Schema Theory
as a
Construction
Grammar

Music Theory Online 21.2 (2015)

Exemplar Models of Categorization

6.1. Premise and Paraphrase.
6.2. Connecting Usage-Based Constructions with Exemplar Models of Categorization.
6.3. The Exemplar-Based Character of Apprenticeship.
6.4. A Link Between Recall and Imagination.
6.5. Exemplar-Based Pedagogy.

Premise

[6.1.1] “Categorization does not involve necessary and sufficient conditions, but rather generalizations over exemplars, typically including prototypical exemplars and conventionalized extensions (Rosch 1973, Fillmore 1977, Rosch and Mervis 1975, Haiman 1985, Lakoff 1987, Brugman 1988, Heine 1992, Taylor 1995, Ross and Makin 1999, Talmy 2000, Abbot-Smith and Tomasello 2006, Ibbotson et. al. 2012)” (Goldberg 2013, 16).

Paraphrase

[6.1.2] While psychological research has long shown that human beings organize their perceptual world by abstracting away surface details from many exemplars in order to create a more general category (Medin and Schaffer 1978, Nosofsky 1988), recent evidence indicates that people often retain detailed information about those exemplars (Goldberg 2006, 45). An exemplar-based model of categorization takes into account such research and assumes that “a category is represented by a collection of instance representations” (Goldberg 2006, 46). An ordinary listener’s response to a cadence, for instance, need not be initiated by the firing of a “cadence” neuron in the brain (i.e., a unitary categorical representation). Instead, the mass effect of the vague recall of thousands of memories similar to the cadence in question (i.e., instance/item/exemplar representations) can lead to the rapid identification of the musical patterns as “one of those,” where “those” includes the meanings associated with previously experienced cadences.

Connecting Usage-Based Constructions with Exemplar Models of Categorization

[6.2.1] Joan Bybee (2010, 19), in describing a usage-based approach and its focus on constructions, points out how exemplar theories of categorization are good candidates for the cognitive development of constructions:

  1. Exemplar representations allow specific information about instances of use to be retained in representation.
  2. Exemplar representation provides a natural way to allow frequency of use to determine the strength of exemplars.
  3. Exemplar clusters are categories that exhibit prototype effects. They are organized in terms of members that are more or less central to the category, rather than in terms of categorical features.

Categorization, as a “domain-general cognitive function” that assists in the formation of constructions, occupies the role played in a mainstream generative grammar by an innate Universal Grammar.

[6.2.2] Whereas the requirements for memory were assumed to be limited in generative grammar (the “processing” of transformations substituted for much of the potential role of memory), a massive memory of previous experiences is at the center of construction grammars. In the words of Joan Bybee:

In order to represent the facts of usage, as well as the facts of change that eventually emerge from this usage, we need to conceive of grammar as based on constructions and as having an exemplar representation in which specific instances of use affect representation. The model to be proposed, then, uses a type of exemplar representation with constructions as the basic unit of morphosyntax. (2006, 714)

The Exemplar-Based Character of Apprenticeship

[6.3.1] “The idea of creating a rich store of memories from which one can later draw (as influenced by the artistic moment) can be mapped directly onto the practices of the Neapolitan conservatories” (Gjerdingen 2007b, 115). The institutionalization of craft apprentiship perfected in Naples was adopted by most nineteenth-century conservatories. Whether in Paris, Milan, or Munich, masters had each apprentice practice particular constructions in various keys and figurations, thus creating related sets of exemplars to be stored in the apprentice's memory. Given enough similar exemplars, the apprentice could learn to manipulate a stable schema or construction in the production of new music. At the Moscow Conservatory, Rachmaninoff studied harmony with Arensky and likely worked through many of the literally one thousand harmonic exemplars that Arensky later published (Arensky [1897] 1929). One of those exercises, no. 374 (see Example 16), began with the already ancient construction of the descending chromatic tetrachord in the bass (sometimes called the Lamento Bass), though the passage shown closed not on the A-major chord as dominant of D minor, but on the relative major (F), exploiting the potential for a more open sequence of 4/2–6/3 progressions.

Arensky Harmony Exercise


[6.3.2] Of a reported ten “songs without words” that Rachmaninoff wrote during his studies with Arensky, only one survives, a Lento in D minor (TN ii/11, 1886–87). The opening is shown in Example 17. His effort is worthy of notice on several counts. First, it appears to be a creative recasting of Arensky’s exemplar (Arensky’s bass becomes, an octave higher, Rachmaninoff’s tenor). Second, the sinuous, undulating neighbor tones of the alto voice mark even this early piece as characteristic of his mature style (cf. the opening melody of his D-minor Piano Concerto No. 3 from almost fifty years later). Third, Rachmaninoff follows Arensky’s sequence of 4/2 chords moving to 6/3 chords, but recontextualizes the final F-major triad as a complex combination of nonharmonic tones in D minor. And fourth, the placement of a D pedal point under Arensky’s exemplar and the addition of the appoggiatura A4 on the downbeat of m. 3 are simple yet deft strokes that strongly color the Arensky model.

