Of the many innovations that characterized mid-century modernist composition, arguably the most radical break from tradition came in the form of a panoply of new music notations. Seen early on in works from both American and European schools, these new techniques were used not only to encode ``extended'' instrumental gestures, but crucially to bring about new modes of indeterminate, improvisatory, or otherwise ``open'' play.

Owing to the many contexts in which it has historically been deployed, notation even in its most traditional form has resisted attempts to holistically theorize its use and function. Therefore, these ``neo-notations'' only complicated matters, raising new (and lending urgency to old) questions as to notation's role in literate music-making.

Having encountered these problems firsthand, several prominent composer-scholars leapt at the chance to take on these thorny new problems. Of particular note are two works penned at the height of the furore: Karlheinz Stockhausen's lecture series „Musik und Graphik: Kommentare zu neuen Partituren“ and György Ligeti's paper „Neue Notation -- Kommunikationsmittel oder Selbstweck?,“ both presented at Darmstadt in 1959 and 1965, respectively, which offer fascinatingly distinct takes on notation's changing form and function. However, while the former received welcome exegesis in a 2012 paper by David Gutkin, Ligeti's essay has gone woefully un-cited considering its incisiveness and continuing relevance, 60-plus years on, to now-common techniques among contemporary composers worldwide.

This paper draws particular attention to Ligeti's heretofore unacknowledged scholarship in order to critically contrast two composers' efforts to make sense of these new ``graphic'' notations---disentangling acts of composition from their accessory visual codes. To this end, the paper appraises the typological structures (implicit or explicit) through which each composer conceives of music's inscription, as well as the extent to which, for each, the graphic may be meaningfully emancipated from the acoustic.

Here, I argue that Ligeti's analysis significantly refines Stockhausen's, presenting a structure which obtains across both historical and contemporary music inscription. Further, I argue that once adequately formalized, Ligeti's well-articulated graphic/notation and connotative/denotative distinctions have the potential to serve as the foundation for a robust, holistic ``theory of notation(s)'' more generally.

Carlos Chávez’s music inhabits a complex relationship with modernism, which analysts have tried to capture using present-day tools such as pitch-class set theory (Bauer 2015; Burns 2016, 2023). In adopting these tools, however, analysts have overlooked Chávez’s own theories, which explain his modernism more fully. My paper combines current with historical theories to examine the first movement of Chávez’s Toccata for six percussionists, commissioned in 1940 by John Cage. By excavating Chávez’s theories of motive and of form, I re-evaluate the terms for analyzing marginalized modernist works.

Chávez’s theory of motive emerges from his 1937 book, Toward a New Music, where he argues that Mexico’s pre-Cortesian music was conditioned by its instruments, which in turn participated in a historical teleology toward instruments of the modern "West." Chávez applies this argument in his Toccata movement as he constructs two motives. Both motives produce complexity through a dialogue between an “Indian drum” and several “modern” instruments, first as they complicate the movement’s triple meter, and then as they hocket.

Chávez’s modernism is further illuminated by his sonata theory, detailed in his 1973 analysis of Beethoven’s “Waldstein” Sonata, mvt. i. Chávez’s analysis aligns with several aspects of current sonata theory, such as Caplin's. Yet instead of examining the cadences in Beethoven’s themes, Chávez examines five motives, whose permutations he tracks across the movement. Chávez mobilizes this same emphasis on motive in his Toccata, as he constructs, without harmonic signposts, a modernistic sonata form.

When Chávez finished his Toccata in 1943 and sent it to Cage, Cage replied that he could not “afford” to perform the work, “either financially or aesthetically.” Financially, the Toccata demanded expensive musicians capable of performing rolls, which Cage’s ensemble could not; and aesthetically, it grated against Cage’s modernism, then opposed to both conventional symphonic instruments and sonata form. In a 1974 interview in Mexico, Cage remembered his rejection of Chávez’s Toccata as one of his greatest social embarrassments. My analysis attempts to repair Cage's embarrassment: it attempts to restore Chávez’s popular Toccata to legibility within modernism, by attending closely to the theories Chávez himself expressed.

The collapse of the Soviet Union shattered utopian ideals. Not only did state communism dissolve, but also mourning for its victims was permanently forestalled. The populace remained haunted, decades after the fact, by Joseph Stalin’s Terror, in which millions of citizens were unaccountably “disappeared.” Late-Soviet society never acknowledged the loss or addressed the trauma.

Drawing on Jacques Derrida’s “hauntology,” the study of ghosts as present absences that exert control over the present, cultural historian Alexander Etkind writes of post-Soviet society, “The living and the undead develop an uneasy friendship that needs to be noticed, articulated, and recognized.” Closure is impossible without a proper burial; memories of the missing are a constant preoccupation.

The music of Sofia Gubaidulina (b. 1931) reflects a hauntological attitude in the post-Soviet context. Building on interdisciplinary research in spectrality studies, I characterize hauntology in music through defamiliarized timbres, distorted quotations, and disrupted temporality. I interpret Gubaidulina’s “shadow tuning,” wherein two groups are tuned a quarter-tone apart, as dramatizing an unshakeable desire for mourning.

Gubaidulina emigrated to Germany in 1992, immediately after the fall of the USSR. She developed shadow tuning in her earliest post-Soviet works: String Quartet No. 4 (1993), Music for Flute, Strings, and Percussion (1994), and Quaternion for four cellos (1996). The shadow metaphor is suggestive. Shadows follow a person wherever they go in an “uneasy friendship.” They cannot be exorcised; they must be lived with.

Hauntology provides a hermeneutic lens for analyzing shadow tuning as actualizing a lingering Soviet spectrality. As shown through musical analysis, shadow tuning affords distorted echoes, irreconciliable sound worlds, and liminal positioning. Gubaidulina’s music of the 1990s notices, articulates, and recognizes the ghostly presence that pervades post-Soviet spaces. Shadow tuning consecrates a site of ongoing mourning for losses that will never be restituted.