beholden to no requirements beyond musical utility. However, transformations can arise from symmetry groups, or groups of isomorphisms of a musical space. The theory considers symmetries to act not on spaces in isolation, but on attribute/element isomorphisms, which are maps between a space of musical elements and a space assigning "data" to elements, akin to a coordinate system or generalized scale. A symmetry group typically gives rise to two kinds of transformations: symmetry transformations, or equivalence classes of arrows that act likea symmetry; and intervals, orequivalence classes ofarrows related bysymmetries; element permutations are symmetries while attribute permutationsare intervals. Since subject-centered locationsareattributes,thesymmetry/ interval duality is closely related to the duality of subjective and objective. The resulting theory is fully general, subsuming and extending Lewin's Generalized Interval Systems, applying to the action of any group on any set or space, and universalizing common music-theoretical constructions like chords, set classes, voice-leadingspaces, andtransformational graphs. Itsheds new lighton a range of issues including harmonic dualism,enharmonicism, metric modulation, andthe distinction between path and transformation.
Abstract When Japan first opened its country and economy to Western cultures and influences in the mid-nineteenth century, Isawa Shūji (1851–1917) argued that Western-style songs were the right way to raise the next generations of Japanese children. He grounded much of this argument in claims about Ancient Greek music, writing that it was not only the fundament of Western music, but also an ancient historical link to Japanese music, via a shared origin in India. In order to demonstrate that it “truly corresponded with the music of this country [Japan],” Isawa wrote transcriptions of an Ancient Greek Hymn to Apollo into the traditional music notations for no fewer than five instruments used in gagaku and included them in his 1884 report on Western music, arguably the first of its kind in modern Japan. In this act of transcription, Isawa not only claimed the prestigious heritage of Ancient Greek music for Japan but also argued that Western music was a close cousin of Japanese music. Our article investigates Isawa's claim about the close relations between Japanese and Ancient Greek music: Exactly which coincidences between the two traditions was he observing? We argue that while Isawa's claim was clearly motivated by his goals and ideologies, the comparison was not a musically empty one either—that there were threads of sonic similarity that he followed, which shaped the ways Japan both took in Western music and reconceived of Japanese music for the new era.
Voice-leading parsimony has long been recognized as a strong tonal currency in nineteenthcentury harmony, with further implications for popular music and jazz. Considerations ofcommon-tone voice leading in jazz, however, have tendedto accord insufficient importance to triadic "extensions"such as ninths, elevenths, and thirteenths (Larson 2009; Heyer 2012; McClimon 2016), despite the wealth of literature that deals with the seventh as a consonance in jazz (McFarland 2012; Strunk 2016). We believe that extensions should be given greater attention as they can prove crucial in achieving continuity and coherence in some of the more novel progressions that increasingly appear in recent works from prominent new artists such as Thundercat. Providing a taxonomy of voice-leading patterns, we use simple adapted harmonic reduction techniquestooffera new methodologyforconsideringextensionstothetriadiccore ofjazz harmonies, which allows us to zoom into key progressions and visually track repeating patterns. We compare this type of common-tone coherence to other cases of voice-leading syntax through integrated graphs. Working through songs from Thundercat's album Drunk (2017), our key finding is that, while many of the chord progressions are difficult to explain according to conventional harmonic motion, the upper extensions act as bindingagents that offeran alternative type of cohesion.