Rachmaninoff Lento


[6.3.3] Rachmaninoff’s further exploration of this same D-minor, tenor-range Lamento Bass can be viewed in another early piano piece (Example 15). The date of this work is uncertain, and it might represent Rachmaninoff’s personal recasting of the 12/8 Gigue from the G-minor English Suite by Bach, BWV 808, along with elements of the 9/8 Prelude to the Eflat-major Fugue from The Well-Tempered Clavier, Book II (possibly creating further layers of exemplar modeling). Rachmaninoff repeatedly places the Lamento Bass in the tenor range, each time altering the harmonization. Two of these chromatic descents are shown in Example 18. The first, mm. 7–8, is part of a conventional circle-of-fifths sequence, where every sonority is a type of dominant seventh and the chromatic tetrachord is explicitly marked by quarter-notes with upward stems in the bass staff. Then, in mm. 8–11, half notes with downward stems in the treble staff mark a slower descent of the same tetrachord, one harmonized closer to the models of Examples 16 and 17 above. G-major and F-major sonorities briefly offer potential but unrealized moments of stability within the larger, brooding D-minor context.

Rachmaninoff Piece in D minor

A Link Between Recall and Imagination

[6.4.1] Exemplar models of categorization depend on the massive resources of ordinary human memory. Rachmaninoff’s memory was anything but ordinary—he was reputedly able to reproduce at the keyboard any composition he had ever heard in any medium (Schonberg 1991). As a young man he attended concerts by the leading pianist Anton Rubinstein (van Riesemann 1934, 49–52) and likely heard pieces by Rubinstein played by other students in the house of Nikolai Zverov, where Rachmaninoff lived and studied for years as a piano prodigy. One such piece could easily have been Rubinstein’s popular second Barcarolle, a brief excerpt of which is shown below in Example 19. There Rubinstein writes a bridge passage to connect two sections of his character piece. The passage features parallel 6/3 chords in sinuous descent, all over a tonic pedal point that mutates into a dominant pedal point.

Rubinstein Barcarolle


[6.4.2] What Goldberg terms an “instance” or what Bybee terms an “exemplar” can mean a veridical memory. A young composer, for example, after hearing lots of instances of short bridge passages and turn-around figures, could cluster them by similarity and hence develop a category for them. When later presented with the need to compose a new instance to fulfill the function of that category, he or she might mistake recall for invention, thus producing something uncomfortably close to someone else’s work. Perhaps that is what happened in the composition of Rachmaninoff’s early Prelude in C-sharp Minor. As shown in Example 20, the short interlude that Rachmaninoff wrote to separate thematic statements in his Prelude is so similar to Rubinstein’s as to be almost a quotation.

Rachmaninoff Prelude


[6.4.3] Rachmaninoff, with his fertile imagination, had no need to quote anyone, and quotation for aesthetic purposes played no great role in his output (with the striking exception of the “Dies irae”). Instead, his most famous prelude may be a case where exemplar-based categorization led to recall of a “pre-fab,” as if Rubinstein’s parallel 6/3 chords over a pedal point constituted an idiomatic catchphrase. In a way, of course, it did, since the passage in Rubinstein’s Barcarolle owes much to the still older model of the opening of Chopin’s great Barcarolle, op. 60 (1845–46). As shown in Example 21 (re-notated to bring out the voice leading), Chopin’s sinuous melody, dominant pedal point, 6/3 chords (or their equivalent, with appoggiatura 7s replacing some of the 6s) slowly descending the scale, and the passage’s subsidiary function within the form all make up a complex of features replicated in the usage of Rubinstein and Rachmaninoff (and still later in Liadov’s Barcarolle [1898], written in Chopin’s Fsharp major).

Chopin Barcarolle



[6.4.4] The literary scholar Northrop Frye once said that “poetry can only be made out of other poems; novels out of other novels” (Frye 1957, 97). For a precocious Rachmaninoff living inside the world of professional pianism in the late nineteenth century, it may have seemed that Romantic character pieces were largely “made out of other” Romantic character pieces.

Exemplar-Based Pedagogy

[6.5.1] While a cognitive scientist today might, as an educational reform, recommend teaching categories and concepts through exemplars instead of through rules, professional musicians have learned that way for centuries. Johann Mattheson (Mattheson 1719) placed the following Latin motto on the title page of a book of examination pieces for organists: “Longum iter est per praecepta, breve et efficax per exempla” (The journey is long by rules, but short and efficient by examples). Padre Martini, the lynchpin in an eighteenth-century social network of composers (one that included Mozart), titled his magnum opus on counterpoint “Exemplars in Counterpoint” (Esemplare, o sia Saggio fondamentale pratico di contrappunto sopra il canto fermo, 1774–76). Even Arensky published an entire page of small exemplars intended for replication in modulating sequences (1896; see Gjerdingen 2011). Richard Taruskin (2011) has noted that Lazar Saminsky, a student of Rimsky-Korsakov (as were Arensky and Glazunov), described such exemplars as “Gebrauchs-formulas.” When translated as “usage-formulas,” the connection to a usage-based construction grammar seems obvious.

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